Harder’s denominator problem for $\mathrm{SL}_{2}(\mathbb{Z})$ and its applications

Hohto Bekki Hohto Bekki
Max Planck Institute for Mathematics
Vivatsgasse 7
53111 Bonn
Germany [email protected] and Ryotaro Sakamoto Ryotaro Sakamoto
Department of Mathematics
University of Tsukuba
1-1-1 Tennodai
Tsukuba
Ibaraki 305-8571
Japan [email protected]

Abstract.

The aim of this paper is to give a full detail of the proof given by Harder of a theorem on the denominator of the Eisenstein class for $\mathrm{SL}_{2}(\mathbb{Z})$ and to show that the theorem has some interesting applications including the proof of a recent conjecture by Duke on the integrality of the higher Rademacher symbols. We also present a sharp universal upper bound for the denominators of the values of partial zeta functions associated with narrow ideal classes of real quadratic fields in terms of the denominator of the values of the Riemann zeta function.

Key words and phrases:

Eisenstein class, special values of partial zeta functions, real quadratic fields, Duke’s conjecture

2020 Mathematics Subject Classification:

11F75, 11R42, 11F11, 11F67, 11R23

1. Introduction

1.1. The denominator of Eisenstein classes for $\mathrm{SL}_{2}(\mathbb{Z})$

Let $\Gamma:=\mathrm{SL}_{2}(\mathbb{Z})$ be the modular group, and let $\mathbb{H}:=\{z\in\mathbb{C}\mid\operatorname{Im}(z)>0\}$ be the upper half plane, on which $\Gamma$ acts by the linear fractional transformation. We denote by $Y:=\Gamma\backslash\mathbb{H}$ the modular curve of level $\Gamma$ , by $Y^{\mathrm{BS}}$ its Borel–Serre compactification, and by $\partial Y^{\mathrm{BS}}:=Y^{\mathrm{BS}}\!-\!Y$ the Borel–Serre boundary of $Y$ . For any even integer $n\geq 2$ , we define a left $\Gamma$ -module $\mathcal{M}_{n}:=\mathrm{Sym}^{n}(\mathbb{Z}^{2})$ to be the $n$ -th symmetric power of $\mathbb{Z}^{2}$ , and define $\mathcal{M}_{n}^{\vee}:=\operatorname{Hom}_{\mathbb{Z}}(\mathcal{M}_{n},\mathbb{Z})$ to be its dual $\Gamma$ -module. Then the $\Gamma$ -module $\mathcal{M}_{n}^{\vee}$ naturally defines a sheaf on $Y^{\mathrm{BS}}$ (which we also denote by $\mathcal{M}_{n}^{\vee}$ ), and we can consider the cohomology groups $H^{\bullet}(Y^{\mathrm{BS}},\mathcal{M}_{n}^{\vee})$ and $H^{\bullet}(\partial Y^{\mathrm{BS}},\mathcal{M}_{n}^{\vee})$ . Since $\mathcal{M}_{n}^{\vee}$ has a left action of $M_{2}(\mathbb{Z})$ , these cohomology groups carry the structure of Hecke modules.

The boundary $\partial Y^{\mathrm{BS}}$ is identified with $\Gamma_{\infty}\backslash\mathbb{R}$ , where $\Gamma_{\infty}:=\left\{\begin{pmatrix}1&a\\ 0&1\end{pmatrix}\,\middle|\,a\in\mathbb{Z}\right\}$ . Hence it is easy to see that $\dim_{\mathbb{Q}}H^{1}(\partial Y^{\mathrm{BS}},\mathcal{M}_{n}^{\vee}\otimes\mathbb{Q})=1$ and we have a natural generator $\omega_{n}\in H^{1}(\partial Y^{\mathrm{BS}},\mathcal{M}_{n}^{\vee})/(\text{torsion})$ .

Harder considered in his book [Har] a unique Hecke-equivariant section

H^{1}(\partial Y^{\mathrm{BS}},\mathcal{M}_{n}^{\vee}\otimes\mathbb{Q})\longrightarrow H^{1}(Y^{\mathrm{BS}},\mathcal{M}_{n}^{\vee}\otimes\mathbb{Q})

of the canonical homomorphism $H^{1}(Y^{\mathrm{BS}},\mathcal{M}_{n}^{\vee}\otimes\mathbb{Q})\longrightarrow H^{1}(\partial Y^{\mathrm{BS}},\mathcal{M}_{n}^{\vee}\otimes\mathbb{Q})$ induced by the inclusion map $\partial Y^{\mathrm{BS}}\longhookrightarrow Y^{\mathrm{BS}}$ , and he defined the Eisenstein cohomology class

\mathrm{Eis}_{n}\in H^{1}(Y^{\mathrm{BS}},\mathcal{M}_{n}^{\vee}\otimes\mathbb{Q})

to be the image of $\omega_{n}$ under this section. Then Harder studied the denominator $\Delta(\mathrm{Eis}_{n})$ of the Eisenstein cohomology class $\mathrm{Eis}_{n}$ , that is, the smallest positive integer $\Delta(\mathrm{Eis}_{n})$ such that

\Delta(\mathrm{Eis}_{n})\mathrm{Eis}_{n}\in H^{1}(Y^{\mathrm{BS}},\mathcal{M}_{n}^{\vee})/(\text{torsion})\subset H^{1}(Y^{\mathrm{BS}},\mathcal{M}_{n}^{\vee}\otimes\mathbb{Q}).

See also the dissertation [Wan89] of Wang. As a result, the following theorem was obtained by Harder (see [HP92, Staz 2, §1]).

Theorem 1.1 (Harder [Har, Theorem 5.1.2]).

For any even integer $n\geq 2$ , we have

\Delta(\mathrm{Eis}_{n})=\textrm{the numerator of }\zeta(-1-n),

where $\zeta(s)$ denotes the Riemann zeta function¹¹1In the present paper, the numerator and the denominator of a rational number are always defined to be positive integers..

Remark 1.2.

Haberland had proved a slightly weaker version of Theorem 1.1 in [Hab83, pp. 272–273]. More precisely, let $p$ be a prime number and assume that $p>n$ . Then Haberland obtained

\mathrm{ord}_{p}(\Delta(\mathrm{Eis}_{n}))\leq\mathrm{ord}_{p}(\textrm{the numerator of }\zeta(-1-n)).

Moreover, if we further assume that $p$ divides $\zeta(-1-n)$ and that there exists $\nu\in\{1,\dots,n-1\}$ such that $p\nmid\zeta(-\nu)\zeta(\nu-n)$ , then he obtained the equality

\mathrm{ord}_{p}(\Delta(\mathrm{Eis}_{n}))=\mathrm{ord}_{p}(\textrm{the numerator of }\zeta(-1-n)).

Remark 1.3.

For any prime numbers $p$ and $\ell$ with $\ell\nmid p(p-1)$ , the denominators over $\mathbb{Z}_{\ell}$ of Eisenstein classes for $\Gamma_{1}(p)$ with a character was computed by Kaiser in the diploma thesis [Kai90] (see also the paper [Mah00] of Mahnkopf for the study of Eisenstein classes for $\Gamma_{1}(p^{e}\ell)$ ). Eisenstein classes for $\mathrm{GL}_{2}$ over totally real fields have been studied by Maennel in the dissertation [Mae93]. Eisenstein classes for $\mathrm{GL}_{2}$ over imaginary quadratic fields have been studied by Harder in [Har81, Har82], Weselmann in [Wes88], Berger in [Ber08, Ber09], and Branchereau in [Bra23].

The purpose of the present paper is to report some arithmetic applications of Theorem 1.1 especially to the special values of partial zeta functions of real quadratic fields. However, since the book [Har] (which is available on Harder’s web-page) is still under development, some of the important arguments and references in the proof of Theorem 1.1 are currently not given completely. Taking this situation into account, we decided to also give the detailed proof of Theorem 1.1, which is another main purpose of the present paper.

1.2. Reformulation in terms of the holomorphic Eisenstein series

In view of applications, we interpret the above definition of the Eisenstein class $\mathrm{Eis}_{n}$ and Theorem 1.1 in terms of the holomorphic Eisenstein series and the Eichler–Shimura homomorphism.

In the following, let $n\geq 2$ be an even integer and let $M_{n+2}(\Gamma)$ denotes the space of modular forms of level $\Gamma=\mathrm{SL}_{2}(\mathbb{Z})$ and weight $n+2$ . Then we have the Hecke-equivariant homomorphism

r\colon M_{n+2}(\Gamma)\longrightarrow H^{1}(Y,\mathcal{M}_{n}^{\vee}\otimes\mathbb{C})=H^{1}(Y^{\mathrm{BS}},\mathcal{M}_{n}^{\vee}\otimes\mathbb{C})

called the Eichler–Shimura homomorphism which is defined by certain path integrals on $\mathbb{H}$ (see §2.6 for the precise definition of the Eichler–Shimura homomorphism). Let

E_{n+2}(z):=1+\frac{2}{\zeta(-1-n)}\sum_{k=1}^{\infty}\sigma_{n+1}(k)q^{k}\in M_{n+2}(\Gamma),\quad q:=e^{2\pi iz}

denote the holomorphic Eisenstein series of weight $n+2$ . Then the following is the reformulation of Theorem 1.1 ([Har, Theorem 5.1.2]) which will be proved in the present paper.

Theorem 1.4 (Lemma 2.8, Proposition 2.11, and Theorem 2.13).

(1)

We have $r(E_{n+2})=\mathrm{Eis}_{n}$ , i.e., the class $r(E_{n+2})$ coincides with the Eisenstein class $\mathrm{Eis}_{n}$ .
(2)

The denominator of $r(E_{n+2})$ is equal to the numerator of $\zeta(-1-n)$ .

Remark 1.5.

From the $q$ -expansion of the Eisenstein series $E_{n+2}$ , we see that the denominator of $E_{n+2}$ with respect to the integral structure coming from the $q$ -expansion (the de Rham integral structure) is clearly the numerator of $\zeta(-1-n)$ . An interesting and non-trivial point in Theorem 1.4 is that on the Eisenstein parts $\mathbb{Q}E_{n+2}$ and $\mathbb{Q}\mathrm{Eis}_{n}$ , the Betti integral structure coincides with the de Rham integral structure under the Eichler–Shimura homomorphism. Cf. [Har21, §1.1] and Remark 2.14.

1.3. Strategy of the proof of Theorem 1.4

We review the strategy of the proof of Theorem 1.4, which is based on Harder’s argument in [Har]. First, note that for any prime number $p$ , we have

\mathrm{ord}_{p}(\Delta(\mathrm{Eis}_{n}))=\min\{\delta\in\mathbb{Z}_{\geq 0}\mid p^{\delta}\mathrm{Eis}_{n}\in H^{1}(Y^{\mathrm{BS}},\mathcal{M}_{n}^{\vee}\otimes\mathbb{Z}_{(p)})\}.

Therefore, it suffices to prove that

\mathrm{ord}_{p}(\Delta(\mathrm{Eis}_{n}))=\mathrm{ord}_{p}(\text{the numerator of }\zeta(-1-n))

for each prime number $p$ . Then the proof consists roughly of the following four parts.

(I)

First, we construct in §3 a certain family of $p$ -adically integral homology classes

$\widetilde{T_{p}^{m}(C_{\nu}(\tau))}\in H_{1}(Y^{\mathrm{BS}},\mathcal{M}_{n}\otimes\mathbb{Z}_{(p)})$

for any $\tau\in\mathbb{H}$ , sufficiently large integer $m$ , and $1\leq\nu\leq n-1$ , where $H_{1}(Y^{\mathrm{BS}},\mathcal{M}_{n}\otimes\mathbb{Z}_{(p)})$ is the cosheaf homology group associated with the $\Gamma$ -module $\mathcal{M}_{n}\otimes\mathbb{Z}_{(p)}$ . Note that the homology class $\widetilde{T_{p}^{m}(C_{\nu}(\tau))}$ is independent of the choice of $\tau\in\mathbb{H}$ . See Lemma 3.15.

(II)

Next, in §4, we compute the $p$ -adic limit of the value of the pairing

\lim_{m\to\infty}\langle\mathrm{Eis}_{n},\widetilde{T_{p}^{m!}(C_{\nu}(\tau))}\rangle,

where $\langle~{},~{}\rangle\colon H^{1}(Y^{\mathrm{BS}},\mathcal{M}_{n}^{\vee})\times H_{1}(Y^{\mathrm{BS}},\mathcal{M}_{n})\longrightarrow\mathbb{Z}$ is the pairing induced by $\mathcal{M}_{n}^{\vee}\times\mathcal{M}_{n}\longrightarrow\mathbb{Z};(f,x)\mapsto f(x)$ . More precisely, we will show in Theorem 4.1 and Corollary 7.2 that this $p$ -adic limit can be described in terms of the Kubota–Leopoldt $p$ -adic $L$ -functions, namely, for any integer $\nu\in\{1,\ldots,n-1\}$ we obtain the following interesting formula

\displaystyle\lim_{m\to\infty}\langle\mathrm{Eis}_{n},\widetilde{T_{p}^{m!}(C_{\nu}(\tau))}\rangle=\frac{1-p^{n+1}}{(1-p^{\nu})(1-p^{n-\nu})}D_{p}(n,\nu),

where

D_{p}(n,\nu):=\frac{L_{p}(-\nu,\omega^{1+\nu})L_{p}(\nu-n,\omega^{n-\nu+1})}{L_{p}(-1-n,\omega^{n+2})}-L_{p}(-\nu,\omega^{1+\nu})-L_{p}(\nu-n,\omega^{n-\nu+1})

and $L_{p}(s,\omega^{a})$ denotes the Kubota–Leopoldt $p$ -adic $L$ -function associated with the $a$ -th power of the Teichmüller character $\omega$ .

(III)

In §5, we prove that the homology classes $\widetilde{T_{p}^{m!}(C_{1}(\tau))},\ldots,\widetilde{T_{p}^{m!}(C_{n-1}(\tau))}$ give a generator of the ordinary part of the quotient group $(H_{1}(Y^{\mathrm{BS}},\mathcal{M}_{n})/H_{1}(\partial Y^{\mathrm{BS}},\mathcal{M}_{n}))\otimes\mathbb{Z}_{p}$ (see Proposition 5.7). Hence, since $\langle\mathrm{Eis}_{n},H_{1}(\partial Y^{\mathrm{BS}},\mathcal{M}_{n})\rangle=\mathbb{Z}$ , we obtain

(1.1)

\displaystyle\mathrm{ord}_{p}(\Delta(\mathrm{Eis}_{n}))=\min\{a\in\mathbb{Z}_{\geq 0}\mid p^{a}D(n,\nu)\in\mathbb{Z}_{p}\textrm{ for any integer }1\leq\nu\leq n-1\}.

See Corollary 5.8, and also Proposition 8.1.

(IV)

In §8, we show that the right hand side of (1.1) is equal to the $p$ -adic valuation of the numerator of $\zeta(-1-n)$ . We devote §6 and §7 to the preparation for proving this fact.

1.4. Applications to Duke’s conjecture and to the special values of the partial zeta functions of real quadratic fields

1.4.1. Duke’s conjecture

In the paper [Duk23], Duke defined a certain map

\Psi_{k}:\Gamma=\mathrm{SL}_{2}(\mathbb{Z})\longrightarrow\mathbb{Q}

for each integer $k\geq 2$ called the higher Rademacher symbol which is a generalization of the classical Rademacher symbol, and he conjectured the integrality of the higher Rademacher symbol ([Duk23, Conjecture, p. 4]). As a first application of Theorem 1.4, we prove this conjecture.

Theorem 1.6 (Corollary 9.5).

Duke’s conjecture holds true, that is, for any integer $k\geq 2$ and matrix $\gamma\in\Gamma$ , we have

\Psi_{k}(\gamma)\in\mathbb{Z}.

In fact, Duke proved in [Duk23, Lemma 6] that the higher Rademacher symbols can be written as the integral of the holomorphic Eisenstein series along a certain homology cycles (see Proposition 9.4). Therefore, we can derive Theorem 1.6 directly from Theorem 1.4.

Remark 1.7.

Duke’s conjecture is recently proved also by O’Sullivan in [O’S23] using a more direct method.

1.4.2. The denominators of the partial zeta functions of real quadratic fields

Next, we discuss the denominators of the partial zeta functions associated with narrow ideal classes of orders in real quadratic fields.

Let $F\subset\mathbb{R}$ be a real quadratic field, $\mathcal{O}\subset F$ be an order in $F$ , and $\mathcal{A}\in Cl_{\mathcal{O}}^{+}$ be a narrow ideal class of $\mathcal{O}$ . Then we have the associated partial zeta function

\zeta_{\mathcal{O}}(\mathcal{A},s)=\sum_{\mathfrak{a}\subset\mathcal{O},\mathfrak{a}\in\mathcal{A}}\frac{1}{N\mathfrak{a}^{s}},

which can be continued meromorphically to $\mathbb{C}$ , and it is known that

\zeta_{\mathcal{O}}(\mathcal{A},1-k)\in\mathbb{Q}

for any integer $k\geq 2$ . We also define the positive integer $J_{2k}$ by

J_{2k}:=\text{the denominator of }\zeta(1-2k).

Then in §9.2, we obtain the following as another consequence of Theorem 1.4.

Proposition 1.8 (Corollary 9.12).

Let $k\geq 2$ be an integer. Then the integer $J_{2k}$ gives a universal upper bound for the denominator of $\zeta_{\mathcal{O}}(\mathcal{A},1-k)$ with respect to orders $\mathcal{O}$ and narrow ideal classes $\mathcal{A}\in Cl_{\mathcal{O}}^{+}$ . In other words, we have

J_{2k}\zeta_{\mathcal{O}}(\mathcal{A},1-k)\in\mathbb{Z}

for all orders $\mathcal{O}$ in all real quadratic fields and narrow ideal classes $\mathcal{A}\in Cl_{\mathcal{O}}^{+}$ .

In fact, one can construct a natural map $\mathfrak{z}_{\mathcal{O},k}\colon Cl_{\mathcal{O}}^{+}\longrightarrow H_{1}(Y^{\mathrm{BS}},\mathcal{M}_{2k-2})$ for any integer $k\geq 2$ (see Definition 9.6), and we show in Proposition 9.10 that

\langle\mathrm{Eis}_{2k-2},\mathfrak{z}_{\mathcal{O},k}(\mathcal{A}^{-1})\rangle=(-1)^{k}\frac{\zeta_{\mathcal{O}}(\mathcal{A},1-k)}{\zeta(1-2k)}=\pm\frac{J_{2k}\zeta_{\mathcal{O}}(\mathcal{A},1-k)}{N_{2k}}.

Hence Proposition 1.8 follows from Theorem 1.4. See also Remark 9.13 for the relation between Duke’s conjecture and Proposition 1.8.

Next, we discuss the sharpness of the universal upper bound obtained in Proposition 1.8.

Theorem 1.9 (Corollary 9.16).

The universal upper bound in Proposition 1.8 is sharp, that is, we have

J_{2k}=\min\left\{J\in\mathbb{Z}_{>0}\,\,\middle|\,\,\begin{aligned} J\zeta_{\mathcal{O}}(\mathcal{A},1-k)\in\mathbb{Z}\text{ for all orders $\mathcal{O}$ in all real quadratic fields}\\ \text{and narrow ideal classes $\mathcal{A}\in Cl_{\mathcal{O}}^{+}$ }\end{aligned}\,\right\}.

In order to derive Theorem 1.9 from Theorem 1.4, we need to show that the narrow ideal classes of orders in real quadratic fields produce sufficiently large submodule of the homology group $H_{1}(Y^{\mathrm{BS}},\mathcal{M}_{2k-2})$ , and this will be done in §9.3 using some techniques from Hida theory.

Remark 1.10.

As for the denominator or the integrality of the special values of partial zeta functions of real quadratic fields, or more generally of totally real fields, many works have been done by Coates and Sinnott [CS74a, CS74b, CS77], Deligne and Ribet [DR80], Cassou-Noguès [CN79], Charollois, Dasgupta, and Greenberg [CDG15], Beilinson, Kings, and Levin [BKL18], Bannai, Hagihara, Yamada, and Yamamoto [BHYY22], Bergeron, Charollois, and Garcia [BCG20] etc., by using variety of methods including Hilbert modular forms, Shintani zeta functions [Shi76], Sczech’s Eisenstein cocycles [Scz93], etc. Actually, when $\mathcal{O}=\mathcal{O}_{F}$ , the upper bound in Proposition 1.8 follows from these preceding works. More precisely, the results proved by Coates and Sinnott in [CS77] or Deligne and Ribet in [DR80] show that for any prime number $p$ , we have

2^{-1}(1-p^{2k})\zeta_{\mathcal{O}_{F}}(\mathcal{A},1-k)\in\mathbb{Z}[1/p],

which implies that

J_{2k}\zeta_{\mathcal{O}_{F}}(\mathcal{A},1-k)\in\mathbb{Z}

(see [Zag76, pp. 73, 75]).

One feature of the method in the present paper is that by using Theorem 1.4, we capture not only the upper bound for the denominators of the partial zeta functions associated with any orders, but also the sharpness of the upper bound.

Remark 1.11.

In the paper [Zag77], Zagier proved a certain formula [Zag77, p. 149, Corollaire] which explicitly computes the special values $\zeta_{\mathcal{O}}(\mathcal{A},1-k)$ of partial zeta functions of orders of real quadratic fields at negative integers in a uniform way. Then by using this formula, he obtained a universal upper bound $d_{k}$ of the denominators of the values $\zeta_{\mathcal{O}}(\mathcal{A},1-k)$ and examined its sharpness briefly. More precisely, he observed that the upper bound $d_{k}$ is not sharp and discussed how one can improve this upper bound when $k=2,3$ (see [Zag77, pp. 149–150]). Theorem 1.9 can be seen as the complete answer to this problem of determining the sharp universal upper bound for the denominators of $\zeta_{\mathcal{O}}(\mathcal{A},1-k)$ .

Acknowledgements

We would like to express our deepest gratitude to Günter Harder who explained to us many beautiful ideas and showed us his notes and manuscripts on the proof of his theorem. We would also like to thank him for encouraging us to write the proof of his theorem in this paper. We are also grateful to Herbert Gangl, Christian Kaiser, and Don Zagier for the fruitful discussions and many valuable comments during the study. In particular, Herbert Gangl and Don Zagier suggested us to conduct a numerical experiment to test the sharpness of Proposition 1.8, and helped us to write PARI/GP programs for the experiment, which provided us with some important ideas to prove Theorem 1.9. Thanks are also due to Toshiki Matsusaka who drew our attention to Duke’s conjecture, which became one of the main motivations in this study. This research has been carried out during the first author’s stay at the Max Planck Institute for Mathematics in Bonn.

2. Preliminaries and the Eisenstein class

In this section, we give the definition of the Eisenstein class and explain Theorem 1.4 (see Theorem 2.13).

Throughout this paper, $n\geq 2$ denotes an even integer.

2.1. Definitions of Modular curve and Borel–Serre compactification

Let

\mathbb{H}:=\{z\in\mathbb{C}\mid\operatorname{Im}(z)>0\}

denote the upper half plane, and let

\mathbb{H}^{\mathrm{BS}}:=\mathbb{H}\sqcup\bigsqcup_{r\in\mathbb{P}^{1}(\mathbb{Q})}\mathbb{P}^{1}(\mathbb{R})\setminus\{r\}

be the Borel–Serre compactification of $\mathbb{H}$ (see [Gor05] or [Har, §1,2,7]). We set

\Gamma:=\mathrm{SL}_{2}(\mathbb{Z}).

The group $\Gamma$ acts on $\mathbb{H}$ and $\mathbb{H}^{\mathrm{BS}}$ by the linear fractional transformation as usual. We denote by

\displaystyle Y:=\Gamma\backslash\mathbb{H}\,\,\,\textrm{ and }\,\,\,Y^{\mathrm{BS}}:=\Gamma\backslash\mathbb{H}^{\mathrm{BS}}

the modular curve of level $\mathrm{SL}_{2}(\mathbb{Z})$ and its Borel–Serre compactification, respectively. Moreover, we denote by $\partial Y^{\mathrm{BS}}:=Y^{\mathrm{BS}}\!-\!Y$ the boundary of $Y^{\mathrm{BS}}$ . The boundary $\partial Y^{\mathrm{BS}}$ is homeomorphic to the circle $S^{1}$ and the fundamental group $\pi_{1}(\partial Y^{\mathrm{BS}})$ can be identified with $\Gamma_{\infty}:=\left\{\begin{pmatrix}1&a\\ 0&1\end{pmatrix}\,\middle|\,a\in\mathbb{Z}\right\}$ .

In the following, $\mathbb{H}^{?}$ (resp. $Y^{?}$ ) means that it is either $\mathbb{H}$ or $\mathbb{H}^{\mathrm{BS}}$ (resp. $Y$ or $Y^{\mathrm{BS}}$ ). Any left $\Gamma$ -module $\mathcal{M}$ can be regarded as (co)sheaf on $Y^{?}$ in a natural way, and hence we can consider the homology groups

\displaystyle H_{\bullet}(Y^{?},\mathcal{M}),\,\,\,H_{\bullet}(\partial Y^{\mathrm{BS}},\mathcal{M}),\,\,\,H_{\bullet}(Y^{\mathrm{BS}},\partial Y^{\mathrm{BS}},\mathcal{M}),

which fit into the long exact sequence

\displaystyle\cdots\rightarrow H_{1}(\partial Y^{\mathrm{BS}},\mathcal{M})\rightarrow H_{1}(Y^{\mathrm{BS}},\mathcal{M})\rightarrow H_{1}(Y^{\mathrm{BS}},\partial Y^{\mathrm{BS}},\mathcal{M})\rightarrow H_{0}(\partial Y^{\mathrm{BS}},\mathcal{M})\rightarrow\cdots.

Similarly, we have the cohomology groups

\displaystyle H^{\bullet}(Y^{?},\mathcal{M}),\,\,\,H^{\bullet}(\partial Y^{\mathrm{BS}},\mathcal{M}),\,\,\,H^{\bullet}(Y^{\mathrm{BS}},\partial Y^{\mathrm{BS}},\mathcal{M}),

which fit into the long exact sequence

\displaystyle\cdots\rightarrow H^{1}(Y^{\mathrm{BS}},\partial Y^{\mathrm{BS}},\mathcal{M})\rightarrow H^{1}(Y^{\mathrm{BS}},\mathcal{M})\rightarrow H^{1}(\partial Y^{\mathrm{BS}},\mathcal{M})\rightarrow H^{2}(Y^{\mathrm{BS}},\partial Y^{\mathrm{BS}},\mathcal{M})\rightarrow\cdots.

We note that the inclusion map $Y\longhookrightarrow Y^{\mathrm{BS}}$ induces isomorphisms

H_{\bullet}(Y,\mathcal{M})\stackrel{{\scriptstyle\sim}}{{\longrightarrow}}H_{\bullet}(Y^{\mathrm{BS}},\mathcal{M})\,\,\,\textrm{ and }\,\,\,H^{\bullet}(Y^{\mathrm{BS}},\mathcal{M})\stackrel{{\scriptstyle\sim}}{{\longrightarrow}}H^{\bullet}(Y,\mathcal{M}).

Moreover, if $\mathcal{M}$ has an action of $M_{2}^{+}(\mathbb{Z}):=\{\gamma\in M_{2}(\mathbb{Z})\mid\det(\gamma)>0\}$ , then these homology groups (resp. cohomology groups) carry the structure of Hecke modules, namely, for each prime number $p$ we have a Hecke operator $T_{p}$ (resp. $T_{p}^{\prime}$ ) on these homology groups (resp. cohomology groups), and the above long exact sequences are compatible with the Hecke operators.

In Section 2.2, we give a way to compute these (co)homology groups, and in Section 2.4, we give an explicit description of the Hecke operators.

Remark 2.1.

As a sheaf on $Y$ , the stalk $\mathcal{M}_{x}$ at $x\in Y$ coincides with $\mathcal{M}^{\Gamma_{\tilde{x}}}$ , where $\tilde{x}\in\mathbb{H}$ is a lift of $x\in Y$ and $\Gamma_{\tilde{x}}:=\{\gamma\in\Gamma\mid\gamma\tilde{x}=\gamma\}$ . This fact shows that a short exact sequence of left $\Gamma$ -modules does not give a short exact sequence of sheaves on $Y$ in general, that is, the sheafification functor is not exact. However, a short exact sequence of left $\mathbb{Z}[1/6][\Gamma$ ]-modules induces a short exact sequence of sheaves on $Y^{\mathrm{BS}}$ since the order of $\Gamma_{\tilde{x}}$ divides $6$ .

2.2. Modular symbols and (co)homology

Let $X\in\{\mathbb{H},\mathbb{H}^{\mathrm{BS}},\partial\mathbb{H}^{\mathrm{BS}}\}$ and let $(S_{\bullet}(X),\partial)$ denote the usual singular chain complex of $X$ , i.e., $S_{q}(X)$ is the free abelian group generated by singular $q$ -simplices in $X$ and $\partial\colon S_{q}(X)\longrightarrow S_{q-1}(X)$ is the boundary operator.

The left action of $M_{2}^{+}(\mathbb{Z})$ on $X$ induces a left action of $M_{2}^{+}(\mathbb{Z})$ on $S_{\bullet}(X)$ , and $(S_{\bullet}(X),\partial)$ is actually an $M_{2}^{+}(\mathbb{Z})$ -equivariant complex. Then it is known that for any left $M_{2}^{+}(\mathbb{Z})$ -module $\mathcal{M}$ , which is also seen as a (co)sheaf on $\Gamma\backslash X$ , we have natural isomorphisms

	$\displaystyle H_{\bullet}(\Gamma\backslash X,\mathcal{M})$	$\displaystyle\cong H_{\bullet}((S_{\bullet}(X)\otimes\mathcal{M})_{\Gamma}),$
	$\displaystyle H_{\bullet}(Y^{\mathrm{BS}},\partial Y^{\mathrm{BS}}\mathcal{M})$	$\displaystyle\cong H_{\bullet}(((S_{\bullet}(\mathbb{H}^{\mathrm{BS}})/S_{\bullet}(\partial\mathbb{H}^{\mathrm{BS}})\otimes\mathcal{M})_{\Gamma}),$

where $(-)_{\Gamma}$ denotes the $\Gamma$ -coinvariant functor. Here the left $M_{2}^{+}(\mathbb{Z})$ -action on $S_{\bullet}(X)\otimes\mathcal{M}$ is defined by

\gamma\cdot(\sigma\otimes m):=\gamma\sigma\otimes\gamma m,

where $\sigma\in S_{\bullet}(X)$ , $m\in\mathcal{M}$ , and $\gamma\in M_{2}^{+}(\mathbb{Z})$ . Set

\mathcal{MS}(X):=\operatorname{coker}(S_{2}(X)\stackrel{{\scriptstyle\partial}}{{\longrightarrow}}S_{1}(X)).

For any elements $\alpha,\beta\in X$ , we denote the equivalence class of a path from $\alpha$ to $\beta$ in $\mathcal{MS}(X)$ by

\{\alpha,\beta\}\in\mathcal{MS}(X).

The boundary map $\partial\colon S_{1}(X)\rightarrow S_{0}(X)$ induces a $\Gamma$ -homomorphism $\partial\colon\mathcal{MS}(X)\rightarrow S_{0}(X)$ , and we have a natural isomorphism

\displaystyle H_{1}(\Gamma\backslash X,\mathcal{M})\cong\ker((\mathcal{MS}(X)\otimes\mathcal{M})_{\Gamma}\overset{\partial}{\rightarrow}(S_{0}(X)\otimes\mathcal{M})_{\Gamma}).

Similarly, we also have natural isomorphisms

	$\displaystyle H^{\bullet}(\Gamma\backslash X,\mathcal{M})$	$\displaystyle\cong H^{\bullet}(\operatorname{Hom}_{\mathbb{Z}}(S_{\bullet}(X),\mathcal{M})^{\Gamma}),$
	$\displaystyle H^{\bullet}(Y^{\mathrm{BS}},\partial Y^{\mathrm{BS}}\mathcal{M})$	$\displaystyle\cong H^{\bullet}(\operatorname{Hom}_{\mathbb{Z}}(S_{\bullet}(\mathbb{H}^{\mathrm{BS}})/S_{\bullet}(\partial\mathbb{H}^{\mathrm{BS}}),\mathcal{M})^{\Gamma}),$

where $(-)^{\Gamma}$ denotes the $\Gamma$ -invariant functor. Here the left $M_{2}^{+}(\mathbb{Z})$ -action on $\operatorname{Hom}_{\mathbb{Z}}(S_{\bullet}(X),\mathcal{M})$ is defined by

(\gamma\phi)(\sigma):=\gamma(\phi(\tilde{\gamma}\sigma)),

where $\phi\in\operatorname{Hom}_{\mathbb{Z}}(S_{\bullet}(X),\mathcal{M})$ , $\sigma\in S_{\bullet}(X)$ , and $\tilde{\gamma}$ is the adjugate of $\gamma\in M_{2}^{+}(\mathbb{Z})$ . Since

\ker\left(\operatorname{Hom}_{\mathbb{Z}}(S_{1}(X),\mathcal{M})\rightarrow\operatorname{Hom}_{\mathbb{Z}}(S_{2}(X),\mathcal{M})\right)=\operatorname{Hom}_{\mathbb{Z}}(\mathcal{MS}(X),\mathcal{M}),

we have a natural isomorphism

H^{1}(\Gamma\backslash X,\mathcal{M})\cong\operatorname{coker}(\operatorname{Hom}_{\mathbb{Z}}(S_{0}(X),\mathcal{M})^{\Gamma}\rightarrow\operatorname{Hom}_{\mathbb{Z}}(\mathcal{MS}(X),\mathcal{M})^{\Gamma}).

2.3. $M_{2}^{+}(\mathbb{Z})$ -modules $\mathcal{M}_{n}$ and $\mathcal{M}_{n}^{\flat}$

For any $2\times 2$ matrix $\gamma=\begin{pmatrix}a&b\\ c&d\end{pmatrix}$ , we denote the adjugate of $\gamma$ by

\widetilde{\gamma}:=\begin{pmatrix}d&-b\\ -c&a\end{pmatrix}.

Note that if $\gamma\in\Gamma$ , then we have $\widetilde{\gamma}=\gamma^{-1}$ .

Let $\mathbb{Z}[X_{1},X_{2}]$ denote the ring of polynomials of two variables over $\mathbb{Z}$ , and we equip $\mathbb{Z}[X_{1},X_{2}]$ with a left action of $M_{2}^{+}(\mathbb{Z})$ by

	$\displaystyle(\gamma P)(X_{1},X_{2}):=$	$\displaystyle P(dX_{1}-bX_{2},-cX_{1}+aX_{2})$
	$\displaystyle=$	$\displaystyle P\left((X_{1},X_{2})\cdot{}^{t}\widetilde{\gamma}\,\right),$

where $P\in\mathbb{Z}[X_{1},X_{2}]$ and $\gamma\in M_{2}^{+}(\mathbb{Z})$ . For each integer $0\leq\nu\leq n$ , we set

\displaystyle e_{\nu}:=X_{1}^{\nu}X_{2}^{n-\nu}\,\,\,\textrm{ and }\,\,\,e_{\nu}^{\flat}:=(-1)^{n-\nu}\binom{n}{\nu}X_{1}^{n-\nu}X_{2}^{\nu}.

We then define submodules $\mathcal{M}_{n}$ and $\mathcal{M}_{n}^{\flat}$ of $\mathbb{Z}[X_{1},X_{2}]$ by

\displaystyle\mathcal{M}_{n}:=\bigoplus_{\nu=0}^{n}\mathbb{Z}e_{\nu}\,\,\,\textrm{ and }\,\,\,\mathcal{M}_{n}^{\flat}:=\bigoplus_{\nu=0}^{n}\mathbb{Z}e_{\nu}^{\flat}.

The $\mathbb{Z}$ -modules $\mathcal{M}_{n}$ and $\mathcal{M}_{n}^{\flat}$ are closed under the left action of $M_{2}^{+}(\mathbb{Z})$ on $\mathbb{Z}[X_{1},X_{2}]$ . In particular, both $\mathcal{M}_{n}$ and $\mathcal{M}_{n}^{\flat}$ are left $\Gamma$ -modules. We also define the pairing

\langle~{},~{}\rangle\colon\mathcal{M}_{n}^{\flat}\times\mathcal{M}_{n}\rightarrow\mathbb{Z}

\langle e_{\nu}^{\flat},e_{\mu}\rangle=\delta_{\nu,\mu},

where $\delta_{\nu,\mu}$ is the Kronecker delta. The pairing $\langle~{},~{}\rangle$ is perfect and $M_{2}^{+}(\mathbb{Z})$ -equivariant in the sense that for any polynomials $P\in\mathcal{M}_{n}^{\flat}$ and $Q\in\mathcal{M}_{n}$ and matrix $\gamma\in M_{2}^{+}(\mathbb{Z})$ , we have

\langle P,\gamma Q\rangle=\langle\widetilde{\gamma}P,Q\rangle.

Hence the pairing $\langle~{},~{}\rangle$ induces an $M_{2}^{+}(\mathbb{Z})$ -equivariant isomorphism

\mathcal{M}_{n}^{\flat}\stackrel{{\scriptstyle\sim}}{{\longrightarrow}}\mathcal{M}_{n}^{\vee}:=\operatorname{Hom}_{\mathbb{Z}}(\mathcal{M}_{n},\mathbb{Z}),m^{\prime}\mapsto\left(m\mapsto\langle m^{\prime},m\rangle\right).

Here the left action of $M_{2}^{+}(\mathbb{Z})$ on $\mathcal{M}_{n}^{\vee}=\operatorname{Hom}_{\mathbb{Z}}(\mathcal{M}_{n},\mathbb{Z})$ is given by

(\gamma\phi)(Q)=\phi(\tilde{\gamma}Q),

where $\phi\in\operatorname{Hom}_{\mathbb{Z}}(\mathcal{M}_{n},\mathbb{Z})$ , $Q\in\mathcal{M}_{n}$ , and $\gamma\in M_{2}^{+}(\mathbb{Z})$ .

Remark 2.2.

The left actions of $M_{2}^{+}(\mathbb{Z})$ on $\mathcal{M}_{n}$ and $\mathcal{M}_{n}^{\flat}$ are slightly different from the left actions used in Harder’s book [Har, (1.57)]. However, since

\begin{pmatrix}&-1\\ 1&\end{pmatrix}^{-1}\begin{pmatrix}a&b\\ c&d\end{pmatrix}\begin{pmatrix}&-1\\ 1&\end{pmatrix}=\begin{pmatrix}d&-b\\ -c&a\end{pmatrix},

they are isomorphic as left $M_{2}^{+}(\mathbb{Z})$ -modules. Therefore, there are no essential differences.

2.4. Hecke operators

Let $X\in\{\mathbb{H},\mathbb{H}^{\mathrm{BS}},\partial\mathbb{H}\}$ and let $\mathcal{M}$ be a left $M_{2}^{+}(\mathbb{Z})$ -module. In this subsection, we define the Hecke operators on $H_{\bullet}(\Gamma\backslash X,\mathcal{M})$ and $H^{\bullet}(\Gamma\backslash X,\mathcal{M})$ , explicitly.

For each prime number $p$ , we have the following double coset decomposition:

\Gamma\backslash\Gamma\begin{pmatrix}p&\\ &1\end{pmatrix}\Gamma=\Gamma\begin{pmatrix}p&\\ &1\end{pmatrix}\sqcup\bigsqcup_{j=0}^{p-1}\Gamma\begin{pmatrix}1&j\\ &p\end{pmatrix}.

Hence the endomorphism

\mathcal{M}\longrightarrow\mathcal{M};m\mapsto\begin{pmatrix}p&0\\ 0&1\end{pmatrix}m+\sum_{j=0}^{p-1}\begin{pmatrix}1&j\\ 0&p\end{pmatrix}m

induces an endomorphism of $\mathcal{M}_{\Gamma}$ . Similarly, the endomorphism

\mathcal{M}\longrightarrow\mathcal{M};m\mapsto\widetilde{\begin{pmatrix}p&0\\ 0&1\end{pmatrix}}m+\sum_{j=0}^{p-1}\widetilde{\begin{pmatrix}1&j\\ 0&p\end{pmatrix}}m

induces an endomorphism of $\mathcal{M}^{\Gamma}$ .

Definition 2.3.

Let $p$ be a prime number.

(1)

We define the Hecke operator $T_{p}$ at $p$ on $S_{\bullet}(X)\otimes\mathcal{M}$ by

\displaystyle T_{p}(\sigma\otimes P)

\displaystyle:=\begin{pmatrix}p&0\\ 0&1\end{pmatrix}\sigma\otimes\begin{pmatrix}p&0\\ 0&1\end{pmatrix}P+\sum_{j=0}^{p-1}\begin{pmatrix}1&j\\ 0&p\end{pmatrix}\sigma\otimes\begin{pmatrix}1&j\\ 0&p\end{pmatrix}P

for any simplex $\sigma\in S_{\bullet}(X)$ and element $P\in\mathcal{M}$ . The operator $T_{p}$ induces operators on $\mathcal{MS}(X)\otimes\mathcal{M}$ and $H_{\bullet}(\Gamma\backslash X,\mathcal{M})$ , etc., also written as $T_{p}$ .

(2)

We define the Hecke operator $T_{p}^{\prime}$ at $p$ on $\operatorname{Hom}_{\mathbb{Z}}(S_{\bullet}(X),\mathcal{M})$ by

\displaystyle(\phi|T_{p}^{\prime})(\sigma):=\widetilde{\begin{pmatrix}p&0\\ 0&1\end{pmatrix}}\left(\phi(\begin{pmatrix}p&0\\ 0&1\end{pmatrix}\sigma)\right)+\sum_{j=0}^{p-1}\widetilde{\begin{pmatrix}1&j\\ 0&p\end{pmatrix}}\left(\phi(\begin{pmatrix}1&j\\ 0&p\end{pmatrix}\sigma)\right)

for any homomorphism $\phi\in\operatorname{Hom}_{\mathbb{Z}}(S_{\bullet}(X),\mathcal{M}_{n}^{\flat})$ and simplex $\sigma\in S_{\bullet}(X)$ . The operator $T_{p}^{\prime}$ induces operators on $\operatorname{Hom}_{\mathbb{Z}}(\mathcal{MS}(X),\mathcal{M})$ and $H^{\bullet}(\Gamma\backslash X,\mathcal{M})$ , etc., also written as $T_{p}^{\prime}$ .

For later use, we also define auxiliary operators $U_{p}$ and $V_{p}$ on $S_{\bullet}(X)\otimes\mathcal{M}$ by

	$\displaystyle V_{p}(\sigma\otimes P)$	$\displaystyle:=\begin{pmatrix}p&0\\ 0&1\end{pmatrix}\sigma\otimes\begin{pmatrix}p&0\\ 0&1\end{pmatrix}P,$
	$\displaystyle U_{p}(\sigma\otimes P)$	$\displaystyle:=\sum_{j=0}^{p-1}\begin{pmatrix}1&j\\ 0&p\end{pmatrix}\sigma\otimes\begin{pmatrix}1&j\\ 0&p\end{pmatrix}P,$

so that $T_{p}:=V_{p}+U_{p}$ .

Lemma 2.4.

The composite $V_{p}U_{p}$ acts on $(S_{\bullet}(X)\otimes\mathcal{M}_{n})_{\Gamma}$ , and we have $V_{p}U_{p}=p^{n+1}$ as operators on $(S_{\bullet}(X)\otimes\mathcal{M}_{n})_{\Gamma}$ .

Proof.

Since diagonal matrices act trivially on $X$ , we have

	$\displaystyle V_{p}U_{p}(\sigma\otimes P)$	$\displaystyle=\sum_{j=0}^{p-1}\begin{pmatrix}p&pj\\ 0&p\end{pmatrix}\sigma\otimes\begin{pmatrix}p&pj\\ 0&p\end{pmatrix}P$
		$\displaystyle=p^{n}\sum_{j=0}^{p-1}\begin{pmatrix}1&j\\ 0&1\end{pmatrix}\left(\sigma\otimes P\right)$

for any simplex $\sigma\in S_{\bullet}(X)$ and polynomial $P\in\mathcal{M}_{n}$ . Since $\begin{pmatrix}1&j\\ 0&1\end{pmatrix}\in\Gamma$ for any integer $j$ , we obtain this lemma from this equality. ∎

2.5. Formal duality

Let $X\in\{\mathbb{H},\mathbb{H}^{\mathrm{BS}},\partial\mathbb{H}^{\mathrm{BS}}\}$ . As explained in §2.2, the homology and cohomology groups can be computed as

	$\displaystyle H_{\bullet}(\Gamma\backslash X,\mathcal{M}_{n})$	$\displaystyle\cong H_{\bullet}((S_{\bullet}(X)\otimes\mathcal{M}_{n})_{\Gamma}),$
	$\displaystyle H^{\bullet}(\Gamma\backslash X,\mathcal{M}_{n}^{\flat})$	$\displaystyle\cong H_{\bullet}(\operatorname{Hom}_{\mathbb{Z}}(S_{\bullet}(X),\mathcal{M}_{n}^{\flat})^{\Gamma}).$

The pairing $\langle~{},~{}\rangle\colon\mathcal{M}^{\flat}\times\mathcal{M}\longrightarrow\mathbb{Z}$ induces a pairing

\displaystyle\langle~{},~{}\rangle\colon\operatorname{Hom}_{\mathbb{Z}}(S_{\bullet}(X),\mathcal{M}_{n}^{\flat})\times S_{\bullet}(X)\otimes\mathcal{M}_{n}\longrightarrow\mathbb{Z},

which is computed as

\big{\langle}\phi,\sigma\otimes P\big{\rangle}:=\langle\phi(\sigma),P\rangle.

Note that for any matrix $\gamma\in M_{2}^{+}(\mathbb{Z})$ , we have

\big{\langle}\widetilde{\gamma}\phi,\sigma\otimes P\big{\rangle}=\langle\widetilde{\gamma}\phi(\gamma\sigma),P\rangle=\langle\phi(\gamma\sigma),{\gamma}P\rangle=\big{\langle}\phi,{\gamma}(\sigma\otimes P)\big{\rangle}.

Therefore, we have

\big{\langle}\phi|T_{p}^{\prime},\sigma\otimes P\big{\rangle}=\big{\langle}\phi,T_{p}(\sigma\otimes P)\big{\rangle}.

In particular, we obtain a Hecke-equivariant pairing

\langle~{},~{}\rangle\colon H^{\bullet}(\Gamma\backslash X,\mathcal{M}_{n}^{\flat})\times H_{\bullet}(\Gamma\backslash X,\mathcal{M}_{n})\longrightarrow\mathbb{Z},

which induces an isomorphism

H^{\bullet}(\Gamma\backslash X,\mathcal{M}_{n}^{\flat})/\text{(torsion)}\overset{\sim}{\longrightarrow}\operatorname{Hom}_{\mathbb{Z}}(H_{\bullet}(\Gamma\backslash X,\mathcal{M}_{n}),\mathbb{Z}).

2.6. Eichler–Shimura homomorphism

Let $M_{n+2}(\Gamma)$ denote the space of modular forms of weight $n+2$ and level $\Gamma=\mathrm{SL}_{2}(\mathbb{Z})$ . We define a homomorphism

\displaystyle r\colon M_{n+2}(\Gamma)\longrightarrow\operatorname{Hom}_{\mathbb{Z}}(\mathcal{MS}(\mathbb{H}),\mathcal{M}_{n}^{\flat}\otimes\mathbb{C})

\displaystyle r(f)(\{\alpha,\beta\}):=\int_{\alpha}^{\beta}f(z)(X_{1}-zX_{2})^{n}\,dz.

for any modular form $f\in M_{n+2}(\Gamma)$ and $\{\alpha,\beta\}\in\mathcal{MS}(\mathbb{H})$ . It is well-known that

r(M_{n+2}(\Gamma))\subset\operatorname{Hom}_{\mathbb{Z}}(\mathcal{MS}(\mathbb{H}),\mathcal{M}_{n}^{\flat}\otimes\mathbb{C})^{\Gamma},

and the homomorphism $r$ induces an injective homomorphism (called Eichler–Shimura homomorphism)

\displaystyle r\colon M_{n+2}(\Gamma)\longhookrightarrow H^{1}(Y,\mathcal{M}_{n}^{\flat})\otimes\mathbb{C}=H^{1}(Y^{\mathrm{BS}},\mathcal{M}_{n}^{\flat})\otimes\mathbb{C}.

See [Bel21, Section 5.3] for example.

Remark 2.5.

The definition of the Eichler–Shimura homomorphism shows that, for any element $\sigma=\{\alpha,\beta\}\otimes P\in\mathcal{MS}(\mathbb{H})\otimes\mathcal{M}_{n}$ and modular form $f\in M_{n+2}(\Gamma)$ , the pairing $\langle r(f),\sigma\rangle$ can be computed as

\langle r(f),\sigma\rangle=\int_{\alpha}^{\beta}f(z)P(z,1)\,dz.

Remark 2.6.

For each prime number $p$ , the double coset operator $T_{p}^{\prime\prime}:=\Gamma\begin{pmatrix}1&\\ &p\end{pmatrix}\Gamma$ acts on the space $M_{n+2}(\Gamma)$ of modular forms from the right by using the weight $n+2$ slash operator $|[~{}]_{n+2}$ . One can easily show that

r(f|[\gamma]_{n+2})=\widetilde{\gamma}\cdot r(f)

for any matrix $\gamma\in M_{2}^{+}(\mathbb{Z})$ . Hence the Eichler–Shimura homomorphism is Hecke-equivariant, that is, for all prime numbers $p$ , we have

r(f|T_{p}^{\prime\prime})=r(f)|T_{p}^{\prime}.

In other words, our Hecke operator $T_{p}^{\prime}$ coincides with the usual one via the Eichler–Shimura homomorphism.

The following lemma is well-known (see [Bel21, Theorem 5.3.27] for example).

Lemma 2.7.

The Eichler–Shimura homomorphism induces a Hecke-equivariant isomorphism

M_{n+2}(\Gamma)/S_{n+2}(\Gamma)\stackrel{{\scriptstyle\sim}}{{\longrightarrow}}H^{1}(\partial Y^{\mathrm{BS}},\mathcal{M}_{n}^{\flat})\otimes\mathbb{C}.

Here $S_{n+2}(\Gamma)$ denotes the space of cusp forms of weight $n+2$ and level $\Gamma$ .

2.7. Definition of the Eisenstein class

In this subsection, we define the Eisenstein class $\mathrm{Eis}_{n}$ and explain its basic properties.

We put $i:=\sqrt{-1}$ and let $\sigma_{n+1}(k)$ denote the sum-of-positive-divisors function, namely, $\sigma_{n+1}(k):=\sum_{0<d\mid k}d^{n+1}$ . Let

\displaystyle E_{n+2}(z):=1+\frac{2}{\zeta(-1-n)}\sum_{k=1}^{\infty}\sigma_{n+1}(k)e^{2\pi ikz}\in M_{n+2}(\Gamma)

denote the normalized holomorphic Eisenstein series of weight $n+2$ .

Lemma 2.8.

(1)

For any element $\tau\in\mathbb{H}$ , we have

$\langle r(E_{n+2}),\{\tau,\tau+1\}\otimes e_{0}\rangle=1.$
(2)

For any prime number $p$ , we have

$r(E_{n+2})|{T_{p}^{\prime}}=(1+p^{n+1})r(E_{n+2}).$

Proof.

Claim (1) follows from the fact that the constant term of $E_{n+2}$ is $1$ , and claim (2) follows from the fact that $E_{n+2}|{T_{p}^{\prime\prime}}=(1+p^{n+1})E_{n+2}$ and the Eichler–Shimura homomorphism is Hecke-equivariant. ∎

Definition 2.9.

We define the Eisenstein class $\mathrm{Eis}_{n}\in H^{1}(Y^{\mathrm{BS}},\mathcal{M}_{n}^{\flat})\otimes\mathbb{C}$ by

\mathrm{Eis}_{n}:=r\left(E_{n+2}\right).

Remark 2.10.

The method of defining the Eisenstein class $\mathrm{Eis}_{n}$ in this paper differs from the method in Harder’s book [Har, §3.3.6 (3.74)]. However, thanks to Lemma 2.8, they coincide.

2.8. Main theorem

Proposition 2.11.

The Eisenstein class $\mathrm{Eis}_{n}$ is rational, that is, $\mathrm{Eis}_{n}\in H^{1}(Y^{\mathrm{BS}},\mathcal{M}_{n}^{\flat})\otimes\mathbb{Q}$ .

This proposition is proved in Corollary 4.18.

Definition 2.12.

For any $\Gamma$ -module $\mathcal{M}$ , we define

H^{1}_{\mathrm{int}}(Y^{\mathrm{BS}},\mathcal{M}):=\operatorname{im}\left(H^{1}(Y^{\mathrm{BS}},\mathcal{M})\longrightarrow H^{1}(Y^{\mathrm{BS}},\mathcal{M})\otimes\mathbb{Q}\right).

Thanks to Proposition 2.11, we define the denominator $\Delta(\mathrm{Eis}_{n})$ of the Eisenstein class $\mathrm{Eis}_{n}$ with respect to the integral structure $H^{1}_{\mathrm{int}}(Y^{\mathrm{BS}},\mathcal{M}^{\flat}_{n})$ by

\Delta(\mathrm{Eis}_{n}):=\min\{\Delta\in\mathbb{Z}_{>0}\mid\Delta\mathrm{Eis}_{n}\in H^{1}_{\mathrm{int}}(Y^{\mathrm{BS}},\mathcal{M}_{n}^{\flat})\}.

Then the following is the main theorem which we want to prove in the present paper.

Theorem 2.13 ([Har, Theorem 5.1.2]).

The denominator $\Delta(\mathrm{Eis}_{n})$ of the Eisenstein class $\mathrm{Eis}_{n}$ is equal to the numerator of the special value $\zeta(-1-n)$ of the Riemann zeta function.

Remark 2.14.

(1)

Since $H^{1}(Y^{\mathrm{BS}},\mathcal{M}_{n}^{\flat})\otimes\mathbb{Q}=H^{1}(Y^{\mathrm{BS}},\mathcal{M}_{n})\otimes\mathbb{Q}$ , we have another integral structure $H^{1}_{\mathrm{int}}(Y^{\mathrm{BS}},\mathcal{M}_{n})$ , and one can consider another denominator $\Delta^{\prime}(\mathrm{Eis}_{n})$ of the Eisenstein class $\mathrm{Eis}_{n}$ :

\Delta^{\prime}(\mathrm{Eis}_{n}):=\min\{\Delta\in\mathbb{Z}_{>0}\mid\Delta\mathrm{Eis}_{n}\in H^{1}_{\mathrm{int}}(Y^{\mathrm{BS}},\mathcal{M}_{n})\}.

However, we show in Lemma 6.1 that $\Delta(\mathrm{Eis}_{n})=\Delta^{\prime}(\mathrm{Eis}_{n})$ .

(2)

By using the $q$ -expansion at the cusp $i\infty$ , one can regard $M_{n+2}(\Gamma)$ as a submodule of $\mathbb{C}[[q]]$ , and we obtain the de Rham rational structure of $M_{n+2}(\Gamma)$ by $M_{n+2}(\Gamma)\cap\mathbb{Q}[[q]]$ . The rationality of the critical values of the $L$ -function associated with a cusp form is obtained by studying the gap between the de Rham and Betti rational structures via the Eichler–Shimura homomorphism $r$ . However, Proposition 2.11 shows that the Eisenstein parts of the two rational structures coincide. Moreover, Theorem 2.13 says that the Eisenstein parts of the two integral structures $M_{n+2}(\Gamma)\cap\mathbb{Z}[[q]]$ and $H^{1}_{\mathrm{int}}(Y^{\mathrm{BS}},\mathcal{M}_{n}^{\flat})$ coincide, namely,

r(\mathbb{Q}E_{n+2}(z)\cap\mathbb{Z}[[q]])=\mathbb{Q}\mathrm{Eis}_{n}\cap H^{1}_{\mathrm{int}}(Y^{\mathrm{BS}},\mathcal{M}_{n}^{\flat})

since $2$ is a regular prime. Cf. [Har21, §1.1].

3. Construction of the cycle $\widetilde{T_{p}^{m}(C_{\nu}(\tau))}$

Fix a prime number $p$ . In this section, we construct a special homology cycle

\widetilde{T_{p}^{m}(C_{\nu}(\tau))}\in H_{1}(Y^{\mathrm{BS}},\mathcal{M}_{n}\otimes\mathbb{Z}_{(p)})

that is used to compute the $p$ -part of the denominator $\Delta(\mathrm{Eis}_{n})$ of the Eisenstein class $\mathrm{Eis}_{n}$ .

For any integer $1\leq\nu\leq n-1$ and element $\tau\in\mathbb{H}^{\mathrm{BS}}$ , we set

C_{\nu}(\tau):=\left\{-\frac{1}{\tau},\tau\right\}\otimes e_{\nu}\in\mathcal{MS}(\mathbb{H}^{\mathrm{BS}})\otimes\mathcal{M}_{n},

where recall that $e_{\nu}=X_{1}^{\nu}X_{2}^{n-\nu}$ .

3.1. Computation of $T_{p}^{m}(C_{\nu}(\tau))$

Recall also the operators

	$\displaystyle V_{p}(\sigma\otimes P)$	$\displaystyle:=\begin{pmatrix}p&0\\ 0&1\end{pmatrix}\sigma\otimes\begin{pmatrix}p&0\\ 0&1\end{pmatrix}P,$
	$\displaystyle U_{p}(\sigma\otimes P)$	$\displaystyle:=\sum_{j=0}^{p-1}\begin{pmatrix}1&j\\ 0&p\end{pmatrix}\sigma\otimes\begin{pmatrix}1&j\\ 0&p\end{pmatrix}P$

on $S_{\bullet}(\mathbb{H}^{\mathrm{BS}})\otimes\mathcal{M}_{n}$ . We have $T_{p}=V_{p}+U_{p}$ . For each integer $m\geq 0$ , set

\displaystyle W_{m}:=\sum_{k=0}^{m}U_{p}^{k}V_{p}^{m-k}.

Note that $W_{0}$ is the identity map. For any (commutative) ring $R$ and cycle $C\in S_{\bullet}(\mathbb{H}^{\mathrm{BS}})\otimes(\mathcal{M}_{n}\otimes R)$ , we denote by $[C]$ the image of $C$ in $(S_{\bullet}(\mathbb{H}^{\mathrm{BS}})\otimes(\mathcal{M}_{n}\otimes R))_{\Gamma}$ .

Lemma 3.1.

Let $m\geq 1$ be an integer and $C\in S_{\bullet}(\mathbb{H}^{\mathrm{BS}})\otimes\mathcal{M}_{n}$ .

(1)

We have $T_{p}([W_{m}(C)])=[W_{m+1}(C)]+p^{n+1}[W_{m-1}(C)]$ and $T_{p}([W_{0}(C)])=[W_{1}(C)]$ .
(2)

We have

\displaystyle T_{p}^{m}([C])=\sum_{A=0}^{\lfloor m/2\rfloor}C(m-A,A)p^{(n+1)A}[W_{m-2A}(C)],

where $\lfloor m/2\rfloor$ is the greatest integer less than or equal to $m/2$ and

C(A,B):=\left(1-\frac{B}{A+1}\right)\binom{A+B}{B}=\binom{A+B}{B}-\binom{A+B}{B-1}\in\mathbb{Z}.

Here we assume $\binom{a}{b}=0$ if $b<0$ .

Proof.

Claim (1) follows from the fact that $V_{p}U_{p}=p^{n+1}$ proved in Lemma 2.4. Let us prove claim (2). For notational simplicity, we put

w_{m}:=[W_{m}(C)].

Then claim (1) shows that we can write

T_{p}^{m}([C])=\sum_{k=0}^{m}a_{k}^{(m)}w_{m-k}

with $a_{k}^{(m)}\in\mathbb{Z}$ such that

	$\displaystyle a_{0}^{(m)}$	$\displaystyle=a_{0}^{(m-1)}=\cdots=a_{0}^{(1)}=a_{0}^{(0)}=1,$
	$\displaystyle a_{1}^{(m)}$	$\displaystyle=a_{1}^{(m-1)}=\cdots=a_{1}^{(1)}=0,$
	$\displaystyle a_{k}^{(m)}$	$\displaystyle=a_{k}^{(m-1)}+p^{n+1}a_{k-2}^{(m-1)},\,\,\,\quad(2\leq k\leq m-1)$
	$\displaystyle a_{m}^{(m)}$	$\displaystyle=p^{n+1}a_{m-2}^{(m-1)}.$

Therefore, we have $a_{2k+1}^{(m)}=0$ for any integer $0\leq k\leq(m-1)/2$ , and hence

T_{p}^{m}([C])=\sum_{k=0}^{\lfloor m/2\rfloor}a_{2k}^{(m)}w_{m-2k}.

Let us show that $a_{2k}^{(m)}=C(m-k,k)p^{k(n+1)}$ for any integer $0\leq k\leq m/2$ by induction on $m$ . When $m=1$ or $k=0$ , this claim is clear. If $m>1$ and $1\leq k\leq(m-1)/2$ , then the induction hypothesis shows that

	$\displaystyle a_{2k}^{(m)}$	$\displaystyle=a_{2k}^{(m-1)}+p^{n+1}a_{2k-2}^{(m-1)}$
		$\displaystyle=C(m-1-k,k)p^{k(n+1)}+C(m-k,k+1)p^{k(n+1)}$
		$\displaystyle=C(m-k,k)p^{k(n+1)}.$

Moreover, if $m$ is even, i.e., $m=2t$ , then we have

\displaystyle a_{m}^{(m)}

\displaystyle=p^{n+1}a^{(m-1)}_{m-2}=C(t,t-1)p^{t(n+1)}=\frac{1}{t+1}\frac{(2t)!}{(t!)(t!)}p^{t(n+1)}=C(t,t)p^{t(n+1)}.

∎

By definition, we have

	$\displaystyle W_{m-2A}(C_{\nu}(\tau))$	$\displaystyle=\sum_{k=0}^{m-2A}U_{p}^{k}V_{p}^{m-2A-k}(C_{\nu}(\tau))$
		$\displaystyle=\sum_{k=0}^{m-2A}U_{p}^{k}\begin{pmatrix}p^{m-2A-k}&0\\ 0&1\end{pmatrix}\left(\left\{-\frac{1}{\tau},\tau\right\}\otimes e_{\nu}\right)$
		$\displaystyle=\sum_{k=0}^{m-2A}p^{(n-\nu)(m-2A-k)}U_{p}^{k}\left(\left\{-\frac{p^{m-2A-k}}{\tau},p^{m-2A-k}\tau\right\}\otimes e_{\nu}\right).$

Definition 3.2.

Take elements $\tau_{0},\tau_{1}\in\mathbb{H}^{\mathrm{BS}}$ . For any integers $\nu$ , $j$ , and $k$ satisfying $1\leq\nu\leq n-1$ and $0\leq j\leq p^{k}-1$ , we define

	$\displaystyle C_{\nu,k,j}(\tau_{0},\tau_{1})$	$\displaystyle:=\begin{pmatrix}1&j\\ 0&p^{k}\end{pmatrix}\left\{-\frac{1}{\tau_{0}},\tau_{1}\right\}\otimes\begin{pmatrix}1&j\\ 0&p^{k}\end{pmatrix}e_{\nu}$
		$\displaystyle=\left\{\frac{-\frac{1}{\tau_{0}}+j}{p^{k}},\frac{\tau_{1}+j}{p^{k}}\right\}\otimes\begin{pmatrix}1&j\\ 0&p^{k}\end{pmatrix}e_{\nu}\in\mathcal{M}\mathcal{S}(\mathbb{H}^{\mathrm{BS}})\otimes\mathcal{M}_{n}.$

Note that we have

U_{p}^{k}\left(\left\{-\frac{1}{\tau_{0}},\tau_{1}\right\}\otimes e_{\nu}\right)=\sum_{j=0}^{p^{k}-1}C_{\nu,k,j}(\tau_{0},\tau_{1}).

To sum up, we obtain the following corollary.

Corollary 3.3.

We have

	$\displaystyle T_{p}^{m}([C_{\nu}(\tau)])$
	$\displaystyle=\sum_{A=0}^{\lfloor m/2\rfloor}C(m-A,A)p^{(n+1)A}\sum_{k=0}^{m-2A}p^{(n-\nu)(m-2A-k)}\sum_{j=0}^{p^{k}-1}\left[C_{\nu,k,j}\left(\frac{\tau}{p^{m-2A-k}},p^{m-2A-k}\tau\right)\right].$

3.2. Computation of the boundary $\partial C_{\nu,k,j}\left(\tau_{0},\tau_{1}\right)$

Next, we compute the boundary $\partial C_{\nu,k,j}\left(\tau_{0},\tau_{1}\right)$ .

Definition 3.4.

(1)

For any integers $j$ and $N$ with $p\nmid j$ and $N>0$ , we denote by $d_{N}(j)$ and $b_{N}(j)$ the integers uniquely determined by

$1\leq d_{N}(j)<p^{N}\,\,\,\textrm{ and }\,\,\,jd_{N}(j)-p^{N}b_{N}(j)=1.$

We also put $d_{0}(j):=0$ and $b_{0}(j):=-1$ for any integer $j$ .
(2)

For any integers $k$ and $j$ , we set

$l_{k}(j):=\min\{\mathrm{ord}_{p}(j),k\}.$

Note that $l_{k}(0)=k$ . We also put $j^{\prime}:=j/p^{l_{k}(j)}$ .

In the following, for integers $j$ and $k$ with $0\leq j\leq p^{k}-1$ , we often write as

(3.1)

\displaystyle\begin{split}l&:=l_{k}(j),\\ j^{\prime}&:=j/p^{l_{k}(j)},\\ d&:=d_{k-l_{k}(j)}(j^{\prime}),\\ b&:=b_{k-l_{k}(j)}(j^{\prime})\end{split}

for simplicity.

Definition 3.5.

For any integers $\nu$ , $j$ , and $k$ with $1\leq\nu\leq n-1$ and $0\leq j\leq p^{k}-1$ , we define homogeneous polynomials $E_{\nu,k,j}^{(1)}$ and $E_{\nu,k,j}^{(0)}$ in $\mathcal{M}_{n}$ by

	$\displaystyle E_{\nu,k,j}^{(1)}(X_{1},X_{2})$	$\displaystyle:=(p^{k}X_{1}-jX_{2})^{\nu}X_{2}^{n-\nu},$
	$\displaystyle E_{\nu,k,j}^{(0)}(X_{1},X_{2})$	$\displaystyle:=(-1)^{\nu+1}(p^{l}X_{2})^{\nu}(p^{k-l}X_{1}+dX_{2})^{n-\nu}.$

Lemma 3.6.

We have

\displaystyle\partial[C_{\nu,k,j}(\tau_{0},\tau_{1})]=\left[\left\{\frac{\tau_{1}+j}{p^{k}}\right\}\otimes E_{\nu,k,j}^{(1)}+\left\{\frac{p^{l}\tau_{0}-d}{p^{k-l}}\right\}\otimes E_{\nu,k,j}^{(0)}\right]

in $(S_{0}(\mathbb{H}^{\mathrm{BS}})\otimes\mathcal{M}_{n})_{\Gamma}$ .

Proof.

By definition, we have

\displaystyle\partial[C_{\nu,k,j}(\tau_{0},\tau_{1})]=\left[\begin{pmatrix}1&j\\ 0&p^{k}\end{pmatrix}\left\{\tau_{1}\right\}\otimes\begin{pmatrix}1&j\\ 0&p^{k}\end{pmatrix}e_{\nu}-\begin{pmatrix}1&j\\ 0&p^{k}\end{pmatrix}\left\{-\frac{1}{\tau_{0}}\right\}\otimes\begin{pmatrix}1&j\\ 0&p^{k}\end{pmatrix}e_{\nu}\right].

The definition of $E_{\nu,k,j}^{(1)}$ shows that

\begin{pmatrix}1&j\\ 0&p^{k}\end{pmatrix}\left\{\tau_{1}\right\}\otimes\begin{pmatrix}1&j\\ 0&p^{k}\end{pmatrix}e_{\nu}=\left\{\frac{\tau_{1}+j}{p^{k}}\right\}\otimes E_{\nu,k,j}^{(1)}.

Moreover, we have

	$\displaystyle\begin{pmatrix}1&j\\ 0&p^{k}\end{pmatrix}\left\{-\frac{1}{\tau_{0}}\right\}\otimes\begin{pmatrix}1&j\\ 0&p^{k}\end{pmatrix}e_{\nu}$	$\displaystyle=\begin{pmatrix}1&j\\ 0&p^{k}\end{pmatrix}\begin{pmatrix}0&-1\\ 1&0\end{pmatrix}\left\{\tau_{0}\right\}\otimes\begin{pmatrix}1&j\\ 0&p^{k}\end{pmatrix}e_{\nu}$
		$\displaystyle=\begin{pmatrix}j^{\prime}&b\\ p^{k-l}&d\end{pmatrix}\begin{pmatrix}p^{l}&-d\\ 0&p^{k-l}\end{pmatrix}\left\{\tau_{0}\right\}\otimes\begin{pmatrix}1&j\\ 0&p^{k}\end{pmatrix}e_{\nu}.$

Since

\displaystyle\left[\begin{pmatrix}j^{\prime}&b\\ p^{k-l}&d\end{pmatrix}\begin{pmatrix}p^{l}&-d\\ 0&p^{k-l}\end{pmatrix}\left\{\tau_{0}\right\}\otimes\begin{pmatrix}1&j\\ 0&p^{k}\end{pmatrix}e_{\nu}\right]=\left[\begin{pmatrix}p^{l}&-d\\ 0&p^{k-l}\end{pmatrix}\left\{\tau_{0}\right\}\otimes\begin{pmatrix}d&-b\\ -p^{k-l}&j^{\prime}\end{pmatrix}\begin{pmatrix}1&j\\ 0&p^{k}\end{pmatrix}e_{\nu}\right]

and $\begin{pmatrix}d&-b\\ -p^{k-l}&j^{\prime}\end{pmatrix}\begin{pmatrix}1&j\\ 0&p^{k}\end{pmatrix}e_{\nu}=-E_{\nu,k,j}^{(0)}$ , we obtain

\displaystyle\left[\begin{pmatrix}1&j\\ 0&p^{k}\end{pmatrix}\left\{-\frac{1}{\tau_{0}}\right\}\otimes\begin{pmatrix}1&j\\ 0&p^{k}\end{pmatrix}e_{\nu}\right]=-\left[\left\{\frac{p^{l}\tau_{0}-d}{p^{k-l}}\right\}\otimes E_{\nu,k,j}^{(0)}\right].

∎

3.3. A cycle $\widetilde{T_{p}^{m}(C_{\nu}(\tau))}$ in $\mathcal{M}\mathcal{S}(\mathbb{H}^{\mathrm{BS}})\otimes(\mathcal{M}_{n}\otimes\mathbb{Q})$

In this subsection, we construct a cycle $\widetilde{T_{p}^{m}(C_{\nu}(\tau))}$ in $\mathcal{M}\mathcal{S}(\mathbb{H}^{\mathrm{BS}})\otimes(\mathcal{M}_{n}\otimes\mathbb{Q})$ which is a lift of $T_{p}^{m}(C_{\nu}(\tau))$ and is $p$ -adically integral for any sufficiently large integer $m$ .

3.3.1. Bernoulli polynomials

Since a key tool for constructing the cycle $\widetilde{T_{p}^{m}(C_{\nu}(\tau))}$ is the Bernoulli polynomials, we briefly recall the basic properties of the Bernoulli polynomials.

Let $t$ be a non-negative integer. We denote by $B_{t}(x)$ the $t$ -th Bernoulli polynomial and by

B_{t}:=B_{t}(0)

the $t$ -th Bernoulli number. For notational simplicity, we put

\widetilde{B}_{t}(x):=\frac{B_{t}(x)-B_{t}}{t}.

In this paper, we use the following well-known facts without any notice:

	$\displaystyle B_{t}(x)=\sum_{\mu=0}^{t}\binom{t}{\mu}B_{t-\mu}x^{\mu},$
	$\displaystyle\int_{x}^{x+1}B_{t}(x)\,dx=x^{t},$
	$\displaystyle\int_{a}^{x}B_{t}(x)\,dx=\frac{B_{t+1}(x)-B_{t+1}(a)}{t+1},$
	$\displaystyle\mathrm{ord}_{p}(B_{t})\geq-1.$

The last fact is called the von Staudt–Clausen theorem. We note that the second and third facts imply that

\sum_{j=0}^{x-1}j^{t}=\widetilde{B}_{t+1}(x).

3.3.2. $P^{\dagger}$ and $P^{\ddagger}$

Set

\mathcal{M}_{n,(p)}:=\bigoplus_{\mu=0}^{n}\mathbb{Z}_{(p)}X_{1}^{\mu}X_{2}^{n-\mu}\,\,\,\textrm{ and }\,\,\,\mathcal{M}_{n,(p)}^{\circ}:=\bigoplus_{\mu=0}^{n-1}\mathbb{Z}_{(p)}X_{1}^{\mu}X_{2}^{n-\mu},

and let

{\dagger}\colon\mathcal{M}_{n,(p)}^{\circ}\otimes\mathbb{Q}\rightarrow\mathcal{M}_{n,(p)}\otimes\mathbb{Q};P\mapsto P^{{\dagger}}

be the $\mathbb{Q}$ -linear map defined by

(X_{1}^{\mu}X_{2}^{n-\mu})^{{\dagger}}:=X_{2}^{n}\frac{B_{\mu+1}(X_{1}/X_{2})-B_{\mu+1}}{\mu+1}=X_{2}^{n}\widetilde{B}_{\mu+1}(X_{1}/X_{2}).

For any integer $\mu\in\{0,\ldots,n-1\}$ , we have $1+\mathrm{ord}_{p}(\mu+1)\leq n$ , and hence $p^{n}X_{2}^{n}\widetilde{B}_{\mu+1}(X_{1}/X_{2})\in\mathcal{M}_{n,(p)}$ . This fact shows that

{\dagger}(\mathcal{M}_{n,(p)}^{\circ})\subset\frac{1}{p^{n}}\mathcal{M}_{n,(p)}.

Similarly, let

{\ddagger}\colon\mathcal{M}_{n,(p)}^{\circ}\otimes\mathbb{Q}\rightarrow\mathcal{M}_{n}\otimes\mathbb{Q};P\mapsto P^{{\ddagger}}

be the $\mathbb{Q}$ -linear map defined by

(X_{1}^{\mu}X_{2}^{n-\mu})^{{\ddagger}}:=X_{2}^{n}\frac{(X_{1}/X_{2})^{\mu+1}-B_{\mu+1}}{\mu+1}.

The following lemma follows from the definitions of $P^{\dagger}$ and $P^{\ddagger}$ .

Lemma 3.7.

For any polynomial $P(X_{1},X_{2})\in\mathcal{M}_{n}^{\circ}\otimes\mathbb{Q}$ , we have

	$\displaystyle P^{{\dagger}}(X_{1}+X_{2},X_{2})-P^{{\dagger}}(X_{1},X_{2})$	$\displaystyle=P(X_{1},X_{2}),$
	$\displaystyle\int_{x}^{x+1}P^{{\dagger}}(z,1)\,dz$	$\displaystyle=P^{{\ddagger}}(x,1),$
	$\displaystyle\int_{x_{1}}^{x_{2}}P(z,1)\,dz$	$\displaystyle=P^{{\ddagger}}(x_{2},1)-P^{{\ddagger}}(x_{1},1).$

3.3.3. Definitions of polynomials $P^{(1)}_{\nu,k,j}$ and $P^{(0)}_{\nu,k,j}$

Definition 3.8.

For any integers $\nu$ , $j$ , and $k$ with $1\leq\nu\leq n-1$ and $0\leq j\leq p^{k}-1$ , we define polynomials $P_{\nu,k,j}^{(1)}$ and $P_{\nu,k,j}^{(0)}$ in $\mathcal{M}_{n}\otimes\mathbb{Q}$ by

	$\displaystyle P_{\nu,k,j}^{(1)}$	$\displaystyle:=E_{\nu,k,j}^{(1){\dagger}},$
	$\displaystyle P_{\nu,k,j}^{(0)}$	$\displaystyle:=E_{\nu,k,j}^{(0){\dagger}}.$

Lemma 3.9.

For each $i\in\{0,1\}$ , we have $P_{\nu,k,j}^{(i)}\in p^{\min\{0,k-n\}}\mathcal{M}_{n,(p)}$ .

Proof.

This lemma follows from the facts that ${\dagger}(\mathcal{M}_{n,(p)}^{\circ})\subset p^{-n}\mathcal{M}_{n,(p)}$ and $(X_{2}^{n})^{\dagger}=X_{1}X_{2}^{n-1}$ and that the coefficient of $X_{1}^{\mu}X_{2}^{n-\mu}$ in $E_{\nu,k,j}^{(i)}$ is divided by $p^{k}$ if $\mu\geq 1$ . ∎

3.3.4. Definition of the cycle $\widetilde{T_{p}^{m}(C_{\nu}(\tau))}$

Now we can define $\widetilde{C}_{\nu,k,j}(\tau_{0},\tau_{1})$ and $\widetilde{T_{p}^{m}(C_{\nu}(\tau))}$ .

Definition 3.10.

Let $\tau_{0},\tau_{1}\in\mathbb{H}^{\mathrm{BS}}$ . For any integers $\nu$ , $k$ , and $j$ with $1\leq\nu\leq n-1$ and $0\leq j\leq p^{k}-1$ , we define an element $\widetilde{C}_{\nu,k,j}(\tau_{0},\tau_{1})\in\mathcal{M}\mathcal{S}(\mathbb{H}^{\mathrm{BS}})\otimes(\mathcal{M}_{n}\otimes\mathbb{Q})$ by

\displaystyle\widetilde{C}_{\nu,k,j}(\tau_{0},\tau_{1}):=C_{\nu,k,j}(\tau_{0},\tau_{1})-\left\{\frac{\tau_{1}+j}{p^{k}},\frac{\tau_{1}+j}{p^{k}}+1\right\}\otimes P_{\nu,k,j}^{(1)}-\left\{\frac{p^{l}\tau_{0}-d}{p^{k-l}},\frac{p^{l}\tau_{0}-d}{p^{k-l}}+1\right\}\otimes P_{\nu,k,j}^{(0)}.

We also put

\widetilde{C}_{\nu,k,j}^{\rm int}(\tau_{0},\tau_{1}):=p^{\max\{0,n-k\}}\widetilde{C}_{\nu,k,j}(\tau_{0},\tau_{1}).

Note that $\widetilde{C}_{\nu,k,j}^{\rm int}(\tau_{0},\tau_{1})\in\mathcal{M}\mathcal{S}(\mathbb{H}^{\mathrm{BS}})\otimes\mathcal{M}_{n,(p)}$ by Lemma 3.9.

Lemma 3.11.

We have

\partial[\widetilde{C}_{\nu,k,j}^{\rm int}(\tau_{0},\tau_{1})]=0\,\,\,\textrm{ in }\,\,\,(S_{0}(\mathbb{H}^{\mathrm{BS}})\otimes\mathcal{M}_{n,(p)})_{\Gamma}.

In particular, $[\widetilde{C}_{\nu,k,j}^{\rm int}(\tau_{0},\tau_{1})]$ defines a homology class

[\widetilde{C}_{\nu,k,j}^{\rm int}(\tau_{0},\tau_{1})]\in H_{1}(Y^{\mathrm{BS}},\mathcal{M}_{n,(p)}).

Proof.

This lemma follows from Definitions 3.5 and 3.8 and Lemmas 3.6 and 3.7. ∎

Lemma 3.12.

The homology class $[\widetilde{C}_{\nu,k,j}^{\rm int}(\tau_{0},\tau_{1})]$ is independent of the choices of $\tau_{0}$ and $\tau_{1}$ .

Proof.

Let $\tau_{0}^{\prime}$ and $\tau_{1}^{\prime}$ be another pair of points in $\mathbb{H}^{\mathrm{BS}}$ . We will prove that

[\widetilde{C}_{\nu,k,j}^{\rm int}(\tau_{0},\tau_{1})]=[\widetilde{C}_{\nu,k,j}^{\rm int}(\tau_{0}^{\prime},\tau_{1}^{\prime})]\,\,\,\text{ in }\,\,\,H_{1}(Y^{\mathrm{BS}},\mathcal{M}_{n,{(p)}}).

It suffices to construct an element $h\in S_{2}(\mathbb{H}^{\mathrm{BS}})\otimes\mathcal{M}_{n,{(p)}}$ such that

\partial[h]={[\widetilde{C}_{\nu,k,j}^{\rm int}(\tau_{0},\tau_{1})]-[\widetilde{C}_{\nu,k,j}^{\rm int}(\tau_{0}^{\prime},\tau_{1}^{\prime})]}\,\,\,\text{ in }\,\,\,(\mathcal{MS}(\mathbb{H}^{\mathrm{BS}})\otimes\mathcal{M}_{n,{(p)}})_{\Gamma}.

For notational simplicity, set $q:=p^{\max\{0,n-k\}}$ . First, since $\mathbb{H}^{\mathrm{BS}}$ is simply connected, there exist elements $h_{1},h_{2},h_{3}\in S_{2}(\mathbb{H}^{\mathrm{BS}})$ such that

	$\displaystyle\partial h_{1}$	$\displaystyle=\left\{\frac{-\frac{1}{\tau_{0}}+j}{p^{k}},\frac{\tau_{1}+j}{p^{k}}\right\}-\left\{\frac{-\frac{1}{\tau_{0}^{\prime}}+j}{p^{k}},\frac{\tau_{1}^{\prime}+j}{p^{k}}\right\}-\left\{\frac{\tau_{1}^{\prime}+j}{p^{k}}\frac{\tau_{1}+j}{p^{k}}\right\}+\left\{\frac{-\frac{1}{\tau_{0}^{\prime}}+j}{p^{k}},\frac{-\frac{1}{\tau_{0}}+j}{p^{k}},\right\},$
	$\displaystyle\partial h_{2}$	$\displaystyle=\left\{\frac{\tau_{1}+j}{p^{k}},\frac{\tau_{1}+j}{p^{k}}+1\right\}-\left\{\frac{\tau_{1}^{\prime}+j}{p^{k}},\frac{\tau_{1}^{\prime}+j}{p^{k}}+1\right\}-\left\{\frac{\tau_{1}^{\prime}+j}{p^{k}}+1,\frac{\tau_{1}+j}{p^{k}}+1\right\}+\left\{\frac{\tau_{1}^{\prime}+j}{p^{k}},\frac{\tau_{1}+j}{p^{k}}\right\},$
	$\displaystyle\partial h_{3}$	$\displaystyle=\left\{\frac{p^{l}\tau_{0}-d}{p^{k-l}},\frac{p^{l}\tau_{0}-d}{p^{k-l}}+1\right\}-\left\{\frac{p^{l}\tau_{0}^{\prime}-d}{p^{k-l}},\frac{p^{l}\tau_{0}^{\prime}-d}{p^{k-l}}+1\right\}-\left\{\frac{p^{l}\tau_{0}^{\prime}-d}{p^{k-l}}+1,\frac{p^{l}\tau_{0}-d}{p^{k-l}}+1\right\}+\left\{\frac{p^{l}\tau_{0}^{\prime}-d}{p^{k-l}},\frac{p^{l}\tau_{0}-d}{p^{k-l}}\right\}.$

Then we see that

h:={{q}\left(h_{1}\otimes\begin{pmatrix}1&j\\ 0&p^{k}\end{pmatrix}e_{\nu}-h_{2}\otimes P_{\nu,k,j}^{(1)}-h_{3}\otimes P_{\nu,k,j}^{(0)}\right)}

satisfies the desired property. Indeed, we have

\displaystyle\partial h={{q}\left(\widetilde{C}_{\nu,k,j}(\tau_{0},\tau_{1})-\widetilde{C}_{\nu,k,j}(\tau_{0}^{\prime},\tau_{1}^{\prime})-B^{(1)}-B^{(0)}\right)},

where

	$\displaystyle B^{(1)}$	$\displaystyle:=\left\{\frac{\tau_{1}^{\prime}+j}{p^{k}}\frac{\tau_{1}+j}{p^{k}}\right\}\otimes\begin{pmatrix}1&j\\ 0&p^{k}\end{pmatrix}e_{\nu}-\left\{\frac{\tau_{1}^{\prime}+j}{p^{k}}+1,\frac{\tau_{1}+j}{p^{k}}+1\right\}\otimes P_{\nu,k,j}^{(1)}+\left\{\frac{\tau_{1}^{\prime}+j}{p^{k}},\frac{\tau_{1}+j}{p^{k}}\right\}\otimes P_{\nu,k,j}^{(1)},$
	$\displaystyle B^{(0)}$	$\displaystyle:=-\left\{\frac{-\frac{1}{\tau_{0}^{\prime}}+j}{p^{k}},\frac{-\frac{1}{\tau_{0}}+j}{p^{k}},\right\}\otimes\begin{pmatrix}1&j\\ 0&p^{k}\end{pmatrix}e_{\nu}-\left\{\frac{p^{l}\tau_{0}^{\prime}-d}{p^{k-l}}+1,\frac{p^{l}\tau_{0}-d}{p^{k-l}}+1\right\}\otimes P_{\nu,k,j}^{(0)}+\left\{\frac{p^{l}\tau_{0}^{\prime}-d}{p^{k-l}},\frac{p^{l}\tau_{0}-d}{p^{k-l}}\right\}\otimes P_{\nu,k,j}^{(0)}.$

Then the same computation as in the proof of Lemma 3.6 shows that

	$\displaystyle{q}B^{(1)}$	$\displaystyle\equiv\left\{\frac{\tau_{1}^{\prime}+j}{p^{k}}\frac{\tau_{1}+j}{p^{k}}\right\}\otimes{q}\left(E_{\nu,k,j}^{(1)}(X_{1},X_{2})-P_{\nu,k,j}^{(1)}(X_{1}+X_{2},X_{2})+P_{\nu,k,j}^{(1)}(X_{1},X_{2})\right)=0,$
	$\displaystyle{q}B^{(0)}$	$\displaystyle\equiv\left\{\frac{p^{l}\tau_{0}^{\prime}-d}{p^{k-l}}+1,\frac{p^{l}\tau_{0}-d}{p^{k-l}}+1\right\}\otimes{q}\left(E_{\nu,k,j}^{(0)}(X_{1},X_{2})-P_{\nu,k,j}^{(0)}(X_{1}+X_{2},X_{2})+P_{\nu,k,j}^{(0)}(X_{1},X_{2})\right)=0,$

where $\equiv$ means that it is an equality in the $\Gamma$ -coinvariant $(\mathcal{MS}(\mathbb{H}^{\mathrm{BS}})\otimes\mathcal{M}_{n,{(p)}})_{\Gamma}$ . Hence we obtain $\partial[h]=[\widetilde{C}_{\nu,k,j}^{\rm int}(\tau_{0},\tau_{1})]-[\widetilde{C}_{\nu,k,j}^{\rm int}(\tau_{0}^{\prime},\tau_{1}^{\prime})]$ . ∎

Definition 3.13.

For any integer $m>0$ and element $\tau\in\mathbb{H}^{\mathrm{BS}}$ , we define a cycle $\widetilde{T_{p}^{m}(C_{\nu}(\tau))}\in\mathcal{MS}(\mathbb{H}^{\mathrm{BS}})\otimes(\mathcal{M}_{n}\otimes\mathbb{Q})$ by

	$\displaystyle\widetilde{T_{p}^{m}(C_{\nu}(\tau))}$
	$\displaystyle:=\sum_{A=0}^{\lfloor m/2\rfloor}C(m-A,A)p^{(n+1)A}\sum_{k=0}^{m-2A}p^{(n-\nu)(m-2A-k)}\sum_{j=0}^{p^{k}-1}\widetilde{C}_{\nu,k,j}\left(\frac{\tau}{p^{m-2A-k}},p^{m-2A-k}\tau\right).$

Definition 3.14.

For any integer $\nu\in\{0,\ldots,n\}$ , we define a homology class

[C_{\nu}]\in H_{1}(Y^{\rm BS},\partial Y^{\rm BS},\mathcal{M}_{n})

to be the element represented by the cycle

C_{\nu}:=\{0,i\infty\}\otimes e_{\nu},

where $i\infty\in\partial\mathbb{H}^{\mathrm{BS}}$ is a point such that

i\infty:=\lim_{t\in\mathbb{R}_{>0},t\to\infty}it.

Lemma 3.15.

Let $1\leq\nu\leq n-1$ be an integer.

(1)

$\partial[\widetilde{T_{p}^{m}(C_{\nu}(\tau))}]=0$ in $(S_{0}(\mathbb{H}^{\mathrm{BS}})\otimes\mathcal{M}_{n}\otimes\mathbb{Q})_{\Gamma}$ .
(2)

If $m\geq n$ , then we have $\widetilde{T_{p}^{m}(C_{\nu}(\tau))}\in\mathcal{M}\mathcal{S}(\mathbb{H})\otimes\mathcal{M}_{n,(p)}$ . Hence $\widetilde{T_{p}^{m}(C_{\nu}(\tau))}$ defines a $p$ -integral homology class $[\widetilde{T_{p}^{m}(C_{\nu}(\tau))}]$ in $H_{1}(Y^{\mathrm{BS}},\mathcal{M}_{n,(p)})$ which is independent of the choice of $\tau$ .
(3)

If $m\geq n$ , then the image of the homology class $[\widetilde{T_{p}^{m}(C_{\nu}(\tau))}]$ under the homomorphism $H_{1}(Y^{\mathrm{BS}},\mathcal{M}_{n,(p)})\longrightarrow H_{1}(Y^{\rm BS},\partial Y^{\rm BS},\mathcal{M}_{n,(p)})$ is $T_{p}^{m}([C_{\nu}])$ .

Proof.

Claim (1) follows from Lemma 3.11. Let us show claim (2). By Lemma 3.9, we have $p^{\max\{0,n-k\}}\widetilde{C}_{\nu,k,j}(\tau_{0},\tau_{1})\in\mathcal{M}\mathcal{S}(\mathbb{H}^{\mathrm{BS}})\otimes\mathcal{M}_{n,(p)}$ . Moreover, since $1\leq\nu\leq n-1$ and $2\leq n$ , we have

(n+1)A+(n-\nu)(m-2A-k)\geq(n+1)A+m-2A-k\geq m-k

for any non-negative integer $A$ . This fact shows that $\widetilde{T_{p}^{m}(C_{\nu}(\tau))}\in\mathcal{M}\mathcal{S}(\mathbb{H})\otimes\mathcal{M}_{n,(p)}$ if $m\geq n$ . It follows from Lemma 3.12 that the homology class $[\widetilde{T_{p}^{m}(C_{\nu}(\tau))}]$ does not depend on the choice of $\tau$ . Claim (3) follows from the definition of $\widetilde{T_{p}^{m}(C_{\nu}(\tau))}$ and Lemma 3.1.

∎

4. Period

The aim of this section is to compute the value

\displaystyle\langle\mathrm{Eis}_{n},\widetilde{T_{p}^{m!}(C_{\nu}(\tau))}\rangle

and its $p$ -adic limit as $m\to\infty$ . In this paper, we often consider the $p$ -adic limit, and hence the symbol $\lim_{m\to\infty}$ will always mean the $p$ -adic limit. The following is the main result of this section.

Theorem 4.1.

For any integer $\nu\in\{1,\ldots,n-1\}$ , we have

\displaystyle\lim_{m\to\infty}\langle\mathrm{Eis}_{n},\widetilde{T_{p}^{m!}(C_{\nu}(\tau))}\rangle=(1-p^{n+1})\left(\frac{1}{1-p^{n+1}}\frac{\zeta(-\nu)\zeta(\nu-n)}{\zeta(-1-n)}-\frac{\zeta(-\nu)}{1-p^{n-\nu}}-\frac{\zeta(\nu-n)}{1-p^{\nu}}\right).

In fact, we show in §4.3 that

(4.1)

\displaystyle\begin{split}\lim_{m\to\infty}\sum_{k=0}^{m}p^{(n-\nu)(m-k)}&\sum_{j=0}^{p^{k}-1}\langle\mathrm{Eis}_{n},\widetilde{C}_{\nu,k,j}\left(\frac{\tau}{p^{m-k}},p^{m-k}\tau\right)\rangle\\ &=\frac{1}{1-p^{n+1}}\frac{\zeta(-\nu)\zeta(\nu-n)}{\zeta(-1-n)}-\frac{\zeta(-\nu)}{1-p^{n-\nu}}-\frac{\zeta(\nu-n)}{1-p^{\nu}}.\end{split}

Hence Theorem 4.1 follows from (4.1) and the following lemma.

Lemma 4.2.

Suppose that the $p$ -adic limit

\lim_{m\to\infty}\sum_{k=0}^{m}p^{(n-\nu)(m-k)}\sum_{j=0}^{p^{k}-1}\langle\mathrm{Eis}_{n},\widetilde{C}_{\nu,k,j}\left(\frac{\tau}{p^{m-k}},p^{m-k}\tau\right)\rangle

exists. We then have

\displaystyle\lim_{m\to\infty}\langle\mathrm{Eis}_{n},\widetilde{T_{p}^{m!}(C_{\nu}(\tau))}\rangle=(1-p^{n+1})\lim_{m\to\infty}\sum_{k=0}^{m}p^{(n-\nu)(m-k)}\sum_{j=0}^{p^{k}-1}\langle\mathrm{Eis}_{n},\widetilde{C}_{\nu,k,j}\left(\frac{\tau}{p^{m-k}},p^{m-k}\tau\right)\rangle.

Proof.

For notational simplicity, we put

\mathscr{W}^{(m)}:=\sum_{k=0}^{m}p^{(n-\nu)(m-k)}\sum_{j=0}^{p^{k}-1}\langle\mathrm{Eis}_{n},\widetilde{C}_{\nu,k,j}\left(\frac{\tau}{p^{m-k}},p^{m-k}\tau\right)\rangle\,\,\,\textrm{ and }\,\,\,\mathscr{W}:=\lim_{m\to\infty}\mathscr{W}^{(m)}.

We then have

\langle\mathrm{Eis}_{n},\widetilde{T_{p}^{m!}(C_{\nu}(\tau))}\rangle=\sum_{A=0}^{m!/2}C(m!-A,A)p^{(n+1)A}\mathscr{W}^{(m!-2A)}.

Take a positive integer $Q$ . Then there is a positive integer $r$ such that $\mathscr{W}^{(s)}-\mathscr{W}\in p^{Q}\mathbb{Z}_{p}$ for any integer $s\geq r$ . Hence we have

	$\displaystyle\sum_{A=0}^{\lfloor m/2\rfloor}C(m-A,A)p^{(n+1)A}(\mathscr{W}^{(m-2A)}-\mathscr{W})$
	$\displaystyle\equiv\sum_{A=\lfloor(m-r)/2\rfloor+1}^{\lfloor m/2\rfloor}C(m-A,A)p^{(n+1)A}(\mathscr{W}^{(m-2A)}-\mathscr{W})\pmod{p^{Q}\mathbb{Z}_{p}}.$

The sequence $(\mathscr{W}^{(m)}-\mathscr{W})_{m=0}^{\infty}$ is bounded in $\mathbb{Q}_{p}$ , and hence for any sufficiently large integer $m$ , we have

\sum_{A=\lfloor(m-r)/2\rfloor+1}^{\lfloor m/2\rfloor}C(m-A,A)p^{(n+1)A}(\mathscr{W}^{(m-2A)}-\mathscr{W})\in p^{Q}\mathbb{Z}_{p}.

This implies that

\lim_{m\to\infty}\langle\mathrm{Eis}_{n},\widetilde{T_{p}^{m!}(C_{\nu}(\tau))}\rangle=\lim_{m\to\infty}\sum_{A=0}^{m!/2}C(m!-A,A)p^{(n+1)A}\mathscr{W}.

Since $C(m!-A,A)=\binom{m!}{A}-\binom{m!}{A-1}$ (note that $\binom{m}{-1}=0$ ), we have

	$\displaystyle\sum_{A=0}^{m!/2}C(m!-A,A)p^{(n+1)A}$	$\displaystyle=\sum_{A=0}^{m!/2}\binom{m!}{A}p^{(n+1)A}-\sum_{A=0}^{m!/2-1}\binom{m!}{A}p^{(n+1)(A+1)}$
		$\displaystyle=(1-p^{n+1})\sum_{A=0}^{m!/2-1}\binom{m!}{A}p^{(n+1)A}+\binom{m!}{m!/2}p^{(n+1)m!/2}.$

Since

	$\displaystyle\sum_{A=0}^{m!/2-1}\binom{m!}{A}p^{(n+1)A}$	$\displaystyle\equiv\sum_{A=0}^{m!}\binom{m!}{A}p^{(n+1)A}\pmod{p^{m!/2}}$
		$\displaystyle=(1+p^{n+1})^{m!},$

we obtain that

\lim_{m\to\infty}\sum_{A=0}^{m!/2}C(m!-A,A)p^{(n+1)A}=(1-p^{n+1})\lim_{m\to\infty}(1+p^{n+1})^{m!}=1-p^{n+1},

which completes the proof. ∎

Therefore, it remains to prove (4.1), and this will be done in Proposition 4.13.

4.1. $\langle\mathrm{Eis}_{n},\widetilde{C}_{\nu,k,j}\left(\tau_{0},\tau_{1}\right)\rangle$

We start with computing the value $\langle\mathrm{Eis}_{n},\widetilde{C}_{\nu,k,j}\left(\tau_{0},\tau_{1}\right)\rangle$ . In this subsection, we fix integers $k$ and $j$ with $0\leq j\leq p^{k}-1$ . Recall that

l:=l_{k}(j)\,\,\,\textrm{ and }\,\,\,d:=d_{k-l_{k}(j)}(j^{\prime})=d_{k-l_{k}(j)}(j/p^{l_{k}(j)})

are taken as in Definition 3.4 and (3.1) and that

	$\displaystyle\widetilde{C}_{\nu,k,j}\left(\tau_{0},\tau_{1}\right)=\left\{\frac{-\frac{1}{\tau_{0}}+j}{p^{k}},\frac{\tau_{1}+j}{p^{k}}\right\}\otimes(p^{k}X_{1}-jX_{2})^{\nu}X_{2}^{n-\nu}$	$\displaystyle-\left\{\frac{\tau_{1}+j}{p^{k}},\frac{\tau_{1}+j}{p^{k}}+1\right\}\otimes P_{\nu,k,j}^{(1)}$
		$\displaystyle-\left\{\frac{p^{l}\tau_{0}-d}{p^{k-l}},\frac{p^{l}\tau_{0}-d}{p^{k-l}}+1\right\}\otimes P_{\nu,k,j}^{(0)}.$

Hence we have

(4.2)

\displaystyle\begin{split}\langle\mathrm{Eis}_{n},\widetilde{C}_{\nu,k,j}\left(\tau_{0},\tau_{1}\right)\rangle&=\int_{\frac{-\frac{1}{\tau_{0}}+j}{p^{k}}}^{\frac{\tau_{1}+j}{p^{k}}}E_{n+2}(z)(p^{k}z-j)^{\nu}\,dz\\ &\quad-\int_{\frac{\tau_{1}+j}{p^{k}}}^{\frac{\tau_{1}+j}{p^{k}}+1}E_{n+2}(z)P_{\nu,k,j}^{(1)}(z,1)\,dz-\int_{\frac{p^{l}\tau_{0}-d}{p^{k-l}}}^{\frac{p^{l}\tau_{0}-d}{p^{k-l}}+1}E_{n+2}(z)P_{\nu,k,j}^{(0)}(z,1)\,dz\\ &=\frac{1}{p^{k}}\int_{-\frac{1}{\tau_{0}}}^{\tau_{1}}E_{n+2}\left(\frac{z+j}{p^{k}}\right)z^{\nu}\,dz\\ &\quad-\int_{\frac{\tau_{1}+j}{p^{k}}}^{\frac{\tau_{1}+j}{p^{k}}+1}E_{n+2}(z)P_{\nu,k,j}^{(1)}(z,1)\,dz-\int_{\frac{p^{l}\tau_{0}-d}{p^{k-l}}}^{\frac{p^{l}\tau_{0}-d}{p^{k-l}}+1}E_{n+2}(z)P_{\nu,k,j}^{(0)}(z,1)\,dz.\end{split}

The definition of the Eisenstein series $E_{n+2}$ shows that

	$\displaystyle E_{n+2}\left(\frac{z+j}{p^{k}}\right)-1$	$\displaystyle=O(e^{-2\pi\operatorname{Im}(z)})\quad\text{ for }\operatorname{Im}(z)\geq 1,$
	$\displaystyle E_{n+2}\left(\frac{z+j}{p^{k}}\right)-\frac{p^{l(n+2)}}{z^{n+2}}$	$\displaystyle=O\left(\frac{p^{l(n+2)}}{\|z\|^{n+2}}e^{-2\pi p^{2l-k}/\operatorname{Im}(z)}\right)\quad\text{ for }\operatorname{Im}(z)\leq 1,$

where $f(z)=O(g(z))$ means that there is a constant $C$ which does not depend on $k$ and $j$ such that $|f(z)|\leq Cg(z)$ . Set

\displaystyle\mathscr{L}_{k,j}(s):=\int_{0}^{\infty}\left(E_{n+2}\left(\frac{iy+j}{p^{k}}\right)-1\right)y^{s}dy.

We have the following Lemma 4.3 and Proposition 4.4, whose proofs will be given in §4.1.1 and §4.1.3, respectively.

Lemma 4.3.

The function $\mathscr{L}_{k,j}(s)$ converges for $\operatorname{Re}(s)>n+1$ , and continued to a meromorphic function on $\mathbb{C}$ . Moreover, it has at most simple poles at $s=-1$ and $s=n+1$ . In particular, $\mathscr{L}_{k,j}(s)$ is holomorphic at $s=\nu$ for any integer $1\leq\nu\leq n-1$ .

Proposition 4.4.

We have

\langle\mathrm{Eis}_{n},\widetilde{C}_{\nu,k,j}\left(\tau_{0},\tau_{1}\right)\rangle=\frac{i^{\nu+1}}{p^{k}}\mathscr{L}_{k,j}(\nu)-E_{\nu,k,j}^{(1){\ddagger}}\left(\frac{j}{p^{k}},1\right)-E_{\nu,k,j}^{(0){\ddagger}}\left(-\frac{d}{p^{k-l}},1\right).

Note that since $\partial[\widetilde{C}_{\nu,k,j}\left(\tau_{0},\tau_{1}\right)]=0$ by Lemma 3.11, the value $\langle\mathrm{Eis}_{n},\widetilde{C}_{\nu,k,j}\left(\tau_{0},\tau_{1}\right)\rangle$ does not depend on the choices of $\tau_{0}$ and $\tau_{1}$ . Therefore, in the following we take $\tau_{0}=it_{0}$ and $\tau_{1}=it_{1}$ for $t_{0},t_{1}\in\mathbb{R}_{>0}$ .

4.1.1. Computation of the first term of (4.2)

Here we compute the first term of (4.2):

\frac{1}{p^{k}}\int_{\frac{i}{t_{0}}}^{it_{1}}E_{n+2}\left(\frac{z+j}{p^{k}}\right)z^{\nu}\,dz.

This integral is transformed as follows:

	$\displaystyle\frac{1}{p^{k}}\int_{\frac{i}{t_{0}}}^{it_{1}}E_{n+2}\left(\frac{z+j}{p^{k}}\right)z^{\nu}\,dz$
	$\displaystyle=\frac{1}{p^{k}}\int_{\frac{i}{t_{0}}}^{\infty}\left(E_{n+2}\left(\frac{z+j}{p^{k}}\right)-1\right)z^{\nu}\,dz+\frac{1}{p^{k}}\int_{\frac{i}{t_{0}}}^{it_{1}}z^{\nu}\,dz-\frac{1}{p^{k}}\int_{it_{1}}^{\infty}\left(E_{n+2}\left(\frac{z+j}{p^{k}}\right)-1\right)z^{\nu}\,dz$
	$\displaystyle=\frac{i^{\nu+1}}{p^{k}}\left\{\int_{\frac{1}{t_{0}}}^{\infty}\left(E_{n+2}\left(\frac{iy+j}{p^{k}}\right)-1\right)y^{\nu}\,dy+\int_{\frac{1}{t_{0}}}^{t_{1}}y^{\nu}\,dy-\int_{t_{1}}^{\infty}\left(E_{n+2}\left(\frac{iy+j}{p^{k}}\right)-1\right)y^{\nu}\,dy\right\}.$

Set

	$\displaystyle\mathscr{S}_{k,j}(t_{0},t_{1},s)$	$\displaystyle:=\int_{\frac{1}{t_{0}}}^{\infty}\left(E_{n+2}\left(\frac{iy+j}{p^{k}}\right)-1\right)y^{s}\,dy+\int_{\frac{1}{t_{0}}}^{t_{1}}y^{s}\,dy,$
	$\displaystyle\mathscr{R}_{k,j}^{(1)}(t_{1},s)$	$\displaystyle:=\int_{t_{1}}^{\infty}\left(E_{n+2}\left(\frac{iy+j}{p^{k}}\right)-1\right)y^{s}\,dy,$
	$\displaystyle\mathscr{R}_{k,j}^{(0)}(t_{0},s)$	$\displaystyle:=\int_{0}^{\frac{1}{t_{0}}}\left(E_{n+2}\left(\frac{iy+j}{p^{k}}\right)-\frac{p^{l(n+2)}}{(iy)^{n+2}}\right)y^{s}\,dy.$

Then we have

\displaystyle\frac{1}{p^{k}}\int_{\frac{i}{t_{0}}}^{it_{1}}E_{n+2}\left(\frac{z+j}{p^{k}}\right)z^{\nu}\,dz=\frac{i^{\nu+1}}{p^{k}}\left(\mathscr{S}_{k,j}(t_{0},t_{1},\nu)-\mathscr{R}_{k,j}^{(1)}(t_{1},\nu)\right).

Now, we see that

•

the first terms of $\mathscr{S}_{k,j}(t_{0},t_{1},s)$ and $\mathscr{R}^{(i)}_{k,j}(t_{i},s)$ converge for all $s\in\mathbb{C}$ ,
•

the second term of $\mathscr{S}_{k,j}(t_{0},t_{1},s)$ is meromorphic and has at most simple pole at $s=-1$ .

In addition, we also see that

	$\displaystyle\mathscr{S}_{k,j}(t_{0},t_{1},s)$	$\displaystyle=\mathscr{L}_{k,j}(s)-\mathscr{R}_{k,j}^{(0)}(t_{0},s)-\int_{0}^{\frac{1}{t_{0}}}\frac{p^{l(n+2)}}{(iy)^{n+2}}y^{s}\,dy+\int_{0}^{t_{1}}y^{s}\,dy$
		$\displaystyle=\mathscr{L}_{k,j}(s)-\mathscr{R}_{k,j}^{(0)}(t_{0},s)-\frac{p^{l(n+2)}}{i^{n+2}}\frac{1}{s-n-1}\frac{1}{t_{0}^{s-n-1}}+\frac{1}{s+1}t_{1}^{s+1}.$

In particular, all of these functions are meromorphically continued to $s\in\mathbb{C}$ and are holomorphic at $s=\nu$ . This proves Lemma 4.3, and moreover, we get the following.

Lemma 4.5.

	$\displaystyle\frac{1}{p^{k}}\int_{\frac{i}{t_{0}}}^{it_{1}}E_{n+2}\left(\frac{z+j}{p^{k}}\right)z^{\nu}\,dz$
	$\displaystyle=\frac{i^{\nu+1}}{p^{k}}\left\{\mathscr{L}_{k,j}(\nu)+\frac{t_{1}^{\nu+1}}{\nu+1}+\frac{p^{l(n+2)}}{i^{n+2}}\frac{t_{0}^{n-\nu+1}}{n-\nu+1}-\mathscr{R}_{k,j}^{(1)}(t_{1},\nu)-\mathscr{R}_{k,j}^{(0)}(t_{0},\nu)\right\}.$

4.1.2. Computation of the second and the third terms of (4.2)

Here we compute

\int_{\frac{\tau_{1}+j}{p^{k}}}^{\frac{\tau_{1}+j}{p^{k}}+1}E_{n+2}(z)P_{\nu,k,j}^{(1)}(z,1)\,dz+\int_{\frac{p^{l}\tau_{0}-d}{p^{k-l}}}^{\frac{p^{l}\tau_{0}-d}{p^{k-l}}+1}E_{n+2}(z)P_{\nu,k,j}^{(0)}(z,1)\,dz.

We put

\displaystyle\mathscr{T}_{\nu,k,j}(\tau_{0},\tau_{1}):=\int_{\frac{\tau_{1}+j}{p^{k}}}^{\frac{\tau_{1}+j}{p^{k}}+1}\left(E_{n+2}(z)-1\right)P_{\nu,k,j}^{(1)}(z,1)\,dz+\int_{\frac{p^{l}\tau_{0}-d}{p^{k-l}}}^{\frac{p^{l}\tau_{0}-d}{p^{k-l}}+1}\left(E_{n+2}(z)-1\right)P_{\nu,k,j}^{(0)}(z,1)\,dz.

Note that $\mathscr{T}_{\nu,k,j}(\tau_{0},\tau_{1})\to 0$ as $\tau_{0},\tau_{1}\to\infty$ . On the other hand by Lemma 3.7, we have

	$\displaystyle\int_{\frac{\tau_{1}+j}{p^{k}}}^{\frac{\tau_{1}+j}{p^{k}}+1}P_{\nu,k,j}^{(1)}(z,1)\,dz$	$\displaystyle=E_{\nu,k,j}^{(1){\ddagger}}\left(\frac{\tau_{1}+j}{p^{k}},1\right)$
		$\displaystyle=E_{\nu,k,j}^{(1){\ddagger}}\left(\frac{j}{p^{k}},1\right)+\int_{\frac{j}{p^{k}}}^{\frac{\tau_{1}+j}{p^{k}}}E_{\nu,k,j}^{(1)}(z,1)dz$
		$\displaystyle=E_{\nu,k,j}^{(1){\ddagger}}\left(\frac{j}{p^{k}},1\right)+\int_{\frac{j}{p^{k}}}^{\frac{\tau_{1}+j}{p^{k}}}(p^{k}z-j)^{\nu}\,dz$
		$\displaystyle=E_{\nu,k,j}^{(1){\ddagger}}\left(\frac{j}{p^{k}},1\right)+\frac{1}{p^{k}}\frac{(it_{1})^{\nu+1}}{\nu+1}.$

Similarly we have

\displaystyle\int_{\frac{p^{l}\tau_{0}-d}{p^{k-l}}}^{\frac{p^{l}\tau_{0}-d}{p^{k-l}}+1}P_{\nu,k,j}^{(0)}(z,1)\,dz=E_{\nu,k,j}^{(0){\ddagger}}\left(-\frac{d}{p^{k-l}},1\right)+(-1)^{\nu+1}\frac{p^{l(n+2)}}{p^{k}}\frac{(it_{0})^{n-\nu+1}}{n-\nu+1}.

4.1.3. Proof of Proposition 4.4

By combining the computations in subsections 4.1.1 and 4.1.2, we find

	$\displaystyle\langle\mathrm{Eis}_{n},\widetilde{C}_{\nu,k,j}\left(\tau_{0},\tau_{1}\right)\rangle$
	$\displaystyle=\frac{i^{\nu+1}}{p^{k}}\mathscr{L}_{\nu,k,j}(\nu)+\frac{1}{p^{k}}\frac{(it_{1})^{\nu+1}}{\nu+1}+(-1)^{\nu-1}\frac{p^{l(n+2)}}{p^{k}}\frac{(it_{0})^{n-\nu+1}}{n-\nu+1}$
	$\displaystyle\quad-E_{\nu,k,j}^{(1){\ddagger}}\left(\frac{j}{p^{k}},1\right)-\frac{1}{p^{k}}\frac{(it_{1})^{\nu+1}}{\nu+1}-E_{\nu,k,j}^{(0){\ddagger}}\left(-\frac{d}{p^{k-l}},1\right)-(-1)^{\nu+1}\frac{p^{l(n+2)}}{p^{k}}\frac{(it_{0})^{n-\nu+1}}{n-\nu+1}$
	$\displaystyle\quad-\frac{i^{\nu+1}}{p^{k}}\mathscr{R}_{k,j}^{(1)}(t_{1},\nu)-\frac{i^{\nu+1}}{p^{k}}\mathscr{R}_{k,j}^{(0)}(t_{0},\nu)-\mathscr{T}_{k,j}(\tau_{0},\tau_{1}),$

and hence

	$\displaystyle\langle\mathrm{Eis}_{n},\widetilde{C}_{\nu,k,j}\left(\tau_{0},\tau_{1}\right)\rangle$	$\displaystyle=\frac{i^{\nu+1}}{p^{k}}\mathscr{L}_{k,j}(\nu)-E_{\nu,k,j}^{(1){\ddagger}}\left(\frac{j}{p^{k}},1\right)-E_{\nu,k,j}^{(0){\ddagger}}\left(-\frac{d}{p^{k-l}},1\right)$
		$\displaystyle\quad-\frac{i^{\nu+1}}{p^{k}}\mathscr{R}_{k,j}^{(1)}(t_{1},\nu)-\frac{i^{\nu+1}}{p^{k}}\mathscr{R}_{k,j}^{(0)}(t_{0},\nu)-\mathscr{T}_{k,j}(\tau_{0},\tau_{1}).$

Since the value $\langle\widetilde{C}_{\nu,k,j}\left(\tau_{0},\tau_{1}\right),\mathrm{Eis}_{n}\rangle$ does not depend on $t_{0}$ and $t_{1}$ , we can take the limit $t_{0},t_{1}\to\infty$ . Then the last three terms vanish and we obtain the desired identity. ∎

4.2. Summation over $j$

In this subsection, we compute the sum

\displaystyle\sum_{j=0}^{p^{k}-1}\langle\mathrm{Eis}_{n},\widetilde{C}_{\nu,k,j}\left(\tau_{0},\tau_{1}\right)\rangle=\sum_{j=0}^{p^{k}-1}\left\{\frac{i^{\nu+1}}{p^{k}}\mathscr{L}_{k,j}(\nu)-E_{\nu,k,j}^{(1){\ddagger}}\left(\frac{j}{p^{k}},1\right)-E_{\nu,k,j}^{(0){\ddagger}}\left(-\frac{d}{p^{k-l}},1\right)\right\}.

We keep using the abbreviation

l:=l_{k}(j)\,\,\,\textrm{ and }\,\,\,d:=d_{k-l_{k}(j)}(j^{\prime})=d_{k-l_{k}(j)}(j/p^{l_{k}(j)})

that are actually depending on $k$ and $j$ . Recall that $\widetilde{B}_{t}(x)=(B_{t}(x)-B_{t})/t$ .

Lemma 4.6.

(1)

We have

\displaystyle\frac{i^{\nu+1}}{p^{k}}\sum_{j=0}^{p^{k}-1}\mathscr{L}_{k,j}(\nu)=\frac{(1-p^{(n+1)(k+1)})-(1-p^{(n+1)k})p^{n-\nu}}{1-p^{n+1}}\frac{\zeta(-\nu)\zeta(\nu-n)}{\zeta(-1-n)}.

(2)

We have

\displaystyle\sum_{j=0}^{p^{k}-1}E_{\nu,k,j}^{(1){\ddagger}}\left(\frac{j}{p^{k}},1\right)=\frac{(-1)^{\nu}}{(\nu+1)}\frac{1}{p^{k}}\widetilde{B}_{\nu+2}(p^{k})+\sum_{j=0}^{p^{k}-1}E_{\nu,k,j}^{(1),{\ddagger}}\left(0,1\right).

(3)

We have

	$\displaystyle\sum_{j=0}^{p^{k}-1}E_{\nu,k,j}^{(0){\ddagger}}\left(-\frac{d}{p^{k-l}},1\right)$
	$\displaystyle=\frac{(-1)^{\nu}}{n-\nu+1}\frac{1}{p^{k}}\sum_{l^{\prime}=0}^{k-1}p^{l^{\prime}(\nu+1)}\left(\widetilde{B}_{n-\nu+2}(p^{k-l^{\prime}})-p^{n-\nu+1}\widetilde{B}_{n-\nu+2}(p^{k-l^{\prime}-1})\right)+\sum_{j=0}^{p^{k}-1}E_{\nu,k,j}^{(0){\ddagger}}\left(0,1\right).$

Proof.

Recall that

\displaystyle\mathscr{L}_{k,j}(s)=\int_{0}^{\infty}\left(E_{n+2}\left(\frac{iy+j}{p^{k}}\right)-1\right)y^{s}dy.

Hence we have

	$\displaystyle\frac{1}{p^{k}}\sum_{j=0}^{p^{k}-1}\mathscr{L}_{k,j}(s)$	$\displaystyle=\frac{1}{p^{k}}\sum_{j=0}^{p^{k}-1}\frac{2}{\zeta(-1-n)}\sum_{\mu=1}^{\infty}\sigma_{n+1}(\mu)e^{\frac{2\pi i\mu j}{p^{k}}}\int_{0}^{\infty}e^{-\frac{2\pi\mu y}{p^{k}}}y^{s+1}\frac{dy}{y}$
		$\displaystyle=\frac{1}{p^{k}}\frac{2}{\zeta(-1-n)}\sum_{j=0}^{p^{k}-1}\sum_{\mu=1}^{\infty}\sigma_{n+1}(\mu)e^{\frac{2\pi i\mu j}{p^{k}}}\frac{p^{k(s+1)}}{(2\pi\mu)^{s+1}}\Gamma(s+1)$
		$\displaystyle=\frac{2}{\zeta(-1-n)}\frac{\Gamma(s+1)p^{ks}}{(2\pi)^{s+1}}\sum_{\mu=1}^{\infty}\left(\sum_{j=0}^{p^{k}-1}e^{\frac{2\pi i\mu j}{p^{k}}}\right)\frac{\sigma_{n+1}(\mu)}{\mu^{s+1}}$
		$\displaystyle=\frac{2\Gamma(s+1)}{\zeta(-1-n)(2\pi)^{s+1}}\sum_{\mu=1}^{\infty}\frac{\sigma_{n+1}(p^{k}\mu)}{\mu^{s+1}}.$

For notational simplicity, we put

\mathcal{E}_{p}(s):=1-p^{-s}.

We then have

	$\displaystyle\sum_{a=0}^{\infty}\frac{\sigma_{n+1}(p^{k+a})}{p^{a(s+1)}}$	$\displaystyle=\sum_{a=0}^{\infty}\frac{1}{p^{a(s+1)}}\frac{1-p^{(k+a+1)(n+1)}}{1-p^{n+1}}$
		$\displaystyle=\frac{\mathcal{E}_{p}(1+s)^{-1}-p^{(k+1)(n+1)}\mathcal{E}_{p}(s-n)^{-1}}{1-p^{n+1}}$
		$\displaystyle=\frac{\mathcal{E}_{p}(s-n)-p^{(k+1)(n+1)}\mathcal{E}_{p}(1+s)}{1-p^{n+1}}\mathcal{E}_{p}(1+s)^{-1}\mathcal{E}_{p}(s-n)^{-1}.$

Hence the well-known relation that $\zeta(1+s)\zeta(s-n)=\sum_{a=1}^{\infty}\sigma_{n+1}(a)a^{-(s+1)}$ implies that

\displaystyle\frac{1}{p^{k}}\sum_{j=0}^{p^{k}-1}\mathscr{L}_{k,j}(s)

\displaystyle=\frac{2\Gamma(s+1)}{\zeta(-1-n)(2\pi)^{s+1}}\frac{\mathcal{E}_{p}(s-n)-p^{(k+1)(n+1)}\mathcal{E}_{p}(1+s)}{1-p^{n+1}}\zeta(s+1)\zeta(s-n).

By setting $s=\nu$ and using the functional equation of the Riemann zeta function (see [Hid93, p.29] for example), we find

	$\displaystyle\frac{i^{\nu+1}}{p^{k}}\sum_{j=0}^{p^{k}-1}\mathscr{L}_{k,j}(\nu)$	$\displaystyle=\frac{(\mathcal{E}_{p}(\nu-n)-p^{(k+1)(n+1)}\mathcal{E}_{p}(\nu+1))}{1-p^{n+1}}\frac{\zeta(-\nu)\zeta(\nu-n)}{\zeta(-1-n)}$
		$\displaystyle=\frac{(1-p^{(n+1)(k+1)})-(1-p^{(n+1)k})p^{n-\nu}}{1-p^{n+1}}\frac{\zeta(-\nu)\zeta(\nu-n)}{\zeta(-1-n)}.$

By using Lemma 3.7, we find

	$\displaystyle\sum_{j=0}^{p^{k}-1}E_{\nu,k,j}^{(1),{\ddagger}}\left(\frac{j}{p^{k}},1\right)$	$\displaystyle=\sum_{j=0}^{p^{k}-1}\left(E_{\nu,k,j}^{(1),{\ddagger}}\left(\frac{j}{p^{k}},1\right)-E_{\nu,k,j}^{(1),{\ddagger}}\left(0,1\right)\right)+\sum_{j=0}^{p^{k}-1}E_{\nu,k,j}^{(1),{\ddagger}}\left(0,1\right)$
		$\displaystyle=\sum_{j=0}^{p^{k}-1}\int_{0}^{\frac{j}{p^{k}}}E_{\nu,k,j}^{(1)}(z,1)dz+\sum_{j=0}^{p^{k}-1}E_{\nu,k,j}^{(1),{\ddagger}}\left(0,1\right)$
		$\displaystyle=\sum_{j=0}^{p^{k}-1}\int_{0}^{\frac{j}{p^{k}}}(p^{k}z-j)^{\nu}dz+\sum_{j=0}^{p^{k}-1}E_{\nu,k,j}^{(1),{\ddagger}}\left(0,1\right)$
		$\displaystyle=\frac{(-1)^{\nu}}{(\nu+1)}\frac{1}{p^{k}}\sum_{j=0}^{p^{k}-1}j^{\nu+1}+\sum_{j=0}^{p^{k}-1}E_{\nu,k,j}^{(1),{\ddagger}}\left(0,1\right).$

Therefore, claim (2) follows from the fact that $\sum_{j=0}^{p^{k}-1}j^{\nu+1}=\widetilde{B}_{\nu+2}(p^{k})$ .

Lemma 3.7 shows that

	$\displaystyle E_{\nu,k,j}^{(0){\ddagger}}\left(-\frac{d}{p^{k-l}},1\right)$	$\displaystyle=-(-1)^{\nu}p^{l\nu}\int_{0}^{-\frac{d}{p^{k-l}}}(p^{k-l}z+d)^{n-\nu}dz+E_{\nu,k,j}^{(0){\ddagger}}\left(0,1\right)$
		$\displaystyle=\frac{(-1)^{\nu}}{n-\nu+1}\frac{1}{p^{k}}p^{l(\nu+1)}d^{n-\nu+1}+E_{\nu,k,j}^{(0){\ddagger}}\left(0,1\right).$

Since $d_{0}(0)=0$ , $n-\nu+1\geq 2$ , and the map $a\mapsto d_{N}(a)$ induces a permutation on the set $\{1\leq a<p^{N}\mid(a,p^{N})=1\}$ , we find

	$\displaystyle\frac{1}{p^{k}}\sum_{j=0}^{p^{k}-1}p^{l(\nu+1)}d^{n-\nu+1}$	$\displaystyle=\frac{1}{p^{k}}\sum_{l^{\prime}=0}^{k-1}p^{l^{\prime}(\nu+1)}\sum_{\begin{subarray}{c}d^{\prime}=1\\ (d^{\prime},p^{k-l^{\prime}})=1\end{subarray}}^{p^{k-l^{\prime}}-1}d^{\prime n-\nu+1}$
		$\displaystyle=\frac{1}{p^{k}}\sum_{l^{\prime}=0}^{k-1}p^{l^{\prime}(\nu+1)}\left(\widetilde{B}_{n-\nu+2}(p^{k-l^{\prime}})-p^{n-\nu+1}\widetilde{B}_{n-\nu+2}(p^{k-l^{\prime}-1})\right).$

∎

Lemma 4.7.

(1)

We have

\displaystyle\sum_{j=0}^{p^{k}-1}E_{\nu,k,j}^{(1){\ddagger}}(0,1)=(-1)^{\nu+1}\sum_{\mu=0}^{\nu}\binom{\nu}{\mu}(-1)^{\mu}p^{k\mu}\frac{B_{\mu+1}}{\mu+1}\widetilde{B}_{\nu-\mu+1}(p^{k}).

(2)

We have

\sum_{j=0}^{p^{k}-1}E_{\nu,k,j}^{(0){\ddagger}}\left(0,1\right)=(-1)^{\nu}p^{k\nu}\frac{B_{n-\nu+1}}{n-\nu+1}+(-1)^{\nu}\sum_{\mu=0}^{n-\nu}\begin{pmatrix}n-\nu\\ \mu\end{pmatrix}p^{k\mu}\frac{B_{\mu+1}}{\mu+1}\sum_{l^{\prime}=0}^{k-1}p^{l^{\prime}(\nu-\mu)}\\ \times\left(\widetilde{B}_{n-\nu-\mu+1}(p^{k-l^{\prime}})-p^{n-\nu-\mu}\widetilde{B}_{n-\nu-\mu+1}(p^{k-l^{\prime}-1})\right).

Proof.

Note that $(X_{1}^{\mu}X_{2}^{n-\mu})^{{\ddagger}}(0,1)=-B_{\mu+1}/(\mu+1)$ . Since $E_{\nu,k,j}^{(1)}(X_{1},X_{2})=(p^{k}X_{1}-jX_{2})^{\nu}X_{2}^{n-\nu}$ , we have

E_{\nu,k,j}^{(1){\ddagger}}\left(0,1\right)=\sum_{\mu=0}^{\nu}\binom{\nu}{\mu}p^{k\mu}\left(-\frac{B_{\mu+1}}{\mu+1}\right)(-j)^{\nu-\mu}.

Hence we have

\displaystyle\sum_{j=0}^{p^{k}-1}E_{\nu,k,j}^{(1){\ddagger}}(0,1)=(-1)^{\nu+1}\sum_{\mu=0}^{\nu}\binom{\nu}{\mu}(-1)^{\mu}p^{k\mu}\frac{B_{\mu+1}}{\mu+1}\widetilde{B}_{\nu-\mu+1}(p^{k}).

Since $E_{\nu,k,j}^{(0)}(X_{1},X_{2})=(-1)^{\nu+1}(p^{l}X_{2})^{\nu}(p^{k-l}X_{1}+dX_{2})^{n-\nu}$ , we have

E_{\nu,k,j}^{(0){\ddagger}}(0,1)=(-1)^{\nu+1}p^{l\nu}\sum_{\mu=0}^{n-\nu}\binom{n-\nu}{\mu}p^{\mu(k-l)}\left(-\frac{B_{\mu+1}}{\mu+1}\right)d^{n-\nu-\mu}.

First note that in the case where $j=0$ , since $d=d_{0}(0)=0$ , we find

E_{\nu,k,0}^{(0){\ddagger}}(0,1)=(-1)^{\nu}p^{k\nu}\frac{B_{n-\nu+1}}{n-\nu+1}.

Then by using the same argument as in the proof of Lemma 4.6(3), we also obtain

	$\displaystyle\sum_{j=1}^{p^{k}-1}E_{\nu,k,j}^{(0){\ddagger}}(0,1)$	$\displaystyle=(-1)^{\nu}\sum_{l^{\prime}=0}^{k-1}\sum_{\begin{subarray}{c}d^{\prime}=1\\ (d^{\prime},p^{k-l^{\prime}})=1\end{subarray}}^{p^{k-l^{\prime}}-1}p^{l^{\prime}\nu}\sum_{\mu=0}^{n-\nu}\binom{n-\nu}{\mu}p^{\mu(k-l^{\prime})}\frac{B_{\mu+1}}{\mu+1}d^{\prime n-\nu-\mu}$
		$\displaystyle=(-1)^{\nu}\sum_{l^{\prime}=0}^{k-1}p^{l^{\prime}\nu}\sum_{\mu=0}^{n-\nu}\binom{n-\nu}{\mu}p^{\mu(k-l^{\prime})}\frac{B_{\mu+1}}{\mu+1}$
		$\displaystyle\quad\quad\quad\times\left(\widetilde{B}_{n-\nu-\mu+1}(p^{k-l^{\prime}})-p^{n-\nu-\mu}\widetilde{B}_{n-\nu-\mu+1}(p^{k-l^{\prime}-1})\right).$

This completes the proof. ∎

4.3. Summation over $k$ and the $p$ -adic limits

In this subsection, we compute the value

\mathscr{W}^{(m)}:=\sum_{k=0}^{m}p^{(n-\nu)(m-k)}\sum_{j=0}^{p^{k}-1}\langle\mathrm{Eis}_{n},\widetilde{C}_{\nu,k,j}\left(\frac{\tau}{p^{m-k}},p^{m-k}\tau\right)\rangle

and its $p$ -adic limit as $m\to\infty$ . This enables us to complete the proof of Theorem 4.1.

We keep the notation in the previous sections. Proposition 4.4 shows that

\langle\mathrm{Eis}_{n},\widetilde{C}_{\nu,k,j}\left(\tau_{0},\tau_{1}\right)\rangle=\frac{i^{\nu+1}}{p^{k}}\mathscr{L}_{k,j}(\nu)-E_{\nu,k,j}^{(1){\ddagger}}\left(\frac{j}{p^{k}},1\right)-E_{\nu,k,j}^{(0){\ddagger}}\left(-\frac{d}{p^{k-l}},1\right),

and hence

\mathscr{W}^{(m)}=\sum_{k=0}^{m}p^{(n-\nu)(m-k)}\sum_{j=0}^{p^{k}-1}\left(\frac{i^{\nu+1}}{p^{k}}\mathscr{L}_{k,j}(\nu)-E_{\nu,k,j}^{(1){\ddagger}}\left(\frac{j}{p^{k}},1\right)-E_{\nu,k,j}^{(0){\ddagger}}\left(-\frac{d}{p^{k-l}},1\right)\right).

We set

	$\displaystyle\mathscr{W}_{1}^{(m)}$	$\displaystyle:=\sum_{k=0}^{m}p^{(n-\nu)(m-k)}\sum_{j=0}^{p^{k}-1}\frac{i^{\nu+1}}{p^{k}}\mathscr{L}_{k,j}(\nu),$
	$\displaystyle\mathscr{W}_{2}^{(m)}$	$\displaystyle:=\sum_{k=0}^{m}p^{(n-\nu)(m-k)}\sum_{j=0}^{p^{k}-1}E_{\nu,k,j}^{(1){\ddagger}}\left(\frac{j}{p^{k}},1\right),$
	$\displaystyle\mathscr{W}_{3}^{(m)}$	$\displaystyle:=\sum_{k=0}^{m}p^{(n-\nu)(m-k)}\sum_{j=0}^{p^{k}-1}E_{\nu,k,j}^{(0){\ddagger}}\left(-\frac{d}{p^{k-l}},1\right),$

so that we have $\mathscr{W}^{(m)}=\mathscr{W}^{(m)}_{1}-\mathscr{W}^{(m)}_{2}-\mathscr{W}^{(m)}_{3}$ .

Lemma 4.8.

We have

\lim_{m\to\infty}\mathscr{W}_{1}^{(m)}=\frac{1}{1-p^{n+1}}\frac{\zeta(-\nu)\zeta(\nu-n)}{\zeta(-1-n)}.

Proof.

Lemma 4.6(1) shows that

\displaystyle\mathscr{W}_{1}^{(m)}

\displaystyle=\frac{1}{1-p^{n+1}}\frac{\zeta(-\nu)\zeta(\nu-n)}{\zeta(-1-n)}\sum_{k=0}^{m}p^{(n-\nu)(m-k)}((1-p^{(n+1)(k+1)})-(1-p^{(n+1)k})p^{n-\nu}).

Moreover, we have

	$\displaystyle\sum_{k=0}^{m}p^{(n-\nu)(m-k)}$	$\displaystyle((1-p^{(n+1)(k+1)})-(1-p^{(n+1)k})p^{n-\nu})$
		$\displaystyle=p^{(n-\nu)m}(1-p^{n-\nu})\sum_{k=0}^{m}p^{-(n-\nu)k}-p^{(n-\nu)m}(p^{n+1}-p^{n-\nu})\sum_{k=0}^{m}p^{(\nu+1)k}$
		$\displaystyle=p^{(n-\nu)m}(1-p^{n-\nu})\frac{1-p^{-(n-\nu)(m+1)}}{1-p^{-(n-\nu)}}-p^{(n-\nu)m}(p^{n+1}-p^{n-\nu})\frac{1-p^{(\nu+1)(m+1)}}{1-p^{\nu+1}}$
		$\displaystyle=1-p^{(n-\nu)(m+1)}-p^{(n-\nu)m}(p^{n+1}-p^{n-\nu})\frac{1-p^{(\nu+1)(m+1)}}{1-p^{\nu+1}}$
		$\displaystyle\to 1\quad\quad(m\to\infty).$

∎

Lemma 4.9.

Let $s$ and $t$ be integers with $t>0$ .

(1)

For any positive integer $u>0$ , we have

$\lim_{m\to\infty}\sum_{k=0}^{m}\frac{p^{u(m-k)}}{p^{k-s}}\widetilde{B}_{t}(p^{k-s})=\frac{B_{t-1}}{1-p^{u}}.$
(2)

For any integer $u$ and any $\varepsilon>0$ with $u+\varepsilon>0$ , we have

$\lim_{m\to\infty}p^{m\varepsilon}\sum_{k=0}^{m}\frac{p^{u(m-k)}}{p^{k-s}}\widetilde{B}_{t}(p^{k-s})=0.$

Proof.

We have

	$\displaystyle\sum_{k=0}^{m}p^{u(m-k)}\frac{1}{p^{k-s}}\widetilde{B}_{t}(p^{k-s})$	$\displaystyle=\frac{1}{t}\sum_{k=0}^{m}p^{u(m-k)}\sum_{\mu=1}^{t}\binom{t}{\mu}B_{t-\mu}p^{(\mu-1)(k-s)}$
		$\displaystyle=\frac{1}{t}\sum_{\mu=1}^{t}\binom{t}{\mu}B_{t-\mu}\sum_{k=0}^{m}p^{u(m-k)+(\mu-1)(k-s)}$
		$\displaystyle=\frac{1}{t}\sum_{\mu=1}^{t}\binom{t}{\mu}B_{t-\mu}p^{um-(\mu-1)s}\frac{1-p^{(\mu-1-u)(m+1)}}{1-p^{\mu-1-u}}$
		$\displaystyle=\frac{1}{t}\sum_{\mu=1}^{t}\binom{t}{\mu}B_{t-\mu}\frac{p^{um-(\mu-1)s}-p^{-(\mu-1)s+(m+1)(\mu-1)-u}}{1-p^{\mu-1-u}}.$

Hence claim (2) is clear. Moreover if $u>0$ , then all the terms with $\mu\geq 2$ vanish as $m\to\infty$ , and therefore,

\lim_{m\to\infty}\sum_{k=0}^{m}\frac{p^{u(m-k)}}{p^{k-s}}\widetilde{B}_{t}(p^{k-s})=B_{t-1}\frac{-p^{-u}}{1-p^{-u}}=\frac{B_{t-1}}{1-p^{u}}.

∎

Lemma 4.10.

We have

\lim_{m\to\infty}\mathscr{W}_{2}^{(m)}=\frac{(-1)^{\nu}}{1-p^{n-\nu}}\frac{B_{\nu+1}}{\nu+1}=\frac{1}{1-p^{n-\nu}}\zeta(-\nu).

Proof.

Lemma 4.6(2) shows that

	$\displaystyle\mathscr{W}_{2}^{(m)}$	$\displaystyle=\sum_{k=0}^{m}p^{(n-\nu)(m-k)}\sum_{j=0}^{p^{k}-1}E_{\nu,k,j}^{(1){\ddagger}}\left(\frac{j}{p^{k}},1\right)$
		$\displaystyle=\sum_{k=0}^{m}p^{(n-\nu)(m-k)}\left(\frac{(-1)^{\nu}}{(\nu+1)}\frac{1}{p^{k}}\widetilde{B}_{\nu+2}(p^{k})+\sum_{j=0}^{p^{k}-1}E_{\nu,k,j}^{(1),{\ddagger}}\left(0,1\right)\right).$

Hence Lemma 4.9(1) implies that

\lim_{m\to\infty}\mathscr{W}_{2}^{(m)}=\frac{(-1)^{\nu}}{1-p^{n-\nu}}\frac{B_{\nu+1}}{\nu+1}+\lim_{m\to\infty}\sum_{k=0}^{m}p^{(n-\nu)(m-k)}\sum_{j=0}^{p^{k}-1}E_{\nu,k,j}^{(1),{\ddagger}}(0,1).

Moreover, by Lemma 4.7, we have

	$\displaystyle\sum_{k=0}^{m}p^{(n-\nu)(m-k)}\sum_{j=0}^{p^{k}-1}E_{\nu,k,j}^{(1),{\ddagger}}(0,1)$	$\displaystyle=\sum_{k=0}^{m}p^{(n-\nu)(m-k)}\sum_{\mu=0}^{\nu}(-1)^{\nu+\mu+1}\binom{\nu}{\mu}p^{k(\mu+1)}\frac{B_{\mu+1}}{\mu+1}\frac{\widetilde{B}_{\nu-\mu+1}(p^{k})}{p^{k}}$
		$\displaystyle=\sum_{\mu=0}^{\nu}(-1)^{\nu+\mu+1}\binom{\nu}{\mu}\frac{B_{\mu+1}}{\mu+1}p^{m(\mu+1)}\sum_{k=0}^{m}p^{(n-\nu-\mu-1)(m-k)}\frac{\widetilde{B}_{\nu-\mu+1}(p^{k})}{p^{k}}.$

Since $\mu+1+n-\nu-\mu-1=n-\nu>0$ , Lemma 4.9(2) shows that

\lim_{m\to\infty}p^{m(\mu+1)}\sum_{k=0}^{m}p^{(n-\nu-\mu-1)(m-k)}\frac{\widetilde{B}_{\nu-\mu+1}(p^{k})}{p^{k}}=0.

Hence we conclude that

\lim_{m\to\infty}\mathscr{W}_{2}^{(m)}=\frac{(-1)^{\nu}}{1-p^{n-\nu}}\frac{B_{\nu+1}}{\nu+1}.

∎

Lemma 4.11.

Let $b$ , $s$ , $t$ , and $u$ be integers with $s\geq 0$ and $t\geq 0$ .

(1)

If $s>0$ and $u>0$ , then

\displaystyle\lim_{m\to\infty}\sum_{k=0}^{m}p^{u(m-k)}\sum_{l^{\prime}=0}^{k-1}p^{l^{\prime}s}\left(\frac{\widetilde{B}_{t+1}(p^{k-l^{\prime}})}{p^{k-l^{\prime}}}-p^{b}\frac{\widetilde{B}_{t+1}(p^{k-l^{\prime}-1})}{p^{k-l^{\prime}-1}}\right)=\frac{B_{t}}{1-p^{s}}\frac{1-p^{b}}{1-p^{u}}.

(2)

For any $\varepsilon>0$ with $u+\varepsilon>0$ , we have

\displaystyle\lim_{m\to\infty}p^{\varepsilon m}\sum_{k=0}^{m}p^{u(m-k)}\sum_{l^{\prime}=0}^{k-1}p^{l^{\prime}s}\left(\frac{\widetilde{B}_{t+1}(p^{k-l^{\prime}})}{p^{k-l^{\prime}}}-p^{b}\frac{\widetilde{B}_{t+1}(p^{k-l^{\prime}-1})}{p^{k-l^{\prime}-1}}\right)=0.

Proof.

We have

	$\displaystyle\sum_{k=0}^{m}p^{u(m-k)}\sum_{l^{\prime}=0}^{k-1}p^{l^{\prime}s}\left(\frac{\widetilde{B}_{t+1}(p^{k-l^{\prime}})}{p^{k-l^{\prime}}}-p^{b}\frac{\widetilde{B}_{t+1}(p^{k-l^{\prime}-1})}{p^{k-l^{\prime}-1}}\right)$
	$\displaystyle=\frac{1}{t+1}\sum_{k=0}^{m}p^{u(m-k)}\sum_{l^{\prime}=0}^{k-1}p^{l^{\prime}s}\sum_{\mu=1}^{t+1}\binom{t+1}{\mu}B_{t+1-\mu}\times(p^{(k-l^{\prime})(\mu-1)}-p^{(k-l^{\prime}-1)(\mu-1)+b})$
	$\displaystyle=\frac{1}{t+1}\sum_{\mu=1}^{t+1}\binom{t+1}{\mu}B_{t+1-\mu}\times(1-p^{b-(\mu-1)})p^{um}\sum_{k=0}^{m}p^{k(\mu-1-u)}\sum_{l^{\prime}=0}^{k-1}p^{l^{\prime}(s-\mu+1)}$
	$\displaystyle=\frac{1}{t+1}\sum_{\mu=1}^{t+1}\binom{t+1}{\mu}B_{t+1-\mu}\times(1-p^{b-(\mu-1)})p^{um}\sum_{k=0}^{m}p^{k(\mu-1-u)}\frac{1-p^{k(s-\mu+1)}}{1-p^{s-\mu+1}}$
	$\displaystyle=\frac{1}{t+1}\sum_{\mu=1}^{t+1}\binom{t+1}{\mu}\frac{B_{t+1-\mu}}{1-p^{s-\mu+1}}\times(1-p^{b-(\mu-1)})\left(\frac{p^{um}-p^{(m+1)(\mu-1)-u}}{1-p^{\mu-1-u}}-\frac{p^{um}-p^{(m+1)s-u}}{1-p^{s-u}}\right),$

which implies this lemma in the same way as in the proof of Lemma 4.9. ∎

Lemma 4.12.

We have

\lim_{m\to\infty}\mathscr{W}_{3}^{(m)}=\frac{(-1)^{\nu}}{1-p^{\nu}}\frac{B_{n-\nu+1}}{n-\nu+1}=\frac{1}{1-p^{\nu}}\zeta(\nu-n).

Proof.

Recall that in Lemma 4.6(3) we have shown that

\sum_{j=0}^{p^{k}-1}E_{\nu,k,j}^{(0){\ddagger}}\left(-\frac{d}{p^{k-l}},1\right)\\ =\frac{(-1)^{\nu}}{n-\nu+1}\frac{1}{p^{k}}\sum_{l^{\prime}=0}^{k-1}p^{l^{\prime}(\nu+1)}\left(\widetilde{B}_{n-\nu+2}(p^{k-l^{\prime}})-p^{n-\nu+1}\widetilde{B}_{n-\nu+2}(p^{k-l^{\prime}-1})\right)+\sum_{j=0}^{p^{k}-1}E_{\nu,k,j}^{(0){\ddagger}}(0,1).

Since $1\leq\nu\leq n-1$ , Lemma 4.11(1) shows that

	$\displaystyle\sum_{k=0}^{m}p^{(n-\nu)(m-k)}\sum_{l^{\prime}=0}^{k-1}p^{l^{\prime}\nu}\left(\frac{\widetilde{B}_{n-\nu+2}(p^{k-l^{\prime}})}{p^{k-l^{\prime}}}-p^{n-\nu}\frac{\widetilde{B}_{n-\nu+2}(p^{k-l^{\prime}-1})}{p^{k-l^{\prime}-1}}\right)$
	$\displaystyle\to\frac{B_{n-\nu+1}}{1-p^{\nu}}\quad\quad(m\to\infty).$

By Lemma 4.7, we have

	$\displaystyle\sum_{j=0}^{p^{k}-1}E_{\nu,k,j}^{(0){\ddagger}}\left(0,1\right)=(-1)^{\nu}p^{k\nu}\frac{B_{n-\nu+1}}{n-\nu+1}$	$\displaystyle+(-1)^{\nu}\sum_{\mu=0}^{n-\nu}\begin{pmatrix}n-\nu\\ \mu\end{pmatrix}p^{k\mu}\frac{B_{\mu+1}}{\mu+1}\sum_{l^{\prime}=0}^{k-1}p^{l^{\prime}(\nu-\mu)}$
		$\displaystyle\quad\times\left(\widetilde{B}_{n-\nu-\mu+1}(p^{k-l^{\prime}})-p^{n-\nu-\mu}\widetilde{B}_{n-\nu-\mu+1}(p^{k-l^{\prime}-1})\right).$

Since $(n-\nu)(m-k)+k\mu=(n-\nu-\mu)(m-k)+m\mu$ and $0\leq\mu\leq n-\nu$ , all the terms with $\mu\geq 1$ vanish when $m\to\infty$ , and hence

	$\displaystyle\lim_{m\to\infty}\sum_{k=0}^{m}p^{(n-\nu)(m-k)}\sum_{j=0}^{p^{k}-1}E_{\nu,k,j}^{(0){\ddagger}}(0,1)$
	$\displaystyle=\lim_{m\to\infty}\sum_{k=0}^{m}(-1)^{\nu}p^{(n-\nu)(m-k)+k\nu}\frac{B_{n-\nu+1}}{n-\nu+1}$
	$\displaystyle+(-1)^{\nu}B_{1}\lim_{m\to\infty}\sum_{k=0}^{m}p^{(n-\nu)(m-k)}\sum_{l^{\prime}=0}^{k-1}p^{l^{\prime}\nu}\left(\widetilde{B}_{n-\nu+1}(p^{k-l^{\prime}})-p^{n-\nu}\widetilde{B}_{n-\nu+1}(p^{k-l^{\prime}-1})\right).$

Since

\displaystyle\lim_{m\to\infty}\sum_{k=0}^{m}p^{(n-\nu)(m-k)+k\nu}=\lim_{m\to\infty}\frac{p^{(n-\nu)(m+1)}-p^{\nu(m+1)}}{p^{n-\nu}-p^{\nu}}=0,

the first limit vanishes. Moreover, Lemma 4.11(2) implies that

	$\displaystyle\sum_{k=0}^{m}p^{(n-\nu)(m-k)}\sum_{l^{\prime}=0}^{k-1}p^{l^{\prime}\nu}\left(\widetilde{B}_{n-\nu+1}(p^{k-l^{\prime}})-p^{n-\nu}\widetilde{B}_{n-\nu+1}(p^{k-l^{\prime}-1})\right)$
	$\displaystyle=p^{m}\sum_{k=0}^{m}p^{(n-\nu-1)(m-k)}\sum_{l^{\prime}=0}^{k-1}p^{l^{\prime}(\nu-1)}\left(\frac{\widetilde{B}_{n-\nu+1}(p^{k-l^{\prime}})}{p^{k-l^{\prime}}}-p^{n-\nu-1}\frac{\widetilde{B}_{n-\nu+1}(p^{k-l^{\prime}-1})}{p^{k-l^{\prime}-1}}\right)$
	$\displaystyle\to 0\quad\quad(m\to\infty),$

which implies this lemma. ∎

By Lemmas 4.8, 4.10, and 4.12, we obtain the following.

Proposition 4.13.

We have

	$\displaystyle\lim_{m\to\infty}\sum_{k=0}^{m}p^{(n-\nu)(m-k)}$	$\displaystyle\sum_{j=0}^{p^{k}-1}\langle\mathrm{Eis}_{n},\widetilde{C}_{\nu,k,j}\left(\frac{\tau}{p^{m-k}},p^{m-k}\tau\right)\rangle$
		$\displaystyle=\frac{1}{1-p^{n+1}}\frac{\zeta(-\nu)\zeta(\nu-n)}{\zeta(-1-n)}-\frac{\zeta(-\nu)}{1-p^{n-\nu}}-\frac{\zeta(\nu-n)}{1-p^{\nu}}.$

By the argument in the beginning of §4, this completes the proof of Theorem 4.1.

4.4. Rationality of the Eisenstein class $\mathrm{Eis}_{n}$

The computations (in the proofs of) Proposition 4.4 and Lemmas 4.6 and 4.7 imply the following proposition as a special case.

Proposition 4.14.

For any integer $\nu\in\{1,\ldots,n-1\}$ , we have

\displaystyle\langle\mathrm{Eis}_{n},\widetilde{C_{\nu}(\tau)}\rangle=\frac{\zeta(-\nu)\zeta(\nu-n)}{\zeta(-1-n)}-\zeta(-\nu)-\zeta(\nu-n)\in\mathbb{Q}.

Proof.

By Proposition 4.4, we have

\langle\mathrm{Eis}_{n},\widetilde{C_{\nu}(\tau)}\rangle=\langle\mathrm{Eis}_{n},\widetilde{C}_{\nu,0,0}\rangle=i^{\nu+1}\mathscr{L}_{0,0}(\nu)-E_{\nu,0,0}^{(1){\ddagger}}\left(0,1\right)-E_{\nu,0,0}^{(0){\ddagger}}\left(0,1\right).

By Lemma 4.6(1), we have

i^{\nu+1}\mathscr{L}_{0,0}(\nu)=\frac{\zeta(-\nu)\zeta(\nu-n)}{\zeta(-1-n)}.

Moreover, in the proof of Lemma 4.7, we showed that

E_{\nu,0,0}^{(1){\ddagger}}\left(0,1\right)=\zeta(-\nu)\,\,\,\textrm{ and }\,\,\,E_{\nu,0,0}^{(0){\ddagger}}\left(0,1\right)=\zeta(\nu-n).

∎

The following lemma will be well-known to experts. For instance, Harder mentioned in [Har, §5.1.3] that this is proved by Gebertz in her diploma thesis. Here we give a proof for the completeness of the paper.

Lemma 4.15.

The relative homology group $H_{1}(Y^{\mathrm{BS}},\partial Y^{\mathrm{BS}},\mathcal{M}_{n})$ is generated by the set $\{[C_{\nu}]\mid 0\leq\nu\leq n\}$ .

Proof.

Note that the relative homology group $H_{1}(Y^{\mathrm{BS}},\partial Y^{\mathrm{BS}},\mathcal{M}_{n})$ can be computed as

	$\displaystyle H_{1}(Y^{\mathrm{BS}},\partial Y^{\mathrm{BS}},\mathcal{M}_{n})$
	$\displaystyle=\ker\left(((\mathcal{MS}(\mathbb{H}^{\mathrm{BS}})/\mathcal{MS}(\partial\mathbb{H}^{\mathrm{BS}}))\otimes\mathcal{M}_{n})_{\Gamma}\overset{\partial}{\rightarrow}((S_{0}(\mathbb{H}^{\mathrm{BS}})/S_{0}(\partial\mathbb{H}^{\mathrm{BS}}))\otimes\mathcal{M}_{n})_{\Gamma}\right).$

Let $[\sigma]\in H_{1}(Y^{\mathrm{BS}},\partial Y^{\mathrm{BS}},\mathcal{M}_{n})$ be a class represented by a $1$ -chain

\sigma=\sum_{j}\{a_{j},b_{j}\}\otimes P_{j}\in(\mathcal{MS}(\mathbb{H}^{\mathrm{BS}})/\mathcal{MS}(\partial\mathbb{H}^{\mathrm{BS}}))\otimes\mathcal{M}_{n},

where $a_{j},b_{j}\in\mathbb{H}^{\mathrm{BS}}$ and $P_{j}\in\mathcal{M}_{n}$ . The condition that

\partial\sigma=0\,\,\,\text{ in }\,\,\,((S_{0}(\mathbb{H}^{\mathrm{BS}})/S_{0}(\partial\mathbb{H}^{\mathrm{BS}}))\otimes\mathcal{M}_{n})_{\Gamma}

implies that

(4.3)

\displaystyle\sum_{j}\{b_{j}\}\otimes P_{j}-\{a_{j}\}\otimes P_{j}=\sum_{k}(\gamma_{k}-1)(\{d_{k}\}\otimes Q_{k})+\sum_{l}\{c_{l}\}\otimes R_{l}

in $S_{0}(\mathbb{H}^{\mathrm{BS}})\otimes\mathcal{M}_{n}$ for some $\gamma_{k}\in\Gamma,d_{k}\in\mathbb{H}^{\mathrm{BS}},c_{l}\in\partial\mathbb{H}^{\mathrm{BS}}$ , and $Q_{k},R_{l}\in\mathcal{M}_{n}$ . Then we can rewrite the identity (4.3) as

\displaystyle\sum_{\tau\in\mathbb{H}^{\mathrm{BS}}}\{\tau\}\otimes\left(\sum_{\begin{subarray}{c}j\\ b_{j}=\tau\end{subarray}}P_{j}-\sum_{\begin{subarray}{c}j\\ a_{j}=\tau\end{subarray}}P_{j}-\sum_{\begin{subarray}{c}k\\ \gamma_{k}d_{k}=\tau\end{subarray}}\gamma_{k}Q_{k}+\sum_{\begin{subarray}{c}k\\ d_{k}=\tau\end{subarray}}Q_{k}-\sum_{\begin{subarray}{c}l\\ c_{l}=\tau\end{subarray}}R_{l}\right)=0

in $S_{0}(\mathbb{H}^{\mathrm{BS}})\otimes\mathcal{M}_{n}$ . Since $S_{0}(\mathbb{H}^{\mathrm{BS}})=\bigoplus_{\tau\in\mathbb{H}^{\mathrm{BS}}}\mathbb{Z}\{\tau\}$ , this shows that for any $\tau\in\mathbb{H}^{\mathrm{BS}}$ , we have

(4.4)

\displaystyle\sum_{\begin{subarray}{c}j\\ b_{j}=\tau\end{subarray}}P_{j}-\sum_{\begin{subarray}{c}j\\ a_{j}=\tau\end{subarray}}P_{j}-\sum_{\begin{subarray}{c}k\\ \gamma_{k}d_{k}=\tau\end{subarray}}\gamma_{k}Q_{k}+\sum_{\begin{subarray}{c}k\\ d_{k}=\tau\end{subarray}}Q_{k}-\sum_{\begin{subarray}{c}l\\ c_{l}=\tau\end{subarray}}R_{l}=0\,\,\,\textrm{ in }\,\,\,\mathcal{M}_{n}.

Now, let $i\infty\in\partial\mathbb{H}^{\mathrm{BS}}$ denote the point defined by

i\infty:=\lim_{t\in\mathbb{R}_{>0},t\to\infty}it.

Then the identity (4.4) implies that in $\mathcal{MS}(\mathbb{H}^{\mathrm{BS}})\otimes\mathcal{M}_{n}$ , we have

(4.5)

\displaystyle\sum_{\tau\in\mathbb{H}^{\mathrm{BS}}}\{i\infty,\tau\}\otimes\left(\sum_{\begin{subarray}{c}j\\ b_{j}=\tau\end{subarray}}P_{j}-\sum_{\begin{subarray}{c}j\\ a_{j}=\tau\end{subarray}}P_{j}-\sum_{\begin{subarray}{c}k\\ \gamma_{k}d_{k}=\tau\end{subarray}}\gamma_{k}Q_{k}+\sum_{\begin{subarray}{c}k\\ d_{k}=\tau\end{subarray}}Q_{k}-\sum_{\begin{subarray}{c}l\\ c_{l}=\tau\end{subarray}}R_{l}\right)=0.

Using the identity (4.5), in $((\mathcal{MS}(\mathbb{H}^{\mathrm{BS}})/\mathcal{MS}(\partial\mathbb{H}^{\mathrm{BS}}))\otimes\mathcal{M}_{n})_{\Gamma}$ , we compute

	$\displaystyle[\sigma]$	$\displaystyle=\sum_{j}[\{i\infty,b_{j}\}\otimes P_{j}]-[\{i\infty,a_{j}\}\otimes P_{j}]$
		$\displaystyle=\sum_{k}[\{i\infty,\gamma_{k}d_{k}\}\otimes\gamma_{k}Q_{k}]-\sum_{k}[\{i\infty,d_{k}\}\otimes Q_{k}]+\sum_{l}[\{i\infty,c_{l}\}\otimes R_{l}]$
		$\displaystyle=\sum_{k}[(\gamma_{k}-1)\left(\{i\infty,d_{k}\}\otimes Q_{k}\right)]+\sum_{k}[\{i\infty,\gamma_{k}i\infty\}\otimes\gamma_{k}Q_{k}]+\sum_{l}[\{i\infty,c_{l}\}\otimes R_{l}]$
		$\displaystyle=\sum_{k}[\{i\infty,\gamma_{k}i\infty\}\otimes\gamma_{k}Q_{k}]+\sum_{l}[\{i\infty,c_{l}\}\otimes R_{l}].$

Moreover, (as we are considering the relative homology classes) we may replace $c_{l}$ with any point in the same connected component of $\partial\mathbb{H}^{\mathrm{BS}}$ as $c_{l}$ , and in particular, we may replace $c_{l}$ by $g_{l}i\infty$ for some $g_{l}\in\Gamma$ . Thus we conclude that the class $[\sigma]\in H_{1}(Y^{\mathrm{BS}},\partial Y^{\mathrm{BS}},\mathcal{M}_{n})$ is represented by a $1$ -chain of the form

\sigma^{\prime}=\sum_{m}\{i\infty,\gamma_{m}^{\prime}i\infty\}\otimes R_{m}^{\prime}

for some $\gamma_{m}^{\prime}\in\Gamma$ and $R_{m}^{\prime}\in\mathcal{M}_{n}$ . Now the lemma follows from the facts that the group $\Gamma=\mathrm{SL}_{2}(\mathbb{Z})$ is generated by matrices $\begin{pmatrix}0&-1\\ 1&0\end{pmatrix}$ and $\begin{pmatrix}1&1\\ 0&1\end{pmatrix}$ and that $[\{i\infty,\gamma_{1}\gamma_{2}i\infty\}\otimes P]=[\{i\infty,\gamma_{1}i\infty\}\otimes P]+[\{i\infty,\gamma_{2}i\infty\}\otimes\gamma_{1}^{-1}P]$ for any $\gamma_{1},\gamma_{2}\in\Gamma$ and $P\in\mathcal{M}_{n}$ . ∎

Recall that $\Gamma_{\infty}$ is the subgroup of $\Gamma$ generated by $\begin{pmatrix}1&1\\ 0&1\end{pmatrix}$ . Since $\mathbb{P}^{1}(\mathbb{R})=\mathbb{R}\cup\{\infty\}$ , the boundary $\partial Y^{\mathrm{BS}}$ can be identified with $\Gamma_{\infty}\backslash\mathbb{R}$ . Hence we obtain the following lemma.

Lemma 4.16.

We have an identification $H_{0}(\partial Y^{\mathrm{BS}},\mathcal{M}_{n})=(\mathcal{M}_{n})_{\Gamma_{\infty}}$ , and hence $H_{0}(\partial Y^{\mathrm{BS}},\mathcal{M}_{n})\otimes\mathbb{Q}$ is a $1$ -dimensional $\mathbb{Q}$ -vector space generated by $[e_{n}]$ , where $e_{n}=X_{1}^{n}$ .

Lemma 4.17.

The kernel of the boundary homomorphism

\partial\colon H_{1}(Y^{\mathrm{BS}},\partial Y^{\mathrm{BS}}\mathcal{M}_{n})\otimes\mathbb{Q}\longrightarrow H_{0}(\partial Y^{\mathrm{BS}},\mathcal{M}_{n})\otimes\mathbb{Q}

is generated by the set $\{[C_{\nu}]\mid 1\leq\nu\leq n-1\}$ .

Proof.

Let $\sigma\in\ker(H_{1}(Y^{\mathrm{BS}},\partial Y^{\mathrm{BS}}\mathcal{M}_{n})\otimes\mathbb{Q}\longrightarrow H_{0}(\partial Y^{\mathrm{BS}},\mathcal{M}_{n})\otimes\mathbb{Q})$ . By Lemma4.15, we can write as $\sigma=\sum_{\nu=0}^{n}a_{\nu}[C_{\nu}]$ for some rational numbers $a_{0},\dots,a_{n}\in\mathbb{Q}$ . Then by using Lemma 4.16, we find that

0=\partial\sigma=\sum_{\nu=0}^{n}a_{\nu}[\{i\infty\}\otimes(e_{\nu}-(-1)^{n-\nu}e_{n-\nu})]=-(a_{0}-a_{n})[\{i\infty\}\otimes e_{n}].

Therefore, Lemma 4.16 shows that $a_{0}=a_{n}$ . On the other hand we see that

[C_{0}]=-[C_{\nu}]\,\,\,\textrm{ in }\,\,\,H_{1}(Y^{\mathrm{BS}},\partial Y^{\mathrm{BS}},\mathcal{M}_{n}).

This proves the lemma. ∎

Corollary 4.18.

We have $\mathrm{Eis}_{n}\in H^{1}(Y^{\mathrm{BS}},\mathcal{M}_{n})\otimes\mathbb{Q}$ .

Proof.

By Lemma 4.17, the homology group $H_{1}(Y^{\mathrm{BS}},\mathcal{M}_{n})\otimes\mathbb{Q}$ is generated by the image of $H_{1}(\partial Y^{\mathrm{BS}},\mathcal{M}_{n})$ and the set $\{[\widetilde{C}_{\nu}(\tau)]\mid 1\leq\nu\leq n-1\}$ . Therefore, by Lemma 2.8 and Proposition 4.14 we have

\langle\mathrm{Eis}_{n},H_{1}(Y^{\mathrm{BS}},\mathcal{M}_{n})\otimes\mathbb{Q}\rangle\subset\mathbb{Q},

which implies $\mathrm{Eis}_{n}\in H^{1}(Y^{\mathrm{BS}},\mathcal{M}_{n})\otimes\mathbb{Q}$ . ∎

5. Denominator of an ordinary cohomology class

In order to study the denominator $\Delta(\mathrm{Eis}_{n})$ of the Eisenstein class $\mathrm{Eis}_{n}$ , in this section, we interpret the denominator $\Delta(\mathrm{Eis}_{n})$ in terms of the values $\langle\mathrm{Eis}_{n},[\widetilde{T_{p}^{m}C_{\nu}(\tau)}]\rangle$ of the pairing between the Eisenstein class $\mathrm{Eis}_{n}$ and the cycles $[\widetilde{T_{p}^{m}C_{\nu}(\tau)}]\in H_{1}(Y^{\mathrm{BS}},\mathcal{M}_{n,(p)})$ .

5.1. Definition of the ordinary part

Let $p$ be a prime number and $M$ a finitely generated $\mathbb{Z}_{p}$ -module with an endomorphism $f\colon M\longrightarrow M$ . In this subsection, we introduce the notion of the $f$ -ordinary part of $M$ .

Since $M$ is a finitely generated $\mathbb{Z}_{p}$ -module, the $p$ -adic limit

e_{f}:=\lim_{m\to\infty}f^{m!}\in\mathrm{End}_{\mathbb{Z}_{p}}(M)

always exists, and $e_{f}^{2}=e_{f}$ . We define the $f$ -ordinary part $M_{\rm ord}$ of $M$ by

M_{\rm ord}:=e_{f}M,

and we say that $m\in M$ is ( $f$ -)ordinary if $m\in M_{\mathrm{ord}}$ , that is, $e_{f}m=m$ . We also put $M_{\textrm{nilp}}:=(1-e_{f})M$ . We then have $M=M_{\mathrm{ord}}\oplus M_{\rm nilp}$ .

The following lemma follows from the fact that $e_{f}^{2}=e_{f}$ ,

Lemma 5.1.

The functor $M\mapsto M_{\mathrm{ord}}$ is exact.

5.2. Denominator of a cohomology class

Recall

H^{1}_{\rm int}(Y^{\mathrm{BS}},\mathcal{M}_{n}^{\flat})=\operatorname{im}\left(H^{1}(Y^{\mathrm{BS}},\mathcal{M}_{n}^{\flat})\longrightarrow H^{1}(Y^{\mathrm{BS}},\mathcal{M}_{n}^{\flat})\otimes\mathbb{Q}\right).

Definition 5.2.

For any cohomology class $c\in H^{1}(Y^{\mathrm{BS}},\mathcal{M}_{n}^{\flat})\otimes\mathbb{Q}$ , we define the denominator $\Delta(c)\in\mathbb{Z}_{>0}$ of $c$ by

\Delta(c):=\min\{\Delta\in\mathbb{Z}_{>0}\mid\Delta c\in H^{1}_{\rm int}(Y^{\mathrm{BS}},\mathcal{M}_{n}^{\flat})\},

and for each prime number $p$ , we set

\delta_{p}(c):=\mathrm{ord}_{p}(\Delta(c))\,\,\,\textrm{ and }\,\,\,\Delta_{p}(c):=p^{\delta_{p}(c)}.

Lemma 5.3.

Let $c\in H^{1}(Y^{\mathrm{BS}},\mathcal{M}_{n})\otimes\mathbb{Q}$ be a cohomology class. We have

\Delta(c)=\min\{\Delta\in\mathbb{Z}_{>0}\mid\Delta\langle c,H_{1}(Y^{\mathrm{BS}},\mathcal{M}_{n})\rangle\subset\mathbb{Z}\}.

Moreover, for any prime number $p$ , we have

\delta_{p}(c)=\min\{\delta\in\mathbb{Z}_{\geq 0}\mid p^{\delta}\langle c,H_{1}(Y^{\mathrm{BS}},\mathcal{M}_{n}\otimes\mathbb{Z}_{p})\rangle\subset\mathbb{Z}_{p}\}.

Proof.

This lemma follows immediately from the formal duality (see §2.5):

H^{\bullet}(\Gamma\backslash X,\mathcal{M}_{n}^{\flat})/\text{(torsion)}\overset{\sim}{\longrightarrow}\operatorname{Hom}_{\mathbb{Z}}(H_{\bullet}(\Gamma\backslash X,\mathcal{M}_{n}),\mathbb{Z}).

∎

5.3. Denominator of an ordinary cohomology class

In this subsection, we fix a prime number $p$ .

Definition 5.4.

(1)

We put $\mathcal{M}_{n,p}:=\mathcal{M}_{n}\otimes\mathbb{Z}_{p}$ and $\mathcal{M}_{n,p}^{\flat}:=\mathcal{M}_{n}^{\flat}\otimes\mathbb{Z}_{p}$ .

(2)

For any finitely generated $\mathbb{Z}_{p}$ -algebra, we put

	$\displaystyle H_{\bullet}^{\mathrm{ord}}(Y^{\mathrm{BS}},\mathcal{M}_{n,p}\otimes R)$	$\displaystyle:=e_{T_{p}}H_{\bullet}(Y^{\mathrm{BS}},\mathcal{M}_{n,p}\otimes R),$
	$\displaystyle H_{\bullet}^{\mathrm{ord}}(Y^{\mathrm{BS}},\partial Y^{\mathrm{BS}},\mathcal{M}_{n,p}\otimes R)$	$\displaystyle:=e_{T_{p}}H_{\bullet}(Y^{\mathrm{BS}},\partial Y^{\mathrm{BS}},\mathcal{M}_{n,p}\otimes R),$
	$\displaystyle H_{\bullet}^{\mathrm{ord}}(\partial Y^{\mathrm{BS}},\mathcal{M}_{n,p}\otimes R)$	$\displaystyle:=e_{T_{p}}H_{\bullet}(\partial Y^{\mathrm{BS}},\mathcal{M}_{n,p}\otimes R).$

(3)

For any finitely generated $\mathbb{Z}_{p}$ -algebra $R$ , we put

	$\displaystyle H^{\bullet}_{\mathrm{ord}}(Y^{\mathrm{BS}},\mathcal{M}_{n,p}^{\flat}\otimes R)$	$\displaystyle:=e_{T_{p}^{\prime}}H^{\bullet}(Y^{\mathrm{BS}},\mathcal{M}_{n,p}^{\flat}\otimes R),$
	$\displaystyle H^{\bullet}_{\mathrm{ord}}(Y^{\mathrm{BS}},\partial Y^{\mathrm{BS}},\mathcal{M}_{n,p}^{\flat}\otimes R)$	$\displaystyle:=e_{T_{p}^{\prime}}H^{\bullet}(Y^{\mathrm{BS}},\partial Y^{\mathrm{BS}},\mathcal{M}_{n,p}^{\flat}\otimes R),$
	$\displaystyle H^{\bullet}_{\mathrm{ord}}(\partial Y^{\mathrm{BS}},\mathcal{M}_{n,p}^{\flat}\otimes R)$	$\displaystyle:=e_{T_{p}^{\prime}}H^{\bullet}(\partial Y^{\mathrm{BS}},\mathcal{M}_{n,p}^{\flat}\otimes R).$

Lemma 5.5.

Let $c\in H^{1}(Y^{\mathrm{BS}},\mathcal{M}_{n,p})\otimes\mathbb{Q}_{p}$ be a cohomology class. For any homology class $C\in H_{1}(Y^{\mathrm{BS}},\mathcal{M}_{n,p})\otimes\mathbb{Q}_{p}$ , we have

\langle c,e_{T_{p}}C\rangle=\lim_{m\to\infty}\langle c,T_{p}^{m!}C\rangle=\lim_{m\to\infty}\langle c|T_{p}^{\prime m!},C\rangle=\langle e_{T_{p}^{\prime}}c,C\rangle.

In particular, if $c$ is ordinary, that is, $e_{T_{p}^{\prime}}c=c$ , then

\langle c,C\rangle=\langle c,e_{T_{p}}C\rangle,

and hence

\delta_{p}(c)=\min\{\delta\in\mathbb{Z}_{\geq 0}\mid p^{\delta}\langle c,H_{1}^{\mathrm{ord}}(Y^{\mathrm{BS}},\mathcal{M}_{n,p})\rangle\subset\mathbb{Z}_{p}\}.

Proof.

This lemma follows from the facts that the pairing $\langle~{},~{}\rangle$ is continuous and $\langle c|T_{p}^{\prime},C\rangle=\langle c,T_{p}C\rangle$ . ∎

Recall the identification $H_{0}(\partial Y^{\mathrm{BS}},\mathcal{M}_{n,p})=(\mathcal{M}_{n,p})_{\Gamma_{\infty}}$ by Lemma 4.16.

Lemma 5.6.

The $\mathbb{Z}_{p}$ -module $H_{0}^{\mathrm{ord}}(\partial Y^{\mathrm{BS}},\mathcal{M}_{n,p})=e_{T_{p}}(\mathcal{M}_{n,p})_{\Gamma_{\infty}}$ is free of rank $1$ and is generated by $e_{T_{p}}[e_{n}]$ .

Proof.

Let $k\in\{0,\ldots,n\}$ be an integer. Since $e_{k}=X_{1}^{k}X_{2}^{n-k}$ , we have

T_{p}([e_{k}])=p^{n-k}[e_{k}]+\sum_{j=0}^{p-1}\sum_{k^{\prime}=0}^{k}\binom{k}{k^{\prime}}p^{k^{\prime}}(-j)^{k-k^{\prime}}[e_{k^{\prime}}].

In particular, we have $T_{p}([e_{0}])=(p^{n}+p)[e_{0}]$ , and hence $e_{T_{p}}[e_{0}]=0$ . Therefore, inductively, we obtain that $e_{T_{p}}[e_{k}]=0$ for any integer $k\in\{0,\ldots,n-1\}$ and that $e_{T_{p}}[e_{n}]=[e_{n}]$ , which implies that the $\mathbb{Z}_{p}$ -module $e_{T_{p}}(\mathcal{M}_{n,p})_{\Gamma_{\infty}}$ is generated by $e_{T_{p}}[e_{n}]$ . Hence by Lemma 4.16, $e_{T_{p}}(\mathcal{M}_{n,p})_{\Gamma_{\infty}}$ is a free $\mathbb{Z}_{p}$ -module of rank $1$ . ∎

Recall $[C_{\nu}]=[\{0,i\infty\}\otimes e_{\nu}]\in H_{1}(Y^{\mathrm{BS}},\partial Y^{\mathrm{BS}},\mathcal{M}_{n})$ .

Proposition 5.7.

For any integer $m\geq 0$ , the $\mathbb{Z}_{p}$ -module $H_{1}^{\mathrm{ord}}(Y^{\mathrm{BS}},\mathcal{M}_{n,p})$ is generated by (the image of) $H_{1}^{\mathrm{ord}}(\partial Y^{\mathrm{BS}},\mathcal{M}_{n,p})$ and a set of lifts of $e_{T_{p}}T_{p}^{m}[C_{\nu}]$ $(1\leq\nu\leq n-1)$ .

Proof.

By definition, we have an exact sequence of Hecke modules

\displaystyle 0\longrightarrow H_{1}(\partial Y^{\mathrm{BS}},\,

\displaystyle\mathcal{M}_{n,p})\longrightarrow H_{1}(Y,\mathcal{M}_{n,p})\longrightarrow H_{1}(Y^{\mathrm{BS}},\partial Y^{\mathrm{BS}},\mathcal{M}_{n,p})\stackrel{{\scriptstyle\partial}}{{\longrightarrow}}H_{0}(\partial Y^{\mathrm{BS}},\mathcal{M}_{n,p})\longrightarrow 0.

By Lemma 4.15, the ordinary part $H_{1}^{\mathrm{ord}}(Y^{\mathrm{BS}},\partial Y^{\mathrm{BS}},\mathcal{M}_{n,p})$ of the relative homology group is generated by the set $\{e_{T_{p}}[C_{\nu}]\mid 0\leq\nu\leq n\}$ . Then by the same argument as in Lemma 4.17 using Lemma 5.6 instead of Lemma 4.16, we find that the kernel of the boundary map $\partial\colon H_{1}^{\mathrm{ord}}(Y^{\mathrm{BS}},\partial Y^{\mathrm{BS}},\mathcal{M}_{n,p})\longrightarrow H_{0}^{\mathrm{ord}}(\partial Y^{\mathrm{BS}},\mathcal{M}_{n,p})$ is generated by the set $\{e_{T_{p}}[C_{\nu}]\mid 1\leq\nu\leq n-1\}$ . Since the homomorphism

T_{p}\colon H_{1}^{\mathrm{ord}}(Y^{\mathrm{BS}},\partial Y^{\mathrm{BS}},\mathcal{M}_{n,p})\longrightarrow H_{1}^{\mathrm{ord}}(Y^{\mathrm{BS}},\partial Y^{\mathrm{BS}},\mathcal{M}_{n,p})

is an isomorphism and the boundary map $\partial$ is Hecke equivariant, it follows that the kernel of the boundary map is generated by the set $\{e_{T_{p}}T_{p}^{m}[C_{\nu}]\mid 1\leq\nu\leq n-1\}$ . This fact together with the above exact sequence implies this proposition. ∎

Corollary 5.8.

Let $m\geq n$ be an integer. For each integer $\nu\in\{1,\ldots,n-1\}$ , recall the cycle $[\widetilde{T_{p}^{m}C_{\nu}(\tau)}]\in H_{1}(Y^{\mathrm{BS}},\mathcal{M}_{n,(p)})$ defined in Definition 3.13 (see also Lemma 3.15(2)). Then for any ordinary cohomology class $c\in H^{1}_{\rm ord}(Y^{\mathrm{BS}},\mathcal{M}_{n,p})\otimes\mathbb{Q}_{p}$ satisfying $\langle c,H_{1}(\partial Y^{\rm BS},\mathcal{M}_{n})\rangle\subset\mathbb{Z}_{p}$ , we have

\delta_{p}(c)=\min\{\delta\in\mathbb{Z}_{\geq 0}\mid p^{\delta}\langle c,[\widetilde{T_{p}^{m}C_{\nu}(\tau)}]\rangle\in\mathbb{Z}_{p}\textrm{ for any integer }1\leq\nu\leq n-1\}.

Proof.

By Lemma 3.15 (3), we see that $e_{T_{p}}[\widetilde{T_{p}^{m}C_{\nu}(\tau)}]\in H_{1}^{\mathrm{ord}}(Y^{\mathrm{BS}},\mathcal{M}_{n,p})$ maps to $e_{T_{p}}T_{p}^{m}[C_{\nu}(\tau)]\in H_{1}^{\mathrm{ord}}(Y^{\mathrm{BS}},\partial Y^{\mathrm{BS}},\mathcal{M}_{n,p})$ under the homomorphism

H_{1}^{\mathrm{ord}}(Y^{\mathrm{BS}},\mathcal{M}_{n,p})\longrightarrow H_{1}^{\mathrm{ord}}(Y^{\mathrm{BS}},\partial Y^{\mathrm{BS}},\mathcal{M}_{n,p}).

Therefore, Proposition 5.7 shows that the $\mathbb{Z}_{p}$ -module $H_{1}^{\mathrm{ord}}(Y^{\mathrm{BS}},\mathcal{M}_{n,p})$ is generated by the image of $H_{1}^{\mathrm{ord}}(\partial Y^{\rm BS},\mathcal{M}_{n})$ and the set $\{e_{T_{p}}[\widetilde{T_{p}^{m}C_{\nu}(\tau)}]\mid 1\leq\nu\leq n-1\}$ . On the other hand, by Lemma 5.5, we have

\langle c,e_{T_{p}}[\widetilde{T_{p}^{m}C_{\nu}(\tau)}]\rangle=\langle c,[\widetilde{T_{p}^{m}C_{\nu}(\tau)}]\rangle.

Now, since $\langle c,H_{1}(\partial Y^{\mathrm{BS}},\mathcal{M}_{n})\rangle\subset\mathbb{Z}_{p}$ by assumption, Lemma 5.5 shows that

\delta_{p}(c)=\min\{\delta\in\mathbb{Z}_{\geq 0}\mid p^{\delta}\langle c,[\widetilde{T_{p}^{m}C_{\nu}(\tau)}]\rangle\in\mathbb{Z}_{p}\textrm{ for any integer }1\leq\nu\leq n-1\}.

∎

Corollary 5.9.

For any integer $m\geq n$ , we have

\delta_{p}(\mathrm{Eis}_{n})=\min\{\delta\in\mathbb{Z}_{\geq 0}\mid p^{\delta}{\langle\mathrm{Eis}_{n},[\widetilde{T_{p}^{m}C_{\nu}(\tau)}]\rangle}\in\mathbb{Z}_{p}\textrm{ for any integer }1\leq\nu\leq n-1\}.

Proof.

By Lemma 2.8, we have $\langle\mathrm{Eis}_{n},H_{1}(\partial Y^{\rm BS},\mathcal{M}_{n})\rangle\subset\mathbb{Z}_{p}$ and $e_{T_{p}^{\prime}}\mathrm{Eis}_{n}=\mathrm{Eis}_{n}$ , this corollary follows from Corollary 5.8. ∎

6. Relation between the denominators of the Eisenstein classes

Recall that $\Delta_{p}(\mathrm{Eis}_{n})$ denotes the $p$ -part of the denominator $\Delta(\mathrm{Eis}_{n})$ of the Eisenstein class (see Definition 5.2). In this section, we fix a prime number $p\geq 5$ and discuss another expression for the denominator $\Delta(\mathrm{Eis}_{n})$ of the Eisenstein class $\mathrm{Eis}_{n}$ . Moreover, we study a relation of the denominators $\Delta_{p}(\mathrm{Eis}_{n})$ and $\Delta_{p}(\mathrm{Eis}_{n^{\prime}})$ of the Eisenstein classes when $n$ and $n^{\prime}$ are $p$ -adically close.

6.1. Structure of the ordinary part of cohomology groups

In this subsection, we study the structure of the ordinary part of cohomology groups. Results similar to those obtained in this subsection can be found in the paper [Hid86] of Hida. In the papers [Hid86, Hid88], Hida studied the ordinary part of cohomology groups for $\Gamma^{\prime}\backslash\mathbb{H}$ in the case that $\Gamma^{\prime}/\{\pm 1\}$ is torsion-free. However, in the present paper we consider the group $\Gamma=\mathrm{SL}_{2}(\mathbb{Z})$ which has torsion elements other than $\pm\begin{pmatrix}1&\\ &1\end{pmatrix}$ . Hence, for the completeness of the present paper, we give the details of the proof of all the necessary facts.

Note that since we assume that $p\geq 5$ , any short exact sequence of $\Gamma$ -modules induces a long exact sequence in cohomology.

Lemma 6.1.

The inclusion map $\mathcal{M}_{n,p}^{\flat}\longhookrightarrow\mathcal{M}_{n,p}$ induces an isomorphism $H^{\bullet}_{\mathrm{ord}}(Y^{\mathrm{BS}},\mathcal{M}_{n,p}^{\flat})\stackrel{{\scriptstyle\sim}}{{\longrightarrow}}H^{\bullet}_{\mathrm{ord}}(Y^{\mathrm{BS}},\mathcal{M}_{n,p})$ .

Proof.

It suffices to prove that $e_{T_{p}^{\prime}}H^{i}(Y^{\mathrm{BS}},\mathcal{M}_{n,p}/\mathcal{M}_{n,p}^{\flat})=0$ for any integer $i\geq 0$ . Since $X_{1}^{n},X_{2}^{n}\in\mathcal{M}_{n,p}^{\flat}$ , any element in $\mathcal{M}_{n,p}/\mathcal{M}_{n,p}^{\flat}$ can be represented by a polynomial of the form $X_{1}X_{2}f(X_{1},X_{2})$ , where $f\in\mathcal{M}_{n-2,p}$ . Hence the fact that

	$\displaystyle\widetilde{\begin{pmatrix}p&\\ &1\end{pmatrix}}\cdot X_{1}X_{2}f(X_{1},X_{2})$	$\displaystyle=pX_{1}X_{2}f(pX_{1},X_{2}),$
	$\displaystyle\widetilde{\begin{pmatrix}1&j\\ &p\end{pmatrix}}\cdot X_{1}X_{2}f(X_{1},X_{2})$	$\displaystyle=p(X_{1}+jX_{2})X_{2}f(X_{1}+jX_{2},pX_{2})$

shows that $e_{T_{p}^{\prime}}c=0$ for any element $c\in\operatorname{Hom}_{\mathbb{Z}}(S_{\bullet}(\mathbb{H}^{\mathrm{BS}}),\mathcal{M}_{n,p}/\mathcal{M}_{n,p}^{\flat})$ . In particular, we have $e_{T_{p}^{\prime}}H^{i}(Y^{\mathrm{BS}},\mathcal{M}_{n,p}/\mathcal{M}_{n,p}^{\flat})=0$ . ∎

Thanks to Lemma 6.1, in the following, we focus on the ordinary cohomology groups with coefficient $\mathcal{M}_{n,p}$ .

Lemma 6.2.

For any polynomial $f(X_{1},X_{2})\in\mathcal{M}_{n}/p\mathcal{M}_{n}$ , we have $f|({T_{p}^{\prime}})^{2}\in\mathbb{F}_{p}X_{2}^{n}$ .

Proof.

By definition, we have

\displaystyle(f|{T_{p}^{\prime}})(X_{1},X_{2})=f(0,X_{2})+\sum_{j=0}^{p-1}f(X_{1}+jX_{2},0).

Hence we see that $(f|{T_{p}^{\prime}})(X_{1},0)=\sum_{j=0}^{p-1}f(X_{1},0)=0$ , and we obtain

(f|({T_{p}^{\prime}})^{2})(X_{1},X_{2})=(f|{T_{p}^{\prime}})(0,X_{2})\in\mathbb{F}_{p}X_{2}^{n}.

∎

The boundary $\partial Y^{\mathrm{BS}}$ is of real dimension $1$ , and hence $H^{2}(\partial Y^{\mathrm{BS}},\mathcal{M})$ vanishes for any $\Gamma$ -module $\mathcal{M}$ . Therefore, for any integer $r\geq 0$ , the short exact sequence $0\longrightarrow\mathcal{M}_{n,p}\stackrel{{\scriptstyle\times p^{r}}}{{\longrightarrow}}\mathcal{M}_{n,p}\longrightarrow\mathcal{M}_{n,p}/p^{r}\mathcal{M}_{n,p}\longrightarrow 0$ induces an isomorphism

(6.1)

\displaystyle H^{1}_{\mathrm{ord}}(\partial Y^{\mathrm{BS}},\mathcal{M}_{n,p})\otimes\mathbb{Z}_{p}/(p^{r})\stackrel{{\scriptstyle\sim}}{{\longrightarrow}}H^{1}_{\mathrm{ord}}(\partial Y^{\mathrm{BS}},\mathcal{M}_{n,p}/p^{r}\mathcal{M}_{n,p}).

Lemma 6.3.

The ordinary part $H^{1}_{\mathrm{ord}}(\partial Y^{\mathrm{BS}},\mathcal{M}_{n,p})$ is torsion-free.

Proof.

By using the exact sequence of $M_{2}^{+}(\mathbb{Z})$ -modules

0\longrightarrow\mathcal{M}_{n,p}\stackrel{{\scriptstyle\times p}}{{\longrightarrow}}\mathcal{M}_{n,p}\longrightarrow\mathcal{M}_{n,p}/p\mathcal{M}_{n,p}\longrightarrow 0,

we obtain an isomorphism of Hecke modules

\operatorname{coker}\left(\mathcal{M}_{n,p}^{\Gamma_{\infty}}/p\mathcal{M}_{n,p}^{\Gamma_{\infty}}\longrightarrow(\mathcal{M}_{n,p}/p\mathcal{M}_{n,p})^{\Gamma_{\infty}}\right)\stackrel{{\scriptstyle\sim}}{{\longrightarrow}}H^{1}(\partial Y^{\mathrm{BS}},\mathcal{M}_{n,p})[p],

where for an abelian group $M$ we write $M[p]:=\ker(M\stackrel{{\scriptstyle\times p}}{{\longrightarrow}}M)$ for the subgroup of $p$ -torsion elements of $M$ . A direct computation shows that $\mathcal{M}_{n,p}^{\Gamma_{\infty}}=\mathbb{Z}_{p}X_{2}^{n}$ . Hence Lemma 6.2 implies that $H^{1}_{\mathrm{ord}}(\partial Y^{\mathrm{BS}},\mathcal{M}_{n,p})[p]=\operatorname{coker}\left(\mathbb{F}_{p}X_{2}^{n}\longrightarrow e_{T_{p}^{\prime}}(\mathcal{M}_{n,p}/p\mathcal{M}_{n,p})^{\Gamma_{\infty}}\right)=0$ . ∎

Corollary 6.4.

(1)

The ordinary part $H^{1}_{\mathrm{ord}}(\partial Y^{\mathrm{BS}},\mathcal{M}_{n,p})$ is a free $\mathbb{Z}_{p}$ -module of rank $1$ .
(2)

We have a canonical isomorphism $H^{1}_{\mathrm{int}}(\partial Y^{\mathrm{BS}},\mathcal{M}_{n})\otimes\mathbb{Z}_{p}\stackrel{{\scriptstyle\sim}}{{\longrightarrow}}H^{1}_{\mathrm{ord}}(\partial Y^{\mathrm{BS}},\mathcal{M}_{n,p})$ .
(3)

We have $c|T_{\ell}^{\prime}=(1+\ell^{n+1})c$ for any element $c\in H^{1}_{\mathrm{ord}}(\partial Y^{\mathrm{BS}},\mathcal{M}_{n,p})$ and prime number $\ell$ .

Proof.

By Lemma 6.3, we have a surjective homomorphism

H^{1}_{\mathrm{int}}(\partial Y^{\mathrm{BS}},\mathcal{M}_{n})\otimes\mathbb{Z}_{p}\longrightarrow H^{1}_{\mathrm{ord}}(\partial Y^{\mathrm{BS}},\mathcal{M}_{n,p}).

Lemma 2.7 shows that $H^{1}_{\mathrm{int}}(\partial Y^{\mathrm{BS}},\mathcal{M}_{n})\cong\mathbb{Z}$ and $c|T_{\ell}^{\prime}=(1+\ell^{n+1})c$ for any element $c\in H^{1}_{\mathrm{int}}(\partial Y^{\mathrm{BS}},\mathcal{M}_{n})$ and prime number $\ell$ . These facts imply this corollary. ∎

Proposition 6.5.

The ordinary part $H^{1}_{\rm ord}(Y^{\mathrm{BS}},\mathcal{M}_{n,p})$ is torsion-free.

Proof.

Since $H^{0}(Y^{\mathrm{BS}},\mathcal{M}_{n})=\mathcal{M}_{n}^{\Gamma}=0$ , the exact sequence

0\longrightarrow\mathcal{M}_{n,p}\stackrel{{\scriptstyle\times p}}{{\longrightarrow}}\mathcal{M}_{n,p}\longrightarrow\mathcal{M}_{n,p}/p\mathcal{M}_{n,p}\longrightarrow 0

implies that

H^{0}_{\mathrm{ord}}(Y^{\mathrm{BS}},\mathcal{M}_{n,p}/p\mathcal{M}_{n,p})=H^{1}_{\mathrm{ord}}(Y^{\mathrm{BS}},\mathcal{M}_{n,p})[p].

Since $\mathbb{F}_{p}X_{2}^{n}\cap H^{0}(Y^{\mathrm{BS}},\mathcal{M}_{n,p}/p\mathcal{M}_{n,p})=(\mathbb{F}_{p}X_{2}^{n})\cap(\mathcal{M}_{n,p}/p\mathcal{M}_{n,p})^{\Gamma}=0$ , Lemma 6.2 shows that the module $H^{0}_{\mathrm{ord}}(Y^{\mathrm{BS}},\mathcal{M}_{n,p}/p\mathcal{M}_{n,p})$ vanishes. ∎

Lemma 6.6.

We have $H^{2}_{\mathrm{ord}}(Y^{\mathrm{BS}},\partial Y^{\mathrm{BS}},\mathcal{M}_{n,p})=H^{2}_{\mathrm{ord}}(Y^{\mathrm{BS}},\mathcal{M}_{n,p})=0$ .

Proof.

Since the boundary $\partial Y^{\mathrm{BS}}$ is homeomorphic to the circle, we have $H^{2}(\partial Y^{\mathrm{BS}},\mathcal{M}_{n,p})=0$ . Hence the canonical homomorphism $H^{2}_{\mathrm{ord}}(Y^{\mathrm{BS}},\partial Y^{\mathrm{BS}},\mathcal{M}_{n,p})\longrightarrow H^{2}_{\mathrm{ord}}(Y^{\mathrm{BS}},\mathcal{M}_{n,p})$ is surjective. Therefore, we only need to show that $H^{2}_{\mathrm{ord}}(Y^{\mathrm{BS}},\partial Y^{\mathrm{BS}},\mathcal{M}_{n,p})=0$ .

Moreover, since $Y$ is a two-dimensional real manifold, we have $H^{3}(Y^{\mathrm{BS}},\partial Y^{\mathrm{BS}},\mathcal{M}_{n,p})=0$ , and hence the short exact sequence $0\longrightarrow\mathcal{M}_{n,p}\stackrel{{\scriptstyle\times p}}{{\longrightarrow}}\mathcal{M}_{n,p}\longrightarrow\mathcal{M}_{n,p}/p\mathcal{M}_{n,p}\longrightarrow 0$ induces an isomorphism

H^{2}(Y^{\mathrm{BS}},\partial Y^{\mathrm{BS}},\mathcal{M}_{n,p})\otimes\mathbb{Z}/(p)\stackrel{{\scriptstyle\sim}}{{\longrightarrow}}H^{2}(Y^{\mathrm{BS}},\partial Y^{\mathrm{BS}},\mathcal{M}_{n,p}/p\mathcal{M}_{n,p}).

Therefore, it suffices to prove that $H^{2}_{\mathrm{ord}}(Y^{\mathrm{BS}},\partial Y^{\mathrm{BS}},\mathcal{M}_{n,p}/p\mathcal{M}_{n,p})=0$ .

For notational simplicity, set $\overline{\mathcal{M}}_{n,p}:=\mathcal{M}_{n,p}/p\mathcal{M}_{n,p}$ . Let

\mathscr{F}\in S_{2}(\mathbb{H}^{\mathrm{BS}})

be a representative of a fundamental class of $H_{2}(Y^{\mathrm{BS}},\partial Y^{\mathrm{BS}},\mathbb{Z})\cong\mathbb{Z}$ . Then it is known that the homomorphism $\operatorname{Hom}_{\mathbb{Z}}(S_{2}(\mathbb{H}^{\mathrm{BS}}),\overline{\mathcal{M}}_{n,p})\longrightarrow\overline{\mathcal{M}}_{n,p};\phi\mapsto\phi(\mathscr{F})$ induces an isomorphism

(6.2)

\displaystyle\mathrm{ev}_{\mathscr{F}}\colon H^{2}(Y^{\mathrm{BS}},\partial Y^{\mathrm{BS}},\overline{\mathcal{M}}_{n,p})\stackrel{{\scriptstyle\sim}}{{\longrightarrow}}(\overline{\mathcal{M}}_{n,p})_{\Gamma}.

See [Shi71, Proposition 8.2] or [Hid93, Proposition 1, §6.1] for example.

We will show that for any $[\phi]\in H^{2}(Y^{\mathrm{BS}},\partial Y^{\mathrm{BS}},\overline{\mathcal{M}}_{n,p})$ , we have $[\phi]|_{T_{p}^{\prime}}=0$ . By (6.2), it suffices to show that $\mathrm{ev}_{\mathscr{F}}([\phi]|_{T_{p}^{\prime}})\equiv 0$ . Here we use $\equiv$ to emphasize that it is an identity in $(\overline{\mathcal{M}}_{n,p})_{\Gamma}$ . We then compute

	$\displaystyle\mathrm{ev}_{\mathscr{F}}([\phi]\|_{T_{p}^{\prime}})$	$\displaystyle\equiv\phi\|_{T_{p}^{\prime}}(\mathscr{F})(X_{1},X_{2})$
		$\displaystyle=\widetilde{\begin{pmatrix}p&0\\ 0&1\end{pmatrix}}\phi\left(\begin{pmatrix}p&0\\ 0&1\end{pmatrix}\mathscr{F}\right)(X_{1},X_{2})+\sum_{j=0}^{p-1}\widetilde{\begin{pmatrix}1&j\\ 0&p\end{pmatrix}}\phi\left(\begin{pmatrix}1&j\\ 0&p\end{pmatrix}\mathscr{F}\right)(X_{1},X_{2})$
		$\displaystyle\equiv\phi\left(\begin{pmatrix}p&0\\ 0&1\end{pmatrix}\mathscr{F}\right)(0,X_{2})+\sum_{j=0}^{p-1}\phi\left(\begin{pmatrix}1&j\\ 0&p\end{pmatrix}\mathscr{F}\right)(X_{1}+jX_{2},0).$

Put

a_{p}:=\phi\left(\begin{pmatrix}p&0\\ 0&1\end{pmatrix}\mathscr{F}\right)(0,1)\in\mathbb{F}_{p},\quad a_{j}:=\phi\left(\begin{pmatrix}1&j\\ 0&p\end{pmatrix}\mathscr{F}\right)(1,0)\in\mathbb{F}_{p}.

Then we find that

	$\displaystyle\mathrm{ev}_{\mathscr{F}}([\phi]\|_{T_{p}^{\prime}})$	$\displaystyle\equiv a_{p}X_{2}^{n}+\sum_{j=0}^{p-1}a_{j}(X_{1}+jX_{2})^{n}$
		$\displaystyle=a_{p}X_{2}^{n}+\sum_{j=0}^{p-1}a_{j}\begin{pmatrix}1&1\\ 0&1\end{pmatrix}^{-j}\begin{pmatrix}0&1\\ -1&0\end{pmatrix}X_{2}^{n}$
		$\displaystyle\equiv a_{p}X_{2}^{n}+\sum_{j=0}^{p-1}a_{j}X_{2}^{n}$
		$\displaystyle=\left(a_{p}+\sum_{j=0}^{p-1}a_{j}\right)\left(\begin{pmatrix}1&-1\\ 0&1\end{pmatrix}-1\right)X_{1}X_{2}^{n-1}$
		$\displaystyle\equiv 0.$

∎

For any $\Gamma$ -module $\mathcal{M}$ , we define the inner cohomology $H^{1}_{!}(Y^{\mathrm{BS}},\mathcal{M})$ by

H^{1}_{!}(Y^{\mathrm{BS}},\mathcal{M}):=\operatorname{im}\left(H^{1}(Y^{\mathrm{BS}},\partial Y^{\mathrm{BS}}\mathcal{M})\longrightarrow H^{1}(Y^{\mathrm{BS}},\mathcal{M})\right)

and, when $\mathcal{M}$ is a finitely generated $\mathbb{Z}_{p}$ -module, we put

H^{1}_{!,\mathrm{ord}}(Y^{\mathrm{BS}},\mathcal{M}):=e_{T_{p}^{\prime}}H^{1}_{!}(Y^{\mathrm{BS}},\mathcal{M}).

Then the following corollary follows from Lemma 6.6 and the isomorphism (6.1).

Corollary 6.7.

Let $r$ be a non-negative integer and $\mathcal{M}\in\{\mathcal{M}_{n,p},\mathcal{M}_{n,p}/p^{r}\mathcal{M}_{n,p}\}$ . Then we have a natural exact sequence of Hecke modules:

\displaystyle 0\longrightarrow H^{1}_{!,\mathrm{ord}}(Y^{\mathrm{BS}},\mathcal{M})\longrightarrow H^{1}_{\mathrm{ord}}(Y^{\mathrm{BS}},\mathcal{M})\longrightarrow H^{1}_{\mathrm{ord}}(\partial Y^{\mathrm{BS}},\mathcal{M})\longrightarrow 0.

Proof.

For notational simplicity, set $\mathcal{M}_{n,p}/p^{r}:=\mathcal{M}_{n,p}/p^{r}\mathcal{M}_{n,p}$ . Consider the natural commutative diagram

The upper row is exact by Lemma 6.6. Moreover, by (6.1), the right vertical map is surjective, and the bottom row is also exact. ∎

Corollary 6.8.

For any integer $r\geq 0$ , the canonical homomorphism $\mathcal{M}_{n,p}\longrightarrow\mathcal{M}_{n,p}/p^{r}\mathcal{M}_{n,p}$ induces isomorphisms

	$\displaystyle H^{1}_{\mathrm{ord}}(Y^{\mathrm{BS}},\mathcal{M}_{n,p})\otimes\mathbb{Z}_{p}/(p^{r})$	$\displaystyle\stackrel{{\scriptstyle\sim}}{{\longrightarrow}}H^{1}_{\mathrm{ord}}(Y^{\mathrm{BS}},\mathcal{M}_{n,p}/p^{r}\mathcal{M}_{n,p}),$
	$\displaystyle H^{1}_{!,\mathrm{ord}}(Y^{\mathrm{BS}},\mathcal{M}_{n,p})\otimes\mathbb{Z}_{p}/(p^{r})$	$\displaystyle\stackrel{{\scriptstyle\sim}}{{\longrightarrow}}H^{1}_{!,\mathrm{ord}}(Y^{\mathrm{BS}},\mathcal{M}_{n,p}/p^{r}\mathcal{M}_{n,p}).$

Proof.

The exact sequence

0\longrightarrow\mathcal{M}_{n,p}\stackrel{{\scriptstyle\times p^{r}}}{{\longrightarrow}}\mathcal{M}_{n,p}\longrightarrow\mathcal{M}_{n,p}/p^{r}\mathcal{M}_{n,p}\longrightarrow 0

shows that we have an exact sequence

\displaystyle 0\longrightarrow H^{1}_{\mathrm{ord}}(Y^{\mathrm{BS}},\mathcal{M}_{n,p})\otimes\mathbb{Z}_{p}/(p^{r}){\longrightarrow}H^{1}_{\mathrm{ord}}(Y^{\mathrm{BS}},\mathcal{M}_{n,p}/p^{r}\mathcal{M}_{n,p})\longrightarrow H^{2}_{\mathrm{ord}}(Y^{\mathrm{BS}},\mathcal{M}_{n,p})[p^{r}]\longrightarrow 0.

Hence by Lemma 6.6, we obtain the first isomorphism.

By Corollary 6.4(1), we see that $\operatorname{Tor}_{1}^{\mathbb{Z}_{p}}(H^{1}_{\mathrm{ord}}(\partial Y^{\mathrm{BS}},\mathcal{M}_{n,p}),\mathbb{Z}_{p}/(p^{r}))=0$ , and hence Corollary 6.7 for $\mathcal{M}=\mathcal{M}_{n,p}$ shows that

\displaystyle H^{1}_{!,\mathrm{ord}}(Y^{\mathrm{BS}},\mathcal{M}_{n,p})\otimes\mathbb{Z}_{p}/(p^{r})=\ker\left(H^{1}_{\mathrm{ord}}(Y^{\mathrm{BS}},\mathcal{M}_{n,p})\otimes\mathbb{Z}_{p}/(p^{r})\longrightarrow H^{1}_{\mathrm{ord}}(\partial Y^{\mathrm{BS}},\mathcal{M}_{n,p})\otimes\mathbb{Z}_{p}/(p^{r})\right).

Hence the second isomorphism follows from the first isomorphism, the isomorphism (6.1), and Corollary 6.7 for $\mathcal{M}=\mathcal{M}_{n,p}/p^{r}\mathcal{M}_{n,p}$ . ∎

Theorem 6.9.

For any positive integers $r$ and $n^{\prime}$ with $n\equiv n^{\prime}\pmod{(p-1)p^{r-1}}$ , we have the following canonical isomorphism of exact sequences which is $T_{\ell}^{\prime}$ -equivalent for any prime number $\ell\neq p$ :

where $\mathcal{M}_{n,p}/p^{r}:=\mathcal{M}_{n,p}/p^{r}\mathcal{M}_{n,p}$ .

Proof.

Theorem 6.9 follows from the results proved by Hida in [Hid86] (see also [Har11]). In the following, we briefly explain how we derive Theorem 6.9 from Hida’s results in [Hid86].

First, note that since $p\geq 5$ , we have canonical isomorphisms between a sheaf cohomology on $Y^{\mathrm{BS}}$ and a group cohomology of $\Gamma$ :

(6.3)

\displaystyle H^{1}(Y^{\mathrm{BS}},\mathcal{M}_{m,p}/p^{r})

\displaystyle\stackrel{{\scriptstyle\sim}}{{\longrightarrow}}H^{1}(\Gamma,\mathcal{M}_{m,p}/p^{r}).

Moreover, the inner cohomology group $H^{1}_{!}(Y^{\mathrm{BS}},\mathcal{M}_{m,p}/p^{r})$ corresponds to the parabolic subgroup $H^{1}_{P}(\Gamma,\mathcal{M}_{m,p}/p^{r})$ of $H^{1}(\Gamma,\mathcal{M}_{m,p}/p^{r})$ under the isomorphism (6.3) (see [Hid86, (4.1a)] for the definition of the parabolic subgroup).

Let $m\in\{n,n^{\prime}\}$ . Hida showed in [Hid86, Proposition 4.7] that we have isomorphisms

(6.4)

\displaystyle\begin{split}e_{T_{p}^{\prime}}H^{1}(\Gamma,\mathcal{M}_{m,p}/p^{r})&\stackrel{{\scriptstyle\sim}}{{\longrightarrow}}e_{U_{p}^{\prime}}H^{1}(\Gamma_{0}(p^{r}),\mathcal{M}_{m,p}/p^{r});x\mapsto e_{U_{p}^{\prime}}\mathrm{res}(x),\\ e_{T_{p}^{\prime}}H^{1}_{P}(\Gamma,\mathcal{M}_{m,p}/p^{r})&\stackrel{{\scriptstyle\sim}}{{\longrightarrow}}e_{U_{p}^{\prime}}H^{1}_{P}(\Gamma_{0}(p^{r}),\mathcal{M}_{m,p}/p^{r});x\mapsto e_{U_{p}^{\prime}}\mathrm{res}(x)\end{split}

which are $T_{\ell}^{\prime}$ -equivariant for any prime number $\ell\neq p$ . Here $\mathrm{res}$ denotes the restriction map.

Let $L_{m,r}$ denote the $\Gamma_{0}(p^{r})$ -module whose underlying abelian group is $\mathbb{Z}_{p}/(p^{r})$ and the $\Gamma_{0}(p^{r})$ -action is given by the homomorphism $\Gamma_{0}(p^{r})\longrightarrow(\mathbb{Z}_{p}/(p^{r}))^{\times};\begin{pmatrix}a&b\\ c&d\end{pmatrix}\mapsto a^{m}\bmod{p^{r}}$ . Then Hida also showed in [Hid86, Corollary 4.5 and (6.8)] that the $\Gamma_{0}(p^{r})$ -homomorphism $i_{r}\colon\mathcal{M}_{m,p}/p^{r}\longrightarrow L_{m,r};f(X_{1},X_{2})\mapsto f(1,0)$ induces Hecke-equivariant isomorphisms

(6.5)

\displaystyle\begin{split}e_{U_{p}^{\prime}}H^{1}(\Gamma_{0}(p^{r}),\mathcal{M}_{m,p}/p^{r})&\stackrel{{\scriptstyle\sim}}{{\longrightarrow}}e_{U_{p}^{\prime}}H^{1}(\Gamma_{0}(p^{r}),L_{m,r});x\mapsto i_{r,*}(x),\\ e_{U_{p}^{\prime}}H^{1}_{P}(\Gamma_{0}(p^{r}),\mathcal{M}_{m,p}/p^{r})&\stackrel{{\scriptstyle\sim}}{{\longrightarrow}}e_{U_{p}^{\prime}}H^{1}_{P}(\Gamma_{0}(p^{r}),L_{m,r});x\mapsto i_{r,*}(x).\end{split}

Since $n\equiv n^{\prime}\pmod{(p-1)p^{r}}$ , we have $L_{n,r}=L_{n^{\prime},r}$ as $\Gamma_{0}(p^{r})$ -modules, by combining the isomorphisms (6.4) and (6.5) for $m=n,n^{\prime}$ , we obtain the following commutative diagram:

where horizontal arrows are isomorphisms and $T_{\ell}^{\prime}$ -equivariant for any prime number $\ell\neq p$ . This completes the proof. ∎

6.2. Another expression for $\Delta_{p}(\mathrm{Eis}_{n})$

Let $p\geq 5$ be a prime number and take a prime number $\ell\neq p$ . Let

\mathcal{H}_{\ell,p}:=\mathbb{Z}_{p}[X]

be the polynomial ring over $\mathbb{Z}_{p}$ , and by using the Hecke operator $T_{\ell}^{\prime}$ at $\ell$ , we regard cohomology groups that appear in the present paper as $\mathcal{H}_{\ell,p}$ -modules. For notational simplicity, we put

x_{\ell,n}:=X-(1+\ell^{n+1})\,\,\,\textrm{ and }\,\,\,\mathcal{B}_{\ell,p,n}:=\mathcal{H}_{\ell,p}/(x_{\ell,n}).

Note that by Corollary 6.4, we have $\mathcal{B}_{\ell,p,n}\stackrel{{\scriptstyle\sim}}{{\longrightarrow}}H^{1}_{\mathrm{ord}}(\partial Y^{\mathrm{BS}},\mathcal{M}_{n,p});1\mapsto e_{T_{p}^{\prime}}[e_{n}]$ as $\mathcal{H}_{\ell,p}$ -modules.

Lemma 6.10.

Let $c\in H^{1}(Y^{\mathrm{BS}},\mathcal{M}_{n})\otimes\mathbb{C}$ be a cohomology class. If $c|T_{\ell}^{\prime}=(1+\ell^{n+1})c$ , then we have $c|T_{\ell^{\prime}}^{\prime}=(1+\ell^{\prime n+1})c$ for any prime number $\ell^{\prime}$ , that is, the cohomology class $c$ is a scalar multiple of $\mathrm{Eis}_{n}$ .

Proof.

It is well-known that one can take a $T_{\ell}^{\prime}$ -Hecke-eigen basis $f_{1},\ldots,f_{t}\in H^{1}(Y^{\mathrm{BS}},\mathcal{M}_{n})\otimes\mathbb{C}$ such that $f_{1}=\mathrm{Eis}_{n}$ and that the elements $f_{2},\ldots,f_{t}$ correspond to either cusp forms or their complex conjugates via the Eichler–Shimura homomorphism. Then the Ramanujan conjecture proved by Deligne shows that the absolute value of the $T_{\ell}^{\prime}$ -eigenvalue of $f_{i}$ ( $2\leq i\leq t$ ) is less than $1+\ell^{n+1}$ , which implies this lemma. ∎

Lemma 6.11.

We have $H^{1}_{\mathrm{ord}}(Y^{\mathrm{BS}},\mathcal{M}_{n,p})[x_{\ell,n}]=\mathbb{Z}_{p}\Delta_{p}(\mathrm{Eis}_{n})\mathrm{Eis}_{n}$ .

Proof.

By Proposition 6.5 and Lemma 6.10, we have $H^{1}_{\mathrm{ord}}(Y^{\mathrm{BS}},\mathcal{M}_{n,p})[x_{\ell,n}]\subset\mathbb{Q}_{p}\cdot\mathrm{Eis}_{n}$ . Hence this lemma follows from the definition of the denominator $\Delta_{p}(\mathrm{Eis}_{n})$ of the Eisenstein class $\mathrm{Eis}_{n}$ and Lemmas 5.5 and 6.1. ∎

Definition 6.12.

We define $[\mathcal{E}_{\ell,p,n}]\in\operatorname{Ext}^{1}_{\mathcal{H}_{\ell,p}}(\mathcal{B}_{\ell,p,n},H^{1}_{!,\mathrm{ord}}(Y^{\mathrm{BS}},\mathcal{M}_{n,p}))$ to be the element corresponding to the exact sequence of $\mathcal{H}_{\ell,p}$ -modules in Corollary 6.7 for $\mathcal{M}=\mathcal{M}_{n,p}$ :

0\longrightarrow H^{1}_{!,\mathrm{ord}}(Y^{\mathrm{BS}},\mathcal{M}_{n,p})\longrightarrow H^{1}_{\mathrm{ord}}(Y^{\mathrm{BS}},\mathcal{M}_{n,p})\longrightarrow\mathcal{B}_{\ell,p,n}\longrightarrow 0.

The following lemma follows directly from Lemma 6.11.

Lemma 6.13.

$\operatorname{Ann}_{\mathbb{Z}_{p}}([\mathcal{E}_{\ell,p,n}])=\Delta_{p}(\mathrm{Eis}_{n})\mathbb{Z}_{p}$ .

Lemma 6.14.

We have a natural identification

H^{1}_{!,\mathrm{ord}}(Y^{\mathrm{BS}},\mathcal{M}_{n,p})\otimes_{\mathcal{H}_{\ell,p}}\mathcal{B}_{\ell,p,n}=\operatorname{Ext}^{1}_{\mathcal{H}_{\ell,p}}(\mathcal{B}_{\ell,p,n},H^{1}_{!,\mathrm{ord}}(Y^{\mathrm{BS}},\mathcal{M}_{n,p})).

Proof.

Since $x_{\ell,n}$ is an regular element of $\mathcal{H}_{\ell,p}$ , we have an exact sequence of $\mathcal{H}_{\ell,p}$ -modules:

0\longrightarrow\mathcal{H}_{\ell,p}\xrightarrow{\times x_{\ell}}\mathcal{H}_{\ell,p}\longrightarrow\mathcal{B}_{\ell,p,n}\longrightarrow 0.

Applying the functor $\operatorname{Hom}_{\mathcal{H}_{\ell,p}}(-,H^{1}_{!,\mathrm{ord}}(Y^{\mathrm{BS}},\mathcal{M}_{n,p}))$ to this short exact sequence, we obtain the desired identification. ∎

Lemma 6.15.

For any positive integers $r$ and $n^{\prime}$ with $n\equiv n^{\prime}\pmod{(p-1)p^{r-1}}$ , we have a natural isomorphism of $\mathcal{H}_{\ell,p}$ -modules:

H^{1}_{!,\mathrm{ord}}(Y^{\mathrm{BS}},\mathcal{M}_{n,p})\otimes_{\mathcal{H}_{\ell,p}}\mathcal{B}_{\ell,p,n}/(p^{r})\cong H^{1}_{!,\mathrm{ord}}(Y^{\mathrm{BS}},\mathcal{M}_{n^{\prime},p})\otimes_{\mathcal{H}_{\ell,p}}\mathcal{B}_{\ell,p,n^{\prime}}/(p^{r}).

Moreover, the image of $[\mathcal{E}_{\ell,p,n}]\bmod{p^{r}}$ is $[\mathcal{E}_{\ell,p,n^{\prime}}]\bmod{p^{r}}$ under this isomorphism (and the identification in Lemma 6.14).

Proof.

This lemma follows from Theorem 6.9 and Corollary 6.8. ∎

Definition 6.16.

We define a polynomial $\Phi_{\ell,n}(t)\in\mathbb{Z}_{p}[t]$ to be the characteristic polynomial associated with $T_{\ell}^{\prime}\colon H^{1}_{!,\mathrm{ord}}(Y^{\mathrm{BS}},\mathcal{M}_{n,p})\longrightarrow H^{1}_{!,\mathrm{ord}}(Y^{\mathrm{BS}},\mathcal{M}_{n,p})$ :

\Phi_{\ell,n}(t):=\det(t\cdot\mathrm{id}-T_{\ell}^{\prime}\mid H^{1}_{!,\mathrm{ord}}(Y^{\mathrm{BS}},\mathcal{M}_{n,p})).

Lemma 6.17.

The $\mathbb{Z}_{p}$ -module $H^{1}_{!,\mathrm{ord}}(Y^{\mathrm{BS}},\mathcal{M}_{n,p})\otimes_{\mathcal{H}_{\ell,p}}\mathcal{B}_{\ell,p,n}$ is annihilated by $\Phi_{\ell,n}(1+\ell^{n+1})$ .

Proof.

By the Cayley–Hamilton theorem, the Hecke module $H^{1}_{!,\mathrm{ord}}(Y^{\mathrm{BS}},\mathcal{M}_{n,p})$ is annihilated by $\Phi_{\ell,n}(T_{\ell}^{\prime})$ . Hence the $\mathbb{Z}_{p}$ -module $H^{1}_{!,\mathrm{ord}}(Y^{\mathrm{BS}},\mathcal{M}_{n,p})\otimes_{\mathcal{H}_{\ell,p}}\mathcal{B}_{\ell,p,n}$ is annihilated by $\Phi_{\ell,n}(1+\ell^{n+1})$ since $\mathcal{B}_{\ell,p,n}=\mathcal{H}_{\ell,p}/(X-(1+\ell^{1+n}))$ . ∎

Lemma 6.18.

Let $r$ be a positive integer satisfying $r>\mathrm{ord}_{p}(\Phi_{\ell,n}(1+\ell^{n+1}))$ . Then for any even integer $n^{\prime}\geq 2$ with $n\equiv n^{\prime}\pmod{(p-1)p^{r-1}}$ , we have

\mathrm{ord}_{p}(\Phi_{\ell,n}(1+\ell^{n+1}))=\mathrm{ord}_{p}(\Phi_{\ell,n^{\prime}}(1+\ell^{n^{\prime}+1})).

Proof.

By Theorem 6.9, we have

\Phi_{\ell,n}(t)\equiv\Phi_{\ell,n^{\prime}}(t)\pmod{p^{r}}.

The fact that $n\equiv n^{\prime}\pmod{(p-1)p^{r-1}}$ implies that $\ell^{n+1}\equiv\ell^{n^{\prime}+1}\pmod{p^{r}}$ , and we obtain $\Phi_{\ell,n}(1+\ell^{n+1})\equiv\Phi_{\ell,n^{\prime}}(1+\ell^{n^{\prime}+1})\pmod{p^{r}}$ . Hence the assumption that $r>\mathrm{ord}_{p}(\Phi_{\ell,n}(1+\ell^{n+1}))$ shows that $\mathrm{ord}_{p}(\Phi_{\ell,n}(1+\ell^{n+1}))=\mathrm{ord}_{p}(\Phi_{\ell,n^{\prime}}(1+\ell^{n^{\prime}+1}))$ . ∎

Proposition 6.19.

Let $r$ be a positive integer satisfying $r>\mathrm{ord}_{p}(\Phi_{\ell,n}(1+\ell^{n+1}))$ . Then for any even integer $n^{\prime}\geq 2$ with $n\equiv n^{\prime}\pmod{(p-1)p^{r-1}}$ , we have

\Delta_{p}(\mathrm{Eis}_{n})=\Delta_{p}(\mathrm{Eis}_{n^{\prime}}).

Proof.

By Lemmas 6.15, 6.17, and 6.18, we have the natural isomorphism

	$\displaystyle H^{1}_{!,\mathrm{ord}}(Y^{\mathrm{BS}},\mathcal{M}_{n,p})\otimes_{\mathcal{H}_{\ell,p}}\mathcal{B}_{\ell,p,n}$	$\displaystyle=H^{1}_{!,\mathrm{ord}}(Y^{\mathrm{BS}},\mathcal{M}_{n,p})\otimes_{\mathcal{H}_{\ell,p}}\mathcal{B}_{\ell,p,n}/(p^{r})$
		$\displaystyle\cong H^{1}_{!,\mathrm{ord}}(Y^{\mathrm{BS}},\mathcal{M}_{n^{\prime},p})\otimes_{\mathcal{H}_{\ell,p}}\mathcal{B}_{\ell,p,n^{\prime}}/(p^{r})$
		$\displaystyle=H^{1}_{!,\mathrm{ord}}(Y^{\mathrm{BS}},\mathcal{M}_{n^{\prime},p})\otimes_{\mathcal{H}_{\ell,p}}\mathcal{B}_{\ell,p,n^{\prime}}.$

Moreover, the image of $[\mathcal{E}_{\ell,p,n}]$ under this isomorphism is $[\mathcal{E}_{\ell,p,n^{\prime}}]$ , and Lemma 6.13 implies that

\displaystyle\Delta_{p}(\mathrm{Eis}_{n})\mathbb{Z}_{p}=\operatorname{Ann}_{\mathbb{Z}_{p}}([\mathcal{E}_{\ell,p,n}])=\operatorname{Ann}_{\mathbb{Z}_{p}}([\mathcal{E}_{\ell,p,n^{\prime}}])=\Delta_{p}(\mathrm{Eis}_{n^{\prime}})\mathbb{Z}_{p}.

∎

7. Kubota–Leopoldt $p$ -adic $L$ -function

Let $p$ be a prime number. In this section, we introduce the Kubota–Leopoldt $p$ -adic $L$ -functions and prove certain congruence properties that will be used in the proof of Theorem 2.13.

Let $\omega\colon\operatorname{Gal}(\overline{\mathbb{Q}}/\mathbb{Q})\longrightarrow\mathbb{Z}_{p}^{\times}$ denote the Teichmüller character, and let $\mathbbm{1}$ denote the trivial character. For any Dirichlet character $\chi$ , we denote by $L_{p}(s,\chi)\in\mathbb{C}_{p}[[s]]$ the Kubota–Leopoldt $p$ -adic $L$ -function attached to $\chi$ .

Proposition 7.1 ([Was97, Theorems 5.11 and 5.12, Exercises 5.11(1)] ).

(1)

For any Dirichlet character $\chi$ , the $p$ -adic $L$ -function $L_{p}(s,\chi)$ converges on $\mathbb{Z}_{p}\!-\!\{1\}$ . Moreover, for any integer $m\geq 2$ , we have

$L_{p}(1-m,\chi)=(1-\chi\omega^{-m}(p)p^{m-1})L(1-m,\chi\omega^{-m}).$

In particular, we have $\mathrm{ord}_{p}(L_{p}(1-m,\omega^{m}))=\mathrm{ord}_{p}(\zeta(1-m))$ .
(2)

We have

$L_{p}(s,\mathbbm{1})\in\frac{p-1}{p(s-1)}+\mathbb{Z}_{p}[[s-1]]$
(3)

If $m\not\equiv 0\pmod{p-1}$ , then we have

$L_{p}(s,\omega^{m})\in\mathbb{Z}_{p}+p\mathbb{Z}_{p}[[s-1]].$

By using the Kubota–Leopoldt $p$ -adic $L$ -functions, Theorem 4.1 can be restated as follows.

Corollary 7.2.

If we put

\displaystyle D_{p}(n,\nu):=\frac{L_{p}(-\nu,\omega^{1+\nu})L_{p}(\nu-n,\omega^{n-\nu+1})}{L_{p}(-1-n,\omega^{n+2})}-L_{p}(-\nu,\omega^{1+\nu})-L_{p}(\nu-n,\omega^{n-\nu+1}),

then for any integer $\nu\in\{1,\ldots,n-1\}$ we have

\displaystyle\lim_{m\to\infty}\langle\mathrm{Eis}_{n},\widetilde{T_{p}^{m!}(C_{\nu}(\tau))}\rangle=\frac{1-p^{n+1}}{(1-p^{\nu})(1-p^{n-\nu})}D_{p}(n,\nu).

For any even integer $m$ , we define a positive integer $N_{m}$ by

N_{m}:=\textrm{the numerator of $\zeta(1-m)$}.

Corollary 7.3.

Let $m\geq 2$ be an even integer.

(1)

If $m\not\equiv 0\pmod{p-1}$ , then we have $\mathrm{ord}_{p}(\zeta(1-m))=\mathrm{ord}_{p}(N_{m})$ .
(2)

If $m\equiv 0\pmod{p-1}$ , then we have $\mathrm{ord}_{p}(N_{m})=0$ .
(3)

Let $r$ and $m^{\prime}$ be positive integers with $m\equiv m^{\prime}\pmod{(p-1)p^{r-1}}$ . If $r>\mathrm{ord}_{p}(N_{m})$ , then

$\mathrm{ord}_{p}(N_{m})=\mathrm{ord}_{p}(N_{m^{\prime}}).$

Proof.

Claims (1) and (2) follows immediately from Proposition 7.1. When $m\equiv 0\pmod{p-1}$ , claim (3) follows from claim (2). When $m\not\equiv 0\pmod{p-1}$ , claim (3) follows from Proposition 7.1. ∎

Corollary 7.4.

Let $x$ be an integer with $x\not\equiv 0\pmod{p-1}$ . For any integer $y$ , we have

\displaystyle\frac{L_{p}(1-x,\omega^{x})L_{p}(1-y,\mathbbm{1})}{L_{p}(1-x-y,\omega^{x})}\in L_{p}(1-y,\mathbbm{1})+\frac{\mathbb{Z}_{p}}{L_{p}(1-x-y,\omega^{x})}.

Proof.

Since $x\not\equiv 0\pmod{p-1}$ , by Proposition 7.1(3), we have

L_{p}(1-x,\omega^{x})\in L_{p}(1-x-y,\omega^{x})+py\mathbb{Z}_{p}.

Moreover, by Proposition 7.1(2), we have $pyL_{p}(1-y,\mathbbm{1})\in 1+p\mathbb{Z}_{p}$ , which shows

\displaystyle\frac{L_{p}(1-x,\omega^{x})L_{p}(1-y,\mathbbm{1})}{L_{p}(1-x-y,\omega^{x})}\in L_{p}(1-y,\mathbbm{1})+\frac{\mathbb{Z}_{p}}{L_{p}(1-x-y,\omega^{x})}.

∎

Corollary 7.5.

For any integers $x$ and $y$ , we have

	$\displaystyle\frac{L_{p}(1-x,\mathbbm{1})L_{p}(1-y,\mathbbm{1})}{L_{p}(1-x-y,\mathbbm{1})}$	$\displaystyle\equiv\frac{p-1}{px}+\frac{p-1}{py}\pmod{\mathbb{Z}_{p}}$
		$\displaystyle\equiv L_{p}(1-x,\mathbbm{1})+L_{p}(1-y,\mathbbm{1})\pmod{\mathbb{Z}_{p}}.$

Proof.

For notational simplicity, we put

R(s):=-\frac{p-1}{ps}\,\,\,\textrm{ and }\,\,\,H(s-1):=L_{p}(s,\mathbbm{1})-R(1-s).

By Proposition 7.1(2), we have $H(s)\in\mathbb{Z}_{p}[[s]]$ , $xR(x),yR(y),(x+y)R(x+y)\in p^{-1}\mathbb{Z}_{p}^{\times}$ , and $R(x+y)^{-1}=R(x)^{-1}+R(y)^{-1}$ . Since $H(s)\in\mathbb{Z}_{p}[[s]]$ , we have

\displaystyle H(x)\in H(x+y)+y\mathbb{Z}_{p},\quad H(y)\in H(x+y)+x\mathbb{Z}_{p}.

Put $\alpha:=1+R(x+y)^{-1}H(x+y)\in 1+p\mathbb{Z}_{p}$ . Then

\displaystyle\frac{L_{p}(1-x,\mathbbm{1})L_{p}(1-y,\mathbbm{1})}{L_{p}(1-x-y,\mathbbm{1})}\in\frac{(R(x)+H(x+y)+y\mathbb{Z}_{p})(R(y)+H(x+y)+x\mathbb{Z}_{p})}{R(x+y)+H(x+y)}

and we have

	$\displaystyle\quad\frac{(R(x)+H(x+y)+y\mathbb{Z}_{p})(R(y)+H(x+y)+x\mathbb{Z}_{p})}{R(x+y)+H(x+y)}$
	$\displaystyle\subset\frac{R(x)^{-1}+R(y)^{-1}}{\alpha}(R(x)R(y)+(R(x)+R(y))H(x+y)+p^{-1}\mathbb{Z}_{p})$
	$\displaystyle=R(x)+R(y)+\mathbb{Z}_{p}.$

∎

8. Proof of Theorem 2.13

Let $p$ be a prime number. As in Corollary 7.2, for any integer $1\leq\nu\leq n-1$ , we define

\displaystyle D_{p}(n,\nu):=\frac{L_{p}(-\nu,\omega^{1+\nu})L_{p}(\nu-n,\omega^{n-\nu+1})}{L_{p}(-1-n,\omega^{n+2})}-L_{p}(-\nu,\omega^{1+\nu})-L_{p}(\nu-n,\omega^{n-\nu+1})

and set

\displaystyle\delta_{p}(n,\nu):=\max\left\{-\mathrm{ord}_{p}(D_{p}(n,\nu)),0\right\}.

Proposition 8.1.

We have

\delta_{p}(\mathrm{Eis}_{n})=\max_{1\leq\nu\leq n-1}\delta_{p}(n,\nu).

Proof.

By Corollary 7.2, we have

\displaystyle\lim_{m\to\infty}\langle\mathrm{Eis}_{n},\widetilde{T_{p}^{m!}(C_{\nu}(\tau))}\rangle=\frac{1-p^{n+1}}{(1-p^{\nu})(1-p^{n-\nu})}D_{p}(n,\nu).

Hence, for any sufficiently large integer $m$ , we have

\mathrm{ord}_{p}\left(\langle\mathrm{Eis}_{n},\widetilde{T_{p}^{m!}(C_{\nu}(\tau))}\rangle\right)=\mathrm{ord}_{p}(D_{p}(n,\nu)),

and this proposition follows from Corollary 5.9. ∎

Recall that $N_{m}$ denotes the numerator of $\zeta(1-m)$ .

Proposition 8.2.

Let $p$ be a prime number.

(1)

$\delta_{p}(\mathrm{Eis}_{n})\leq\mathrm{ord}_{p}(N_{n+2})$ .
(2)

If $p<n$ , then $\delta_{p}(\mathrm{Eis}_{n})=\mathrm{ord}_{p}(N_{n+2})$ .

The proof of Proposition 8.2 is given in §8.1. First, we give the proof of Theorem 2.13 assuming Proposition 8.2, that is, we show that $\Delta(\mathrm{Eis}_{n})=N_{n+2}$ .

Proof of Theorem 2.13.

Take a prime number $p$ . It suffices to show that $\delta_{p}(\mathrm{Eis}_{n})=\mathrm{ord}_{p}(N_{n+2})$ . When $p-1\mid n+2$ , by Proposition 8.2 we have $0\leq\delta_{p}(\mathrm{Eis}_{n})\leq\mathrm{ord}_{p}(N_{n+2})=0$ , and hence we may assume that $n\not\equiv-2\pmod{p-1}$ . Note that $p\geq 5$ in this case. Take a prime number $\ell\neq p$ , and positive integers $r$ and $n^{\prime}$ satisfying

•

$r>\mathrm{ord}_{p}(\Phi_{\ell,n}(1+\ell^{n}))$ ,
•

$p<n^{\prime}$ ,
•

$n\equiv n^{\prime}\pmod{p^{r-1}(p-1)}$ .

Then by Propositions 8.2(2), we have $\delta_{p}(\mathrm{Eis}_{n^{\prime}})=\mathrm{ord}_{p}(N_{n^{\prime}+2})$ , and Proposition 6.19 implies that

\delta_{p}(\mathrm{Eis}_{n})=\delta_{p}(\mathrm{Eis}_{n^{\prime}})=\mathrm{ord}_{p}(N_{n^{\prime}+2}).

Hence Corollary 7.3 shows that $\delta_{p}(\mathrm{Eis}_{n})=\mathrm{ord}_{p}(N_{n^{\prime}+2})=\mathrm{ord}_{p}(N_{n+2})$ . ∎

8.1. Proof of Proposition 8.2

In this subsection, we prove Proposition 8.2. The proof is divided into the following two cases:

•

$p-1\nmid n+2$ ,
•

$p-1\mid n+2$ .

8.1.1. $p-1\nmid n+2$

Lemma 8.3.

If $p-1\nmid n+2$ , then we have $\delta_{p}(\mathrm{Eis}_{n})\leq\mathrm{ord}_{p}(N_{n+2})$ .

Proof.

Take an integer $\nu\in\{1,\ldots,n-1\}$ . When $p-1\nmid 1+\nu$ and $p-1\nmid n-\nu+1$ , both $L_{p}(-\nu,\omega^{1+\nu})$ and $L_{p}(\nu-n,\omega^{n-\nu+1})$ are $p$ -adic integers, and hence we have

\delta_{p}(n,\nu)\leq\mathrm{ord}_{p}(L_{p}(-1-n,\omega^{n+2}))=\mathrm{ord}_{p}(N_{n+2}).

Suppose $p-1\mid 1+\nu$ (resp. $p-1\mid n-\nu+1$ ). Then since $p-1\nmid n+2$ , we see that $n-\nu+1$ (resp. $1+\nu$ ) is not divisible by $p-1$ . Therefore, Corollary 7.4 shows that

\displaystyle D_{p}(n,\nu)\in\frac{\mathbb{Z}_{p}}{L_{p}(-1-n,\omega^{n+2})}+\mathbb{Z}_{p},

which implies that $\delta_{p}(n,\nu)\leq\mathrm{ord}_{p}(N_{n+2})$ . Hence Proposition 8.1 implies this lemma. ∎

Moreover, if $p<n$ , the result of Carlitz concerning the index of irregularity of a prime shows the following lemma.

Lemma 8.4.

If $p-1\nmid n+2$ and $p<n$ , there is an (odd) integer $\nu\in\{1,\ldots,n-1\}$ such that $\delta_{p}(n,\nu)=\mathrm{ord}_{p}(N_{n+2})$ . In particular, we have $\delta_{p}(\mathrm{Eis}_{n})=\mathrm{ord}_{p}(N_{n+2})$ in this case.

Proof.

By Lemma 8.3, for any regular prime $p$ and (odd) integer $\nu\in\{1,\ldots,n-1\}$ , we have $\delta_{p}(n,\nu)=\mathrm{ord}_{p}(N_{n+2})=0$ . Therefore, we may assume that $p$ is an irregular prime. In particular, $p\geq 37$ .

We define the index $d(p)$ of irregularity of the prime number $p$ by

	$\displaystyle d(p)$	$\displaystyle:=\#\{1\leq t\leq p-3\mid t\in 2\mathbb{Z},B_{t}\in p\mathbb{Z}_{p}\}$
		$\displaystyle=\#\{1\leq t\leq p-3\mid t\in 2\mathbb{Z},L_{p}(1-t,\omega^{t})\in p\mathbb{Z}_{p}\}.$

Then by using the result of Carlitz in [Car61, (21)], Skula proved in [Sku80, Theorem 2.2, Remark 2.3] that

d(p)<\frac{p+3}{4}-\frac{\log(2)}{\log(p)}\frac{p-1}{4}.

Hence if $p\geq 47$ , then we have

d(p)<\frac{p-5}{4}.

Moreover, since the only irregular prime smaller than $47$ is $37$ and $d(37)=1<(37-5)/4$ , the inequality $d(p)<(p-5)/4$ holds true.

For any integer $a$ , we define an integer $[a]_{p-1}$ by

0\leq[a]_{p-1}\leq p-2\,\,\,\textrm{ and }\,\,\,[a]_{p-1}\equiv a\pmod{p-1}.

Since $d(p)<(p-5)/4$ , there is an even integer $t\in\{2,4,\ldots,p-3\}$ with $t\neq[n+2]_{p-1}$ such that

L_{p}(1-t,\omega^{t})\in\mathbb{Z}_{p}^{\times}\,\,\,\textrm{ and }\,\,\,L_{p}(1-[n+2-t]_{p-1},\omega^{[n+2-t]_{p-1}})\in\mathbb{Z}_{p}^{\times}.

Furthermore, Proposition 7.1(3) shows that

L_{p}(1-[n+2-t]_{p-1},\omega^{[n+2-t]_{p-1}})-L_{p}(-1+t-n,\omega^{n+2-t})\in p\mathbb{Z}_{p},

and hence we have $L_{p}(-1+t-n,\omega^{n+2-t})\in\mathbb{Z}_{p}^{\times}$ . Therefore, we put $\nu:=t-1$ and get

\delta_{p}(n,\nu)=\mathrm{ord}_{p}(N_{n+2}).

∎

8.1.2. $p-1\mid n+2$

Lemma 8.5.

If $p-1\mid n+2$ , we have $\delta_{p}(\mathrm{Eis}_{n})=0=\mathrm{ord}_{p}(N_{n+2})$ .

Proof.

The fact that $\mathrm{ord}_{p}(N_{n+2})=0$ follows from Corollary 7.3(2). Hence by Proposition 8.1, it suffices to show that $\delta_{p}(n,\nu)=0$ for any integer $1\leq\nu\leq n-1$ . Since $p-1\mid n+2$ , we have $\mathrm{ord}_{p}(L_{p}(-1-n,\omega^{n+2}))<0$ . If $p-1\nmid 1+\nu$ , then we also have $p-1\nmid n-\nu+1$ , and hence we get $\delta_{p}(n,\nu)=0$ since $L_{p}(-\nu,\omega^{1+\nu})$ and $L_{p}(\nu-n,\omega^{n-\nu+1})$ are $p$ -adic integers. When $p-1\mid 1+\nu$ , we also have $p-1\mid n-\nu+1$ , and Corollary 7.5 implies that $D_{p}(n,\nu)\in\mathbb{Z}_{p}$ . Hence, again, we obtain $\delta_{p}(n,\nu)=0$ . ∎

This completes the proof of Proposition 8.2, and in particular of Theorem 2.13.

9. Applications

In this section, we discuss some applications of Theorem 2.13. For notational simplicity, in the following, the (co)homology groups will be denoted by $H^{\bullet}(Y,\mathcal{M}_{n})$ (resp. $H_{\bullet}(Y,\mathcal{M}_{n})$ ) rather than $H^{\bullet}(Y^{\mathrm{BS}},\mathcal{M}_{n})$ (resp. $H_{\bullet}(Y^{\mathrm{BS}},\mathcal{M}_{n})$ ) since they are naturally isomorphic.

First, note that we have the following corollary of Theorem 2.13.

Corollary 9.1.

Let $n\geq 2$ be an even integer and $\gamma\in\Gamma$ a matrix. Take a polynomial $P(X_{1},X_{2})\in\mathcal{M}_{n}$ such that $\gamma P(X_{1},X_{2})=P(X_{1},X_{2})$ . Then for any element $\tau\in\mathbb{H}$ , we have

\displaystyle N_{n+2}\int_{\tau}^{\gamma\tau}E_{n+2}(z)P(z,1)\,dz\in\mathbb{Z}.

Here $N_{n+2}>0$ is the numerator of $\zeta(-1-n)$ .

Proof.

Since $\gamma P=P$ , we have $\partial(\{\tau,\gamma\tau\}\otimes P)=0$ , and hence $\{\tau,\gamma\tau\}\otimes P$ defines an element in the homology group $H_{1}(Y,\mathcal{M}_{n})$ . Therefore, by Theorem 2.13, we obtain

\displaystyle N_{n+2}\int_{\tau}^{\gamma\tau}E_{n+2}(z)P(z,1)\,dz=\langle N_{n+2}\mathrm{Eis}_{n},[\{\tau,\gamma\tau\}\otimes P]\rangle\in\mathbb{Z}.

∎

9.1. Duke’s conjecture

In the paper [Duk23], Duke defined a certain map called the higher Rademacher symbol

\Psi_{k}\colon\Gamma\longrightarrow\mathbb{Q}

for each integer $k\in\mathbb{Z}_{\geq 2}$ which is a generalization of the classical Rademacher symbol and gave a conjecture concerning the integrality of the higher Rademacher symbol $\Psi_{k}$ .

Conjecture 9.2 ([Duk23, Conjecture, p. 4]).

For any integer $k\in\mathbb{Z}_{\geq 2}$ and matrix $\gamma\in\Gamma$ , we have

\Psi_{k}(\gamma)\in\mathbb{Z}.

In the following, we show that Duke’s Conjecture 9.2 follows from Theorem 2.13.

Remark 9.3.

Conjecture 9.2 is recently proved also by O’Sullivan using a more direct method (see [O’S23]).

Here, instead of giving the original definition of the higher Rademacher symbols, we recall an integral representation of the higher Rademacher symbols, also given by Duke in [Duk23], which is equivalent to the original definition and more suitable for our purpose.

Proposition 9.4 ([Duk23, Definition (2.4) and Lemma 6]).

Let $k\in\mathbb{Z}_{\geq 2}$ be an integer. For any matrix $\gamma:=\begin{pmatrix}a&b\\ c&d\end{pmatrix}\in\Gamma$ , we define a binary quadratic polynomial $Q_{\gamma}(X_{1},X_{2})\in\mathcal{M}_{2}$ associated with $\gamma$ by

Q_{\gamma}(X_{1},X_{2}):=-\frac{\operatorname{sgn}(a+d)}{\mathrm{gcd}(c,a-d,b)}(cX_{1}^{2}-(a-d)X_{1}X_{2}-bX_{2}^{2}).

Then for any element $\tau\in\mathbb{H}$ , we have

\Psi_{k}(\gamma)=N_{2k}\int_{\tau}^{\gamma\tau}E_{2k}(z)Q_{\gamma}(z,1)^{k-1}\,dz,

where $N_{2k}>0$ is the numerator of $\zeta(1-2k)$ .

Corollary 9.5.

Duke’s Conjecture 9.2 holds true.

Proof.

By definition, the binary quadratic polynomial $Q_{\gamma}(X_{1},X_{2})$ defined in Proposition 9.4 is $\gamma$ -invariant. Hence Corollary 9.1 and Proposition 9.4 imply that $\Psi_{k}(\gamma)\in\mathbb{Z}$ . ∎

9.2. Partial zeta functions of real quadratic fields

In this subsection, we discuss an application to the denominators of the special values of the partial zeta functions of real quadratic fields.

Let $F$ be a real quadratic field, and let $\mathcal{O}\subset F$ be an order of $F$ with discriminant $D_{\mathcal{O}}$ . We denote by $I_{\mathcal{O}}$ the group of proper fractional $\mathcal{O}$ -ideals and $P_{\mathcal{O}}^{+}\subset I_{\mathcal{O}}$ the subgroup of totally positive principal ideals. We define the narrow ideal class group $Cl_{\mathcal{O}}^{+}$ of $\mathcal{O}$ by

Cl_{\mathcal{O}}^{+}:=I_{\mathcal{O}}/P_{\mathcal{O}}^{+}.

See [Cox13, §7]. We fix an embedding $F\subset\mathbb{R}$ , and for any element $\alpha\in F\subset\mathbb{R}$ , we denote by $\alpha^{\prime}\in F\subset\mathbb{R}$ its conjugate over $\mathbb{Q}$ .

Moreover, let $\mathcal{O}_{+}^{\times}$ denote the group of totally positive units in $\mathcal{O}$ , and let $\varepsilon_{0}\in\mathcal{O}_{+}^{\times}$ denote the generator of $\mathcal{O}_{+}^{\times}$ such that $\varepsilon_{0}>1$ .

Definition 9.6.

We define a map

\displaystyle\mathfrak{z}_{\mathcal{O},k}\colon Cl_{\mathcal{O}}^{+}\longrightarrow H_{1}(Y,\mathcal{M}_{2k-2})

as follows: Let $\mathcal{A}\in Cl_{\mathcal{O}}^{+}$ , and take a representative $\mathfrak{a}\in I_{\mathcal{O}}$ of $\mathcal{A}$ . We also take a basis $\alpha_{1},\alpha_{2}\in\mathfrak{a}$ over $\mathbb{Z}$ such that $\alpha_{1}\alpha^{\prime}_{2}-\alpha^{\prime}_{1}\alpha_{2}>0$ , and let $\gamma_{0}\in\Gamma$ be a matrix such that

\displaystyle\gamma_{0}\begin{pmatrix}\alpha_{1}\\ \alpha_{2}\end{pmatrix}=\begin{pmatrix}\varepsilon_{0}\alpha_{1}\\ \varepsilon_{0}\alpha_{2}\end{pmatrix}.

Moreover, set

\displaystyle N_{\alpha_{1},\alpha_{2}}(X_{1},X_{2}):=-\frac{1}{N\mathfrak{a}}(\alpha_{2}X_{1}-\alpha_{1}X_{2})(\alpha^{\prime}_{2}X_{1}-\alpha^{\prime}_{1}X_{2}).

We see that $N_{\alpha_{1},\alpha_{2}}(X_{1},X_{2})\in\mathcal{M}_{2}$ and that $\gamma_{0}N_{\alpha_{1},\alpha_{2}}(X_{1},X_{2})=N_{\alpha_{1},\alpha_{2}}(X_{1},X_{2})$ . We then define

\displaystyle\mathfrak{z}_{\mathcal{O},k}(\mathcal{A}):=[\{\tau,\gamma_{0}\tau\}\otimes N_{\alpha_{1},\alpha_{2}}(X_{1},X_{2})^{k-1}]\in H_{1}(Y,\mathcal{M}_{2k-2}),

where $\tau$ is an arbitrary element in $\mathbb{H}$ .

Lemma 9.7.

The homology class $\mathfrak{z}_{\mathcal{O},k}(\mathcal{A})$ does not depend on the choices we made.

Proof.

The independence of $\tau\in\mathbb{H}$ is clear. Let $\mathfrak{b}\in\mathcal{A}$ be another representative. Then there exists a totally positive element $\alpha\in F^{\times}$ such that $\mathfrak{b}=\alpha\mathfrak{a}$ . Take a basis $\beta_{1},\beta_{2}\in\mathfrak{a}$ over $\mathbb{Z}$ with $\beta_{1}\beta^{\prime}_{2}-\beta^{\prime}_{1}\beta_{2}>0$ . Then we obtain a matrix $\gamma_{0,\mathfrak{b}}\in\Gamma$ and a binary quadratic polynomial $N_{\alpha\beta_{1},\alpha\beta_{2}}(X_{1},X_{2})$ from the basis $\alpha\beta_{1},\alpha\beta_{2}$ of $\mathfrak{b}$ . Note that since $\alpha$ is totally positive, we have $(\alpha\beta_{1})(\alpha^{\prime}\beta^{\prime}_{2})-(\alpha^{\prime}\beta^{\prime}_{1})(\alpha\beta_{2})>0$ . Let $\gamma\in\mathrm{GL}_{2}(\mathbb{Z})$ be a matrix satisfying

\begin{pmatrix}\beta_{1}\\ \beta_{2}\end{pmatrix}=\gamma\begin{pmatrix}\alpha_{1}\\ \alpha_{2}\end{pmatrix}.

Then the facts that $\alpha_{1}\alpha^{\prime}_{2}-\alpha^{\prime}_{1}\alpha_{2}>0$ and $\beta_{1}\beta^{\prime}_{2}-\beta^{\prime}_{1}\beta_{2}>0$ imply that $\gamma\in\Gamma$ . Since $\gamma_{0,\mathfrak{b}}\gamma\begin{pmatrix}\alpha_{1}\\ \alpha_{2}\end{pmatrix}=\gamma\begin{pmatrix}\varepsilon_{0}\alpha_{1}\\ \varepsilon_{0}\alpha_{2}\end{pmatrix}$ , we have $\gamma^{-1}\gamma_{0,\mathfrak{b}}\gamma=\gamma_{0}$ . Moreover, we have

\begin{pmatrix}X_{1}&X_{2}\end{pmatrix}{}^{t}\widetilde{\gamma}\begin{pmatrix}0&1\\ -1&0\end{pmatrix}\begin{pmatrix}\alpha_{1}\\ \alpha_{2}\end{pmatrix}=\begin{pmatrix}X_{1}&X_{2}\end{pmatrix}\begin{pmatrix}0&1\\ -1&0\end{pmatrix}\gamma\begin{pmatrix}\alpha_{1}\\ \alpha_{2}\end{pmatrix}=\begin{pmatrix}X_{1}&X_{2}\end{pmatrix}\begin{pmatrix}0&1\\ -1&0\end{pmatrix}\begin{pmatrix}\beta_{1}\\ \beta_{2}\end{pmatrix},

which implies that $N_{\alpha\beta_{1},\alpha\beta_{2}}(X_{1},X_{2})=\gamma N_{\alpha_{1},\alpha_{2}}(X_{1},X_{2})$ . Therefore, we have

	$\displaystyle[\{\tau,\gamma_{0,\mathfrak{b}}\tau\}\otimes N_{\alpha\beta_{1},\alpha\beta_{2}}(X_{1},X_{2})^{k-1}]$	$\displaystyle=[\{\tau,\gamma\gamma_{0}\gamma^{-1}\tau\}\otimes\gamma N_{\alpha_{1},\alpha_{2}}(X_{1},X_{2})^{k-1}]$
		$\displaystyle=[\{\gamma^{-1}\tau,\gamma_{0}\gamma^{-1}\tau\}\otimes N_{\alpha_{1},\alpha_{2}}(X_{1},X_{2})^{k-1}]$
		$\displaystyle=[\{\tau,\gamma_{0}\tau\}\otimes N_{\alpha_{1},\alpha_{2}}(X_{1},X_{2})^{k-1}]$

as elements of $H_{1}(Y,\mathcal{M}_{2k-2})$ . ∎

Remark 9.8.

(1)

Since the matrix $\gamma_{0}$ in Definition 9.6 is hyperbolic (i.e., $|\mathrm{trace}(\gamma_{0})|>2$ ), we have $\dim_{\mathbb{Q}}\{Q\in\mathcal{M}_{2}\otimes\mathbb{Q}\mid\gamma Q=Q\}=1$ . This fact together with [Cox13, (7.6)] shows that $N_{\alpha_{1},\alpha_{2}}(X_{1},X_{2})=Q_{\gamma_{0}}(X_{1},X_{2})$ .
(2)

Gauss’s theory concerning binary quadratic forms (see [Cox13, Exercise 7.21] for example) shows that for any hyperbolic element $\gamma\in\Gamma$ , there is an order $\mathcal{O}$ of a real quadratic field and a narrow ideal class $\mathcal{A}\in Cl_{\mathcal{O}}^{+}$ such that

$[\{z,\gamma z\}\otimes Q_{\gamma}(X_{1},X_{2})]\in\mathbb{Z}\mathfrak{z}_{\mathcal{O},2}(\mathcal{A}).$

Definition 9.9.

For each ideal class $\mathcal{A}\in Cl_{\mathcal{O}}^{+}$ , the partial zeta function $\zeta_{\mathcal{O}}(\mathcal{A},s)$ associated with $\mathcal{A}$ is defined by

\displaystyle\zeta_{\mathcal{O}}(\mathcal{A},s):=\sum_{\mathfrak{a}\subset\mathcal{O},\mathfrak{a}\in\mathcal{A}}\frac{1}{(N\mathfrak{a})^{s}},\quad\quad(\operatorname{Re}(s)>1),

and it is well-known that $\zeta_{\mathcal{O}}(\mathcal{A},s)$ can be continued meromorphically to $s\in\mathbb{C}$ and has a simple pole at $s=1$ .

The following integral representation of the special values of the partial zeta function is classically known.

Proposition 9.10.

For any integer $k\in\mathbb{Z}_{\geq 2}$ and ideal class $\mathcal{A}\in Cl_{\mathcal{O}}^{+}$ , we have

\displaystyle\langle\mathrm{Eis}_{2k-2},\mathfrak{z}_{\mathcal{O},k}(\mathcal{A})\rangle=(-1)^{k}\frac{\zeta_{\mathcal{O}}(\mathcal{A}^{-1},1-k)}{\zeta(1-2k)}.

Before we give a proof of Proposition 9.10, we recall (a special case of) the so-called Feynman parametrization.

Lemma 9.11.

Let $x_{1},x_{2},a,b\in\mathbb{C}$ be complex numbers such that $x_{1}a+x_{2}\neq 0$ and $x_{1}b+x_{2}\neq 0$ . Then for any non-negative integers $k_{1}$ and $k_{2}$ , we have

\int_{a}^{b}\frac{\left(b-z\right)^{k_{1}}\left(z-a\right)^{k_{2}}}{(x_{1}z+x_{2})^{2+k_{1}+k_{2}}}\,dz=\frac{k_{1}!k_{2}!}{(k_{1}+k_{2}+1)!}\frac{(b-a)^{k_{1}+k_{2}+1}}{(x_{1}a+x_{2})^{k_{1}+1}(x_{1}b+x_{2})^{k_{2}+1}}.

Proof.

We may assume that $a\neq b$ . By setting $y_{1}=x_{1}a+x_{2}$ and $y_{2}=x_{1}b+x_{2}$ , it suffices to prove that

(b-a)\int_{a}^{b}\frac{\left(b-z\right)^{k_{1}}\left(z-a\right)^{k_{2}}}{((b-z)y_{1}+(z-a)y_{2})^{2+k_{1}+k_{2}}}\,dz=\frac{k_{1}!k_{2}!}{(k_{1}+k_{2}+1)!}\frac{1}{y_{1}^{k_{1}+1}y_{2}^{k_{2}+1}}.

The case where $k_{1}=k_{2}=0$ is clear, i.e., we have

(b-a)\int_{a}^{b}\frac{1}{((b-z)y_{1}+(z-a)y_{2})^{2}}\,dz=\frac{1}{y_{1}y_{2}}.

Then by viewing the both sides as holomorphic functions in $(y_{1},y_{2})\in\mathbb{C}^{\times}\times\mathbb{C}^{\times}$ and applying the differential operator $\left(\frac{\partial}{\partial y_{1}}\right)^{k_{1}}\left(\frac{\partial}{\partial y_{2}}\right)^{k_{2}}$ , we obtain the desired identity. ∎

Proof of Proposition 9.10.

We use the same notations as in Definition 9.6. Since

2\zeta(2k)E_{2k}(z)=\sum_{(0,0)\neq(m,n)\in\mathbb{Z}^{2}}\frac{1}{(mz+n)^{2k}}

and $\mathrm{Eis}_{2k-2}=r(E_{2k})$ , for any element $\tau\in\mathbb{H}$ we have

\displaystyle 2\zeta(2k)\langle\mathrm{Eis}_{2k-2},\mathfrak{z}_{\mathcal{O},k}(\mathcal{A})\rangle=\int_{\tau}^{\gamma_{0}\tau}\sum_{(0,0)\neq(m,n)\in\mathbb{Z}^{2}}\frac{N_{\alpha_{1},\alpha_{2}}(z,1)^{k-1}}{(mz+n)^{2k}}\,dz.

Since $N_{\alpha_{1},\alpha_{2}}(\gamma_{0}z,1)=j(\gamma_{0},z)^{-2}N_{\alpha_{1},\alpha_{2}}(z,1)$ , where $j(\begin{pmatrix}a&b\\ c&d\end{pmatrix},z):=cz+d$ is the factor of automorphy, if we fix a complete set $S_{\gamma_{0}}$ of representatives of $(\mathbb{Z}^{2}\!-\!\{(0,0)\})/\gamma_{0}^{\mathbb{Z}}$ , then we have

	$\displaystyle\int_{\tau}^{\gamma_{0}\tau}\sum_{(0,0)\neq(m,n)\in\mathbb{Z}^{2}}\frac{N_{\alpha_{1},\alpha_{2}}(z,1)^{k-1}}{(mz+n)^{2k}}\,dz$	$\displaystyle=\int_{\tau}^{\gamma_{0}\tau}\sum_{l\in\mathbb{Z}}\sum_{(m,n)\in S_{\gamma_{0}}}\frac{N_{\alpha_{1},\alpha_{2}}(z,1)^{k-1}}{j(\gamma_{0}^{l},z)^{2k}(m(\gamma_{0}^{l}z)+n)^{2k}}\,dz$
		$\displaystyle=\sum_{l\in\mathbb{Z}}\int_{\gamma_{0}^{l}\tau}^{\gamma_{0}^{l+1}\tau}\sum_{(m,n)\in S_{\gamma_{0}}}\frac{N_{\alpha_{1},\alpha_{2}}(\gamma_{0}^{-l}z,1)^{k-1}}{j(\gamma_{0}^{l},\gamma_{0}^{-l}z)^{2k}(mz+n)^{2k}}\,d(\gamma_{0}^{-l}z)$
		$\displaystyle=\sum_{l\in\mathbb{Z}}\int_{\gamma_{0}^{l}\tau}^{\gamma_{0}^{l+1}\tau}\sum_{(m,n)\in S_{\gamma_{0}}}\frac{N_{\alpha_{1},\alpha_{2}}(z,1)^{k-1}}{(mz+n)^{2k}}\,dz.$

Set $\alpha_{0}:=\alpha_{1}/\alpha_{2}\in F\subset\mathbb{R}$ . Then the point $\alpha_{0}\in\mathbb{R}$ (resp. $\alpha^{\prime}_{0}$ ) is the attractive fixed point (resp. repelling fixed point) of the hyperbolic matrix $\gamma_{0}\in\Gamma$ , i.e., we have $\lim_{l\to\infty}\gamma_{0}^{l}\tau=\alpha_{0}$ and $\lim_{l\to\infty}\gamma_{0}^{-l}\tau=\alpha^{\prime}_{0}$ in $\mathbb{P}^{1}(\mathbb{C})$ for any element $\tau\in\mathbb{H}$ . Hence we obtain

	$\displaystyle\sum_{l\in\mathbb{Z}}\int_{\gamma_{0}^{l}\tau}^{\gamma_{0}^{l+1}\tau}\sum_{(m,n)\in S_{\gamma_{0}}}\frac{N_{\alpha_{1},\alpha_{2}}(z,1)^{k-1}}{(mz+n)^{2k}}\,dz$	$\displaystyle=\int_{\alpha^{\prime}_{0}}^{\alpha_{0}}\sum_{(m,n)\in S_{\gamma_{0}}}\frac{N_{\alpha_{1},\alpha_{2}}(z,1)^{k-1}}{(mz+n)^{2k}}\,dz$
		$\displaystyle=\frac{N_{F/\mathbb{Q}}(\alpha_{2})^{k-1}}{(N\mathfrak{a})^{k-1}}\sum_{(m,n)\in S_{\gamma_{0}}}\int_{\alpha^{\prime}_{0}}^{\alpha_{0}}\frac{((\alpha_{0}-z)(z-\alpha^{\prime}_{0}))^{k-1}}{(mz+n)^{2k}}\,dz.$

By using Lemma 9.11, we find

\displaystyle\int_{\alpha^{\prime}_{0}}^{\alpha_{0}}\frac{((\alpha_{0}-z)(z-\alpha^{\prime}_{0}))^{k-1}}{(mz+n)^{2k}}\,dz=\frac{((k-1)!)^{2}}{(2k-1)!}\frac{(\alpha_{0}-\alpha^{\prime}_{0})^{2k-1}}{N_{F/\mathbb{Q}}(m\alpha_{0}+n)^{k}}.

Note that we have the identity $\alpha_{1}\alpha^{\prime}_{2}-\alpha^{\prime}_{1}\alpha_{2}=\sqrt{D_{\mathcal{O}}}N\mathfrak{a}$ , and this shows that

\displaystyle\frac{N_{F/\mathbb{Q}}(\alpha_{2})^{k-1}}{(N\mathfrak{a})^{k-1}}\sum_{(m,n)\in S_{\gamma_{0}}}\frac{(\alpha_{0}-\alpha^{\prime}_{0})^{2k-1}}{N_{F/\mathbb{Q}}(m\alpha_{0}+n)^{k}}=D_{\mathcal{O}}^{k-1/2}(N\mathfrak{a})^{k}\sum_{\alpha\in(\mathfrak{a}\!-\!\{0\})/\mathcal{O}_{+}^{\times}}\frac{1}{N_{F/\mathbb{Q}}(\alpha)^{k}}.

For any subset $X\subset F$ , we put

	$\displaystyle X_{+}$	$\displaystyle:=\{\alpha\in X\mid\alpha>0,\,\alpha^{\prime}>0\},$
	$\displaystyle X_{-}$	$\displaystyle:=\{\alpha\in X\mid\alpha>0,\,\alpha^{\prime}<0\}.$

Let $\mathcal{J}\in Cl_{\mathcal{O}}^{+}$ denote the ideal class containing the principal ideal $(\sqrt{D_{\mathcal{O}}})\subset\mathcal{O}$ . Note that $\mathcal{J}^{-1}=\mathcal{J}$ in $Cl_{\mathcal{O}}^{+}$ . Then we further compute

	$\displaystyle(N\mathfrak{a})^{k}\sum_{\alpha\in(\mathfrak{a}\!-\!\{0\})/\mathcal{O}_{+}^{\times}}\frac{1}{N_{F/\mathbb{Q}}(\alpha)^{k}}$	$\displaystyle=\sum_{\alpha\in\mathfrak{a}_{+}/\mathcal{O}_{+}^{\times}}\frac{2(N\mathfrak{a})^{k}}{N_{F/\mathbb{Q}}(\alpha)^{k}}+\sum_{\alpha\in\mathfrak{a}_{-}/\mathcal{O}_{+}^{\times}}\frac{2(N\mathfrak{a})^{k}}{N_{F/\mathbb{Q}}(\alpha)^{k}}$
		$\displaystyle=\sum_{\alpha\in\mathfrak{a}_{+}/\mathcal{O}_{+}^{\times}}\frac{2(N\mathfrak{a})^{k}}{N_{F/\mathbb{Q}}(\alpha)^{k}}+(-1)^{k}\sum_{\alpha\in(\sqrt{D_{\mathcal{O}}}\mathfrak{a})_{+}/\mathcal{O}_{+}^{\times}}\frac{2N(\sqrt{D_{\mathcal{O}}}\mathfrak{a})^{k}}{N_{F/\mathbb{Q}}(\alpha)^{k}}$
		$\displaystyle=2\left(\zeta_{\mathcal{O}}(\mathcal{A}^{-1},k)+(-1)^{k}\zeta_{\mathcal{O}}(\mathcal{J}\mathcal{A}^{-1},k)\right).$

Now, we recall the functional equations of the partial zeta functions. Set

	$\displaystyle\Lambda_{\mathcal{O}}^{+}(\mathcal{A}^{-1},s)$	$\displaystyle:=\pi^{-s}\Gamma\left(\frac{s}{2}\right)^{2}D_{\mathcal{O}}^{\frac{s-1}{2}}\left(\zeta_{\mathcal{O}}(\mathcal{A}^{-1},s)+\zeta_{\mathcal{O}}(\mathcal{A}^{-1}\mathcal{J},s)\right),$
	$\displaystyle\Lambda_{\mathcal{O}}^{-}(\mathcal{A}^{-1},s)$	$\displaystyle:=\pi^{-s}\Gamma\left(\frac{s+1}{2}\right)^{2}D_{\mathcal{O}}^{\frac{s}{2}}\left(\zeta_{\mathcal{O}}(\mathcal{A}^{-1},s)-\zeta_{\mathcal{O}}(\mathcal{A}^{-1}\mathcal{J},s)\right).$

Then we have

	$\displaystyle\Lambda_{\mathcal{O}}^{+}(\mathcal{A}^{-1},s)$	$\displaystyle=\Lambda_{\mathcal{O}}^{+}(\mathcal{A}^{-1},1-s),$
	$\displaystyle\Lambda_{\mathcal{O}}^{-}(\mathcal{A}^{-1},s)$	$\displaystyle=\Lambda_{\mathcal{O}}^{-}(\mathcal{A}^{-1},1-s).$

See [DIT18, Equations (59), (60)] or [Scz93, p. 545] for example. Although [DIT18] deals only with the maximal orders, we can apply the same argument to general orders. See also [Sie80], [Duk23, (4.19)] or [VZ13, p. 42]. Using these functional equations, we find

D_{\mathcal{O}}^{k-1/2}\left(\zeta_{\mathcal{O}}(\mathcal{A}^{-1},k)+(-1)^{k}\zeta_{\mathcal{O}}(\mathcal{J}\mathcal{A}^{-1},k)\right)=\frac{(2\pi)^{2k}}{2((k-1)!)^{2}}\zeta_{\mathcal{O}}(\mathcal{A}^{-1},1-k).

Therefore, by also using the functional equation for the Riemann zeta function (see, for example, [Hid93, p. 29]), we obtain

\displaystyle\langle\mathrm{Eis}_{2k-2},\mathfrak{z}_{\mathcal{O},k}(\mathcal{A})\rangle=\frac{(2\pi)^{2k}}{2(2k-1)!\zeta(2k)}\zeta_{\mathcal{O}}(\mathcal{A}^{-1},1-k)=(-1)^{k}\frac{\zeta_{\mathcal{O}}(\mathcal{A}^{-1},1-k)}{\zeta(1-2k)}.

∎

We define the positive integer $J_{2k}$ by

J_{2k}:=\text{the denominator of }\zeta(1-2k).

Corollary 9.12.

Let $F$ be a real quadratic field, $\mathcal{O}\subset F$ be an order in $F$ , and let $\mathcal{A}\in Cl_{\mathcal{O}}^{+}$ be a narrow ideal class of $\mathcal{O}$ . Then for any integer $k\geq 2$ , we have

J_{2k}\zeta_{\mathcal{O}}(\mathcal{A},1-k)\in\mathbb{Z}.

Proof.

By Proposition 9.10, we have

N_{2k}\langle\mathrm{Eis}_{2k-2},\mathfrak{z}_{\mathcal{O},k}(\mathcal{A}^{-1})\rangle=\pm N_{2k}\frac{\zeta_{\mathcal{O}}(\mathcal{A},1-k)}{\zeta(1-2k)}=\pm J_{2k}\zeta_{\mathcal{O}}(\mathcal{A},1-k).

Since $N_{2k}\langle\mathrm{Eis}_{2k-2},\mathfrak{z}_{\mathcal{O},k}(\mathcal{A}^{-1})\rangle\in\mathbb{Z}$ by Theorem 2.13, we obtain $J_{2k}\zeta_{\mathcal{O}}(\mathcal{A},1-k)\in\mathbb{Z}$ . ∎

Remark 9.13.

By Proposition 9.4 ([Duk23, Lemma 6]) and Proposition 9.10, we see that Duke’s conjecture 9.2 is equivalent to Corollary 9.12.

Remark 9.14.

As for the denominator of the special values of the Dedekind zeta functions of real quadratic fields, or more generally of totally real fields, the same (a slightly stronger at $p=2$ ) universal upper bound was obtained by Serre in the paper [Ser73, §2, Théorème 6]. Moreover, if we fix a totally real field $F$ , then a more refined description for the denominators and even for the numerators of the special values of the Dedekind zeta function of $F$ is obtained from the classical Iwasawa main conjecture proved by Wiles in [Wil90] (see [Kol04]).

9.3. Sharpness of the universal upper bound in Corollary 9.12

Let $k\geq 2$ be an integer. We define a $\mathbb{Z}$ -submodule $\mathfrak{Z}_{k}\subset H_{1}(Y,\mathcal{M}_{2k-2})$ to be the $\mathbb{Z}$ -submodule generated by homology classes of the form $\mathfrak{z}_{\mathcal{O},k}(\mathcal{A})$ , that is,

\mathfrak{Z}_{k}:=\langle\,\mathfrak{z}_{\mathcal{O},k}(\mathcal{A})\mid\textrm{$\mathcal{O}$ is an order of a real quadratic field and $\mathcal{A}\in Cl_{\mathcal{O}}^{+}$ }\rangle_{\mathbb{Z}}.

This subsection is devoted to proving the following theorem.

Theorem 9.15.

We have $\displaystyle\left\langle\mathrm{Eis}_{2k-2},\mathfrak{Z}_{k}\right\rangle=\frac{1}{N_{2k}}\mathbb{Z}$ .

Theorem 9.15 has the following interesting application.

Corollary 9.16.

The universal bound in Corollary 9.12 is sharp, namely, for any prime number $p$ , there exist an order $\mathcal{O}$ of a real quadratic field and a narrow ideal class $\mathcal{A}\in Cl_{\mathcal{O}}^{+}$ such that

\mathrm{ord}_{p}(J_{2k}\zeta_{\mathcal{O}}(\mathcal{A},1-k))=0.

In other words, we have

J_{2k}=\min\left\{J\in\mathbb{Z}_{>0}\,\,\middle|\,\,\begin{aligned} J\zeta_{\mathcal{O}}(\mathcal{A},1-k)\in\mathbb{Z}\text{ for all orders $\mathcal{O}$ in all real quadratic fields}\\ \text{and narrow ideal classes $\mathcal{A}\in Cl_{\mathcal{O}}^{+}$}\end{aligned}\,\right\}.

Proof.

Let $p$ be a prime number. By the definition of the module $\mathfrak{Z}_{k}$ and Theorem 9.15, one can find an order $\mathcal{O}$ of a real quadratic field and a narrow ideal class $\mathcal{A}\in Cl_{\mathcal{O}}^{+}$ such that

\mathrm{ord}_{p}(\langle\mathrm{Eis}_{2k-2},\mathfrak{z}_{\mathcal{O},k}(\mathcal{A}^{-1})\rangle)=-\mathrm{ord}_{p}(N_{2k}).

Since $\zeta(1-2k)=\pm N_{2k}/J_{2k}$ , Proposition 9.10 shows that

	$\displaystyle 0$	$\displaystyle=\mathrm{ord}_{p}(N_{2k})+\mathrm{ord}_{p}(\langle\mathrm{Eis}_{2k-2},\mathfrak{z}_{\mathcal{O},k}(\mathcal{A}^{-1})\rangle)$
		$\displaystyle=\mathrm{ord}_{p}(N_{2k})-\mathrm{ord}_{p}(\zeta(1-2k))+\mathrm{ord}_{p}(\zeta_{\mathcal{O}}(\mathcal{A},1-k))$
		$\displaystyle=\mathrm{ord}_{p}(J_{2k}\zeta_{\mathcal{O}}(\mathcal{A},1-k)).$

∎

9.3.1. Preparations for proving Theorem 9.15

Let $N\geq 1$ be an integer and define

\Gamma_{1}(N):=\left\{\begin{pmatrix}a&b\\ c&d\end{pmatrix}\in\Gamma\,\middle|\,a-1\equiv c\equiv d-1\equiv 0\pmod{N}\right\}.

We also put

Y_{1}(N):=\Gamma_{1}(N)\backslash\mathbb{H},\,\,\,\,\,\,Y_{1}(N)^{\mathrm{BS}}:=\Gamma_{1}(N)\backslash\mathbb{H}^{\mathrm{BS}},\,\,\,\,\,\,\partial Y_{1}(N)^{\mathrm{BS}}:=Y_{1}(N)^{\mathrm{BS}}\!-\!Y_{1}(N).

We note that the similar facts in §2.1 and §2.2 hold true for the congruence subgroup $\Gamma_{1}(N)$ . Moreover, the Hecke operators $T_{p}$ ( $p\nmid N$ ) and $U_{p}$ ( $p\mid N$ ) act on the homology group $H_{1}(Y_{1}(N),\mathcal{M}_{2k-2})$ .

Let $k\geq 1$ be an integer. For any hyperbolic matrix $\gamma\in\Gamma_{1}(N)$ (i.e., $|\mathrm{trace}(\gamma)|>2$ ), we set

\mathfrak{z}_{\Gamma_{1}(N),k}(\gamma):=[\{z,\gamma z\}\otimes Q_{\gamma}(X_{1},X_{2})^{k-1}]\in H_{1}(Y_{1}(N),\mathcal{M}_{2k-2}).

Definition 9.17.

For any integer $k\geq 1$ , we define a $\mathbb{Z}$ -submodule

\mathfrak{Z}_{\Gamma_{1}(N),k}\subset H_{1}(Y_{1}(N),\mathcal{M}_{2k-2})

\mathfrak{Z}_{\Gamma_{1}(N),k}:=\langle\,\mathfrak{z}_{\Gamma_{1}(N),k}(\gamma)\mid\gamma\in\Gamma_{1}(N)\textrm{ with }|\mathrm{trace}(\gamma)|>2\,\rangle_{\mathbb{Z}}.

Remark 9.18.

By Remark 9.8, we have $\mathfrak{Z}_{\Gamma_{1}(1),k}=\mathfrak{Z}_{k}$ .

Lemma 9.19.

For any integer $N\geq 1$ and prime number $p$ , we have

\langle\,[\{z,\gamma z\}]\mid\gamma\in\Gamma_{1}(Np)\!-\!\Gamma(p)\textrm{ with }|\mathrm{trace}(\gamma)|>2\,\rangle_{\mathbb{Z}}=\mathfrak{Z}_{\Gamma_{1}(Np),1}=H_{1}(Y_{1}(Np),\mathbb{Z}).

Proof.

It suffices to show that

\langle\,\gamma\mid\gamma\in\Gamma_{1}(Np)\!-\!\Gamma(p)\textrm{ with }|\mathrm{trace}(\gamma)|>2\,\rangle=\Gamma_{1}(Np).

Here $\langle~{}\rangle$ means the group generated by the elements inside the bracket. Moreover, since $\Gamma_{1}(2)=\langle\Gamma_{1}(6),\Gamma_{1}(10)\rangle$ , we may assume that $Np\geq 3$ . Put $\gamma:=\begin{pmatrix}1&1\\ Np&1+Np\end{pmatrix}$ . Then the quotient group $\Gamma_{1}(Np)/(\Gamma(p)\cap\Gamma_{1}(Np))$ is generated by the image of $\gamma$ . For any matrix $\gamma^{\prime}\in\Gamma(p)\cap\Gamma_{1}(Np)$ , we have $\gamma^{\prime}\gamma^{1+ap}\not\in\Gamma(p)$ for any integer $a$ . Since $\mathrm{trace}(\gamma)=Np+2>2$ and $\det(\gamma)=1$ , the matrix $\gamma$ is hyperbolic, and one can take a matrix $Q\in\mathrm{GL}_{2}(\mathbb{R})$ such that $Q^{-1}\gamma Q=\begin{pmatrix}\alpha&\\ &\alpha^{-1}\end{pmatrix}$ . If we put $Q^{-1}\gamma^{\prime}Q=:\begin{pmatrix}x&y\\ z&w\end{pmatrix}$ , then we have $\mathrm{trace}(\gamma^{\prime}\gamma^{1+ap})=x\alpha^{1+ap}+w\alpha^{-1-ap}$ . Since $\mathrm{trace}(\gamma^{\prime}\gamma^{1+ap})\equiv 2\pmod{Np}$ and $Np\geq 3$ , we have $\mathrm{trace}(\gamma^{\prime}\gamma^{1+ap})\neq 0$ . Hence the set $\{\mathrm{trace}(\gamma^{\prime}\gamma^{1+ap})\mid a\in\mathbb{Z}\}\subset\mathbb{Z}$ is infinite. Therefore we can find an integer $a$ such that $|\mathrm{trace}(\gamma^{\prime}\gamma^{1+ap})|>2$ . ∎

Next, we recall an important result proved by Hida (see [Hid86, Corollary 4.5] or [Hid88, Corollary 8.2]). Take an integer $N\geq 1$ and a prime number $p$ , and put $q:=p^{\mathrm{ord}_{p}(Np)}$ . Then we have a $\Gamma_{1}(q)$ -homomorphism

j\colon\mathbb{Z}/(q)\longrightarrow\mathcal{M}_{2k-2}\otimes\mathbb{Z}/(q);b\mapsto bX_{2}^{2k-2},

which induces a Hecke-equivariant homomorphism

j_{*}\colon H_{1}(Y_{1}(Np),\mathbb{Z}/(q))\longrightarrow H_{1}(Y_{1}(Np),\mathcal{M}_{2k-2,p}\otimes\mathbb{Z}/(q)).

Proposition 9.20.

When $Np\geq 4$ , the homomorphism $j$ induces a Hecke-equivariant isomorphism

j_{*}\colon H_{1}^{\mathrm{ord}}(Y_{1}(Np),\mathbb{Z}/(q))\stackrel{{\scriptstyle\sim}}{{\longrightarrow}}H_{1}^{\mathrm{ord}}(Y_{1}(Np),\mathcal{M}_{2k-2,p}\otimes\mathbb{Z}/(q)).

Here we define $H_{1}^{\mathrm{ord}}(Y_{1}(Np),-):=e_{U_{p}}H_{1}(Y_{1}(Np),-)$ .

Proof.

The proof of this proposition is essentially the same as that of [Hid93, Theorem 2 in §7.2] for cohomology groups. We note that $\Gamma_{1}(Np)$ is torsion-free since $Np\geq 4$ . Hence any short exact sequence of $\Gamma_{1}(Np)$ -modules induces a long exact sequence in homology.

If we put

\mathcal{C}:=(\mathcal{M}_{2k-2}/\mathbb{Z}X_{2}^{2k-2})\otimes\mathbb{Z}/(q),

then the short exact sequence $0\longrightarrow\mathbb{Z}/(q)\overset{j}{\longrightarrow}\mathcal{M}_{2k-2}\otimes\mathbb{Z}/(q)\longrightarrow\mathcal{C}\longrightarrow 0$ induces an exact sequence of $\mathbb{Z}/(q)$ -modules

\displaystyle H_{2}(Y_{1}(Np),\mathcal{C})\longrightarrow H_{1}(Y_{1}(Np),\mathbb{Z}/(q))\stackrel{{\scriptstyle j_{*}}}{{\longrightarrow}}H_{1}(Y_{1}(N),\mathcal{M}_{2k-2}\otimes\mathbb{Z}/(q))\longrightarrow H_{1}(Y_{1}(Np),\mathcal{C}).

Since the operator $U_{p}$ is defined by $U_{p}=\sum_{u=0}^{p-1}\begin{pmatrix}1&u\\ 0&p\end{pmatrix}$ , we have

	$\displaystyle\begin{pmatrix}1&u\\ 0&p\end{pmatrix}\cdot P(X_{1},X_{2})$	$\displaystyle\equiv P(-uX_{2},X_{2})\pmod{p}$
		$\displaystyle\in\mathbb{F}_{p}X_{2}^{2k-2}$

for any polynomial $P\in\mathcal{M}_{2k-2}$ . In other words, we have

\begin{pmatrix}1&u\\ 0&p\end{pmatrix}\left(\mathcal{M}_{2k-2}/\mathbb{Z}X_{2}^{2k-1}\right)\subset p\left(\mathcal{M}_{2k-2}/\mathbb{Z}X_{2}^{2k-1}\right),

and this shows that $H_{1}^{\mathrm{ord}}(Y_{1}(N),\mathcal{C})=H_{2}^{\mathrm{ord}}(Y_{1}(N),\mathcal{C})=0$ , which completes the proof. ∎

Lemma 9.21.

Let $N$ be an integer and let $p$ be a prime number. Set $q:=p^{\mathrm{ord}_{p}(Np)}$ . Then for any matrix $\gamma\in\Gamma_{1}(Np)\!-\!\Gamma(p)$ , we have

j_{*}(\mathfrak{z}_{\Gamma_{1}(Np),1}(\gamma)\bmod{q})\in(\mathbb{Z}/(q))^{\times}\cdot\mathfrak{z}_{\Gamma_{1}(Np),k}(\gamma).

Proof.

If we put $\gamma:=\begin{pmatrix}a&b\\ c&d\end{pmatrix}$ , then we have

\displaystyle Q_{\gamma}(X_{1},X_{2})=-\frac{\operatorname{sgn}(a+d)}{\mathrm{gcd}(c,a-d,b)}(cX_{1}^{2}-(a-d)X_{1}X_{2}-bX_{2}^{2}).

Since $\gamma\in\Gamma_{1}(Np)\!-\!\Gamma(p)$ , we have $c\equiv a-d\equiv 0\pmod{q}$ and $b\not\equiv 0\pmod{p}$ , which shows that

Q_{\gamma}(X_{1},X_{2})\equiv\pm\frac{b}{\mathrm{gcd}(c,a-d,b)}X_{2}^{2}\pmod{q}

and

	$\displaystyle j_{*}(\mathfrak{z}_{\Gamma_{1}(Np),1}(\gamma)\bmod{q})$	$\displaystyle=\left(\pm\frac{\mathrm{gcd}(c,a-d,b)}{b}\right)^{k-1}\mathfrak{z}_{\Gamma_{1}(Np),k}(\gamma)\pmod{q}$
		$\displaystyle\in(\mathbb{Z}/(q))^{\times}\cdot\mathfrak{z}_{\Gamma_{1}(Np),k}(\gamma).$

∎

Theorem 9.22.

For any integer $N\geq 1$ and prime number $p$ satisfying $Np\geq 4$ , we have

e_{U_{p}}(\mathfrak{Z}_{\Gamma_{1}(Np),k}\otimes\mathbb{Z}_{p})=H_{1}^{\mathrm{ord}}(Y_{1}(Np),\mathcal{M}_{2k-2,p}).

Proof.

We first note that $e_{U_{p}}H_{0}(Y_{1}(Np),\mathcal{M}_{2k-2,p})=e_{U_{p}}((\mathcal{M}_{2k-2,p})_{\Gamma_{1}(Np)})$ vanishes since $U_{p}\cdot X_{2}^{2k-2}=pX_{2}^{2k-2}$ and $\begin{pmatrix}1&u\\ 0&p\end{pmatrix}\cdot P(X_{1},X_{2})\equiv P(-uX_{2},X_{2})\pmod{p}$ . This fact implies that

H_{1}^{\mathrm{ord}}(Y_{1}(Np),\mathcal{M}_{2k-2,p})\otimes\mathbb{Z}/(q)\stackrel{{\scriptstyle\sim}}{{\longrightarrow}}H_{1}^{\mathrm{ord}}(Y_{1}(Np),\mathcal{M}_{2k-2,p}\otimes\mathbb{Z}/(q)).

Here $q:=p^{\mathrm{ord}_{p}(Np)}$ . Hence this theorem follows from Proposition 9.20 and Lemmas 9.19 and 9.21. ∎

9.3.2. Proof of Theorem 9.15

Let $k\geq 2$ be an integer. For any positive integers $M$ and $N$ with $M\mid N$ , we denote by

	$\displaystyle\pi_{*}^{N,M}$	$\displaystyle\colon H_{1}(Y_{1}(N),\mathcal{M}_{2k-2})\longrightarrow H_{1}(Y_{1}(M),\mathcal{M}_{2k-2}),$
	$\displaystyle\pi^{*}_{M,N}$	$\displaystyle\colon H^{1}(Y_{1}(M),\mathcal{M}_{2k-2})\longrightarrow H^{1}(Y_{1}(N),\mathcal{M}_{2k-2})$

the homomorphisms induced by the natural projection $Y_{1}(N)\longrightarrow Y_{1}(M);z\mapsto z$ .

Corollary 9.23.

For any integer $N\geq 1$ and prime number $p$ , we have

\mathbb{Z}_{p}\subset\langle\mathrm{Eis}_{2k-2},\pi^{Np,1}_{*}(e_{U_{p}}(\mathfrak{Z}_{\Gamma_{1}(Np),k}\otimes\mathbb{Z}_{p}))\rangle.

Proof.

Take an element $\tau\in\mathbb{H}$ . Since $U_{p}([\{\tau,\tau+1\}\otimes X_{2}^{2k-2}])=[\{\frac{\tau}{p},\frac{\tau}{p}+1\}\otimes X_{2}^{2k-2}]=[\{\tau,\tau+1\}\otimes X_{2}^{2k-2}]$ , we have

[\{\tau,\tau+1\}\otimes X_{2}^{2k-2}]\in H_{1}^{\mathrm{ord}}(\partial Y_{1}(Np)^{\mathrm{BS}},\mathcal{M}_{2k-2,p})\subset H_{1}^{\mathrm{ord}}(Y_{1}(Np),\mathcal{M}_{2k-2,p}).

On the other hand, since

\pi^{Np,1}_{*}([\{\tau,\tau+1\}\otimes X_{2}^{2k-2}])=[\{\tau,\tau+1\}\otimes X_{2}^{2k-2}]\in H_{1}(Y,\mathcal{M}_{2k-2}),

we have

\langle\mathrm{Eis}_{2k-2},\pi^{Np,1}_{*}([\{\tau,\tau+1\}\otimes X_{2}^{2k-2}])\rangle=1,

Therefore, first when $Np\geq 4$ , Theorem 9.22 implies this lemma. When $Np\leq 3$ , we have $N=1$ and $p|6$ . Then this case follows from the case $N=3$ and $p\mid 6$ since $\pi^{3p,1}_{*}(e_{U_{p}}(\mathfrak{Z}_{\Gamma_{1}(3p),k}\otimes\mathbb{Z}_{p})\subset\pi^{p,1}_{*}(e_{U_{p}}(\mathfrak{Z}_{\Gamma_{1}(p),k}\otimes\mathbb{Z}_{p})$ . ∎

Lemma 9.24.

Let $N\geq 1$ be an integer and let $p$ be a prime number. Then for any homology class $x\in H_{1}(Y_{1}(Np),\mathcal{M}_{2k-2})\otimes\mathbb{C}_{p}$ we have

e_{T_{p}}\pi_{*}^{Np,1}(e_{U_{p}}x)=\pi_{*}^{Np,1}(e_{U_{p}}x).

Proof.

By using the formal duality, it suffices to show that

e_{U_{p}^{\prime}}\pi_{1,Np}^{*}(e_{T_{p}^{\prime}}y)=e_{U_{p}^{\prime}}\pi_{1,Np}^{*}(y)

for any cohomology class $y\in H^{1}(Y,\mathcal{M}_{2k-2})\otimes\mathbb{C}_{p}$ . This claim is well-known (see [Gou92, Lemma 2] for example). ∎

Corollary 9.25.

For any prime number $p\geq 5$ , we have

\langle\mathrm{Eis}_{2k-2},\pi_{*}^{p,1}(e_{U_{p}}(\mathfrak{Z}_{\Gamma_{1}(p),k}\otimes\mathbb{Z}_{p}))\rangle=\frac{1}{N_{2k}}\mathbb{Z}_{p}.

Proof.

Take a prime number $p\geq 5$ . Then Theorem 9.22 shows that

\langle\mathrm{Eis}_{2k-2},\pi_{*}^{p,1}(e_{U_{p}}(\mathfrak{Z}_{\Gamma_{1}(p),k}\otimes\mathbb{Z}_{p}))\rangle=\langle\mathrm{Eis}_{2k-2},\pi_{*}^{p,1}(H_{1}^{\mathrm{ord}}(Y_{1}(p),\mathcal{M}_{2k-2,p}))\rangle.

By Lemma 9.24, we have a natural homomorphism

\pi_{*}^{p,1}\colon H_{1}^{\mathrm{ord}}(Y_{1}(p),\mathcal{M}_{2k-2,p})/(\mathrm{torsion})\longrightarrow H_{1}^{\mathrm{ord}}(Y,\mathcal{M}_{2k-2,p})/(\mathrm{torsion}),

which is the dual of the homomorphism

H^{1}_{\mathrm{ord}}(Y,\mathcal{M}_{2k-2,p})\longrightarrow H^{1}_{\mathrm{ord}}(Y_{1}(p),\mathcal{M}_{2k-2,p});y\mapsto e_{U_{p}^{\prime}}\pi^{*}_{1,p}(y).

The fact that the index $[\Gamma_{0}(p)\colon\Gamma_{1}(p)]=p-1$ is relatively prime to $p$ together with the isomorphism (6.4) implies that $\pi_{*}^{p,1}\colon H_{1}^{\mathrm{ord}}(Y_{1}(p),\mathcal{M}_{2k-2,p})/(\mathrm{torsion})\longrightarrow H_{1}^{\mathrm{ord}}(Y,\mathcal{M}_{2k-2,p})/(\mathrm{torsion})$ is surjective. Hence we have

\displaystyle\langle\mathrm{Eis}_{2k-2},\pi_{*}^{p,1}(H_{1}^{\mathrm{ord}}(Y_{1}(p),\mathcal{M}_{2k-2,p}))\rangle=\langle\mathrm{Eis}_{2k-2},H_{1}^{\mathrm{ord}}(Y,\mathcal{M}_{2k-2,p})\rangle.

Therefore, this corollary follows from Theorem 2.13 and Lemma 5.5. ∎

We note that for any prime number $p$ , the operators $V_{p}:=\begin{pmatrix}p&0\\ 0&1\end{pmatrix}$ and $V_{p}^{\prime}:=\widetilde{\begin{pmatrix}p&0\\ 0&1\end{pmatrix}}$ induce homomorphisms

	$\displaystyle V_{p}$	$\displaystyle\colon H_{1}(Y_{1}(Np),\mathcal{M}_{2k-2})\longrightarrow H_{1}(Y_{1}(N),\mathcal{M}_{2k-2}),$
	$\displaystyle V_{p}^{\prime}$	$\displaystyle\colon H^{1}(Y_{1}(N),\mathcal{M}_{2k-2})\longrightarrow H^{1}(Y_{1}(Np),\mathcal{M}_{2k-2}).$

Lemma 9.26.

For any prime number $p$ and integer $N\geq 1$ , we have

e_{U_{p}^{\prime}}(\pi^{*}_{1,Np}(\mathrm{Eis}_{2k-2}))=\frac{1}{1-p^{2k-1}}\left(\pi^{*}_{1,Np}(\mathrm{Eis}_{2k-2})-\pi^{*}_{1,N}(\mathrm{Eis}_{2k-2})|V_{p}^{\prime}\right).

Proof.

We put

	$\displaystyle\mathrm{Eis}_{2k-2}^{(1)}$	$\displaystyle:=\frac{1}{1-p^{2k-1}}\left(\pi^{}_{1,Np}(\mathrm{Eis}_{2k-2})-\pi^{}_{1,N}(\mathrm{Eis}_{2k-2})\|V_{p}^{\prime}\right),$
	$\displaystyle\mathrm{Eis}_{2k-2}^{(p^{2k-1})}$	$\displaystyle:=\frac{1}{1-p^{2k-1}}\left(-p^{2k-1}\pi^{}_{1,Np}(\mathrm{Eis}_{2k-2})+\pi^{}_{1,N}(\mathrm{Eis}_{2k-2})\|V_{p}^{\prime}\right).$

Since $\pi_{1,Np}^{*}T_{p}^{\prime}=U_{p}^{\prime}\pi_{1,Np}^{*}+V_{p}^{\prime}\pi_{1,N}^{*}$ and $U_{p}^{\prime}V_{p}^{\prime}\pi_{1,N}^{*}=p^{2k-1}\pi_{1,Np}^{*}$ , the relation that $\mathrm{Eis}_{2k-2}|T_{p}^{\prime}=(1+p^{2k-1})\mathrm{Eis}_{2k-2}$ shows that

\displaystyle\mathrm{Eis}_{2k-2}^{(1)}|U_{p}^{\prime}=\mathrm{Eis}_{2k-2}^{(1)}\,\,\,\textrm{ and }\,\,\,\mathrm{Eis}_{2k-2}^{(p^{2k-1})}|U_{p}^{\prime}=p^{2k-1}\mathrm{Eis}_{2k-2}^{(p^{2k-1})}.

Since $\pi^{*}_{1,Np}(\mathrm{Eis}_{2k-2})=\mathrm{Eis}_{2k-2}^{(1)}+\mathrm{Eis}_{2k-2}^{(p^{2k-1})}$ , these facts imply this lemma. ∎

Lemma 9.27.

For any integer $N\geq 1$ and a prime number $p$ , we have

\langle\mathrm{Eis}_{2k-2},\pi^{Np,1}_{*}(e_{U_{p}}(\mathfrak{Z}_{\Gamma_{1}(Np),k}\otimes\mathbb{Z}_{p}))\rangle\subset\langle\mathrm{Eis}_{2k-2},\mathfrak{Z}_{\Gamma_{1}(1),k}\rangle\mathbb{Z}_{p}.

Proof.

Lemma 9.26 implies that

	$\displaystyle\langle e_{U_{p}^{\prime}}\pi^{*}_{Np,1}(\mathrm{Eis}_{2k-2}),\mathfrak{Z}_{\Gamma_{1}(Np),k}\rangle$	$\displaystyle\subset\langle\pi^{}_{1,Np}(\mathrm{Eis}_{2k-2}),\mathfrak{Z}_{\Gamma_{1}(Np),k}\rangle\mathbb{Z}_{p}+\langle V_{p}^{\prime}\pi^{}_{1,N}(\mathrm{Eis}_{2k-2}),\mathfrak{Z}_{\Gamma_{1}(Np),k}\rangle\mathbb{Z}_{p}$
		$\displaystyle\subset\langle\mathrm{Eis}_{2k-2},\mathfrak{Z}_{\Gamma_{1}(1),k}\rangle\mathbb{Z}_{p}+\langle\mathrm{Eis}_{2k-2},\pi_{*}^{N,1}(V_{p}\mathfrak{Z}_{\Gamma_{1}(Np),k})\rangle\mathbb{Z}_{p}.$

Hence we have

	$\displaystyle\langle\mathrm{Eis}_{2k-2},\pi^{Np,1}_{*}(e_{U_{p}}(\mathfrak{Z}_{\Gamma_{1}(Np),k}\otimes\mathbb{Z}_{p}))\rangle$	$\displaystyle=\langle e_{U_{p}^{\prime}}\pi^{*}_{1,Np}(\mathrm{Eis}_{2k-2}),\mathfrak{Z}_{\Gamma_{1}(N),k}\rangle\mathbb{Z}_{p}$
		$\displaystyle\subset\langle\mathrm{Eis}_{2k-2},\mathfrak{Z}_{\Gamma_{1}(1),k}\rangle\mathbb{Z}_{p}+\langle\mathrm{Eis}_{2k-2},\pi_{*}^{N,1}(V_{p}\mathfrak{Z}_{\Gamma_{1}(Np),k})\rangle\mathbb{Z}_{p}.$

Therefore, it suffices to show that $V_{p}\mathfrak{Z}_{\Gamma_{1}(Np),k}\subset\mathfrak{Z}_{\Gamma_{1}(N),k}$ . Let $\gamma\in\Gamma_{1}(Np)$ be a matrix. Then we have $\gamma_{p}:=\begin{pmatrix}p&0\\ 0&1\end{pmatrix}\gamma\begin{pmatrix}p&0\\ 0&1\end{pmatrix}^{-1}\in\Gamma_{1}(N)$ . Moreover, by the definitions of $\mathfrak{z}_{\Gamma_{1}(N),k}$ and $\mathfrak{z}_{\Gamma_{1}(Np),k}$ , we obtain

V_{p}\cdot\mathfrak{z}_{\Gamma_{1}(Np),k}(\gamma)\in\mathbb{Z}\mathfrak{z}_{\Gamma_{1}(N),k}(\gamma_{p}).

In particular, we have $V_{p}\mathfrak{Z}_{\Gamma_{1}(Np),k}\subset\mathfrak{Z}_{\Gamma_{1}(N),k}$ . ∎

Proof of Theorem 9.15.

By Theorem 2.13 and Remark 9.18, we only need to show that

\frac{1}{N_{2k}}\mathbb{Z}_{p}\subset\langle\mathrm{Eis}_{2k-2},\mathfrak{Z}_{\Gamma_{1}(1),k}\rangle\mathbb{Z}_{p}

for any prime number $p$ . When $p\geq 5$ , this claim follows from Corollary 9.25 and Lemma 9.27 applied to $N=1$ . Suppose $p=2$ or $p=3$ . Then Since the primes $2$ and $3$ are regular, these primes does not divide $N_{2k}$ . Hence, Corollary 9.23 and Lemma 9.27 show that

\frac{1}{N_{2k}}\mathbb{Z}_{p}=\mathbb{Z}_{p}\subset\langle\mathrm{Eis}_{2k-2},\mathfrak{Z}_{\Gamma_{1}(1),k}\rangle\mathbb{Z}_{p}.

∎

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Harder’s denominator problem for SL2​(ℤ)\mathrm{SL}_{2}(\mathbb{Z}) and its applications

Abstract.

Key words and phrases:

2020 Mathematics Subject Classification:

1. Introduction

1.1. The denominator of Eisenstein classes for SL2​(ℤ)\mathrm{SL}_{2}(\mathbb{Z})

Theorem 1.1 (Harder [Har, Theorem 5.1.2]).

Remark 1.2.

Remark 1.3.

1.2. Reformulation in terms of the holomorphic Eisenstein series

Theorem 1.4 (Lemma 2.8, Proposition 2.11, and Theorem 2.13).

Remark 1.5.

1.3. Strategy of the proof of Theorem 1.4

1.4. Applications to Duke’s conjecture and to the special values of the partial zeta functions of real quadratic fields

1.4.1. Duke’s conjecture

Theorem 1.6 (Corollary 9.5).

Remark 1.7.

1.4.2. The denominators of the partial zeta functions of real quadratic fields

Proposition 1.8 (Corollary 9.12).

Theorem 1.9 (Corollary 9.16).

Remark 1.10.

Remark 1.11.

Acknowledgements

2. Preliminaries and the Eisenstein class

2.1. Definitions of Modular curve and Borel–Serre compactification

Remark 2.1.

2.2. Modular symbols and (co)homology

2.3. M2+​(ℤ)M_{2}^{+}(\mathbb{Z})-modules ℳn\mathcal{M}_{n} and ℳn♭\mathcal{M}_{n}^{\flat}

Remark 2.2.

2.4. Hecke operators

Definition 2.3.

Lemma 2.4.

Proof.

2.5. Formal duality

2.6. Eichler–Shimura homomorphism

Remark 2.5.

Remark 2.6.

Lemma 2.7.

2.7. Definition of the Eisenstein class

Lemma 2.8.

Proof.

Definition 2.9.

Remark 2.10.

2.8. Main theorem

Proposition 2.11.

Definition 2.12.

Theorem 2.13 ([Har, Theorem 5.1.2]).

Remark 2.14.

3. Construction of the cycle Tpm​(Cν​(τ))~\widetilde{T_{p}^{m}(C_{\nu}(\tau))}

3.1. Computation of Tpm​(Cν​(τ))T_{p}^{m}(C_{\nu}(\tau))

Lemma 3.1.

Proof.

Definition 3.2.

Corollary 3.3.

3.2. Computation of the boundary ∂Cν,k,j​(τ0,τ1)\partial C_{\nu,k,j}\left(\tau_{0},\tau_{1}\right)

Definition 3.4.

Definition 3.5.

Lemma 3.6.

Proof.

3.3. A cycle Tpm​(Cν​(τ))~\widetilde{T_{p}^{m}(C_{\nu}(\tau))} in ℳ​𝒮​(ℍBS)⊗(ℳn⊗ℚ)\mathcal{M}\mathcal{S}(\mathbb{H}^{\mathrm{BS}})\otimes(\mathcal{M}_{n}\otimes\mathbb{Q})

3.3.1. Bernoulli polynomials

3.3.2. P†P^{\dagger} and P‡P^{\ddagger}

Lemma 3.7.

3.3.3. Definitions of polynomials Pν,k,j(1)P^{(1)}_{\nu,k,j} and Pν,k,j(0)P^{(0)}_{\nu,k,j}

Definition 3.8.

Lemma 3.9.

Proof.

3.3.4. Definition of the cycle Tpm​(Cν​(τ))~\widetilde{T_{p}^{m}(C_{\nu}(\tau))}

Definition 3.10.

Lemma 3.11.

Proof.

Lemma 3.12.

Proof.

Definition 3.13.

Definition 3.14.

Lemma 3.15.

Proof.

4. Period

Theorem 4.1.

Lemma 4.2.

Harder’s denominator problem for $\mathrm{SL}_{2}(\mathbb{Z})$ and its applications

1.1. The denominator of Eisenstein classes for $\mathrm{SL}_{2}(\mathbb{Z})$

2.3. $M_{2}^{+}(\mathbb{Z})$ -modules $\mathcal{M}_{n}$ and $\mathcal{M}_{n}^{\flat}$

3. Construction of the cycle $\widetilde{T_{p}^{m}(C_{\nu}(\tau))}$

3.1. Computation of $T_{p}^{m}(C_{\nu}(\tau))$

3.2. Computation of the boundary $\partial C_{\nu,k,j}\left(\tau_{0},\tau_{1}\right)$

3.3. A cycle $\widetilde{T_{p}^{m}(C_{\nu}(\tau))}$ in $\mathcal{M}\mathcal{S}(\mathbb{H}^{\mathrm{BS}})\otimes(\mathcal{M}_{n}\otimes\mathbb{Q})$

3.3.2. $P^{\dagger}$ and $P^{\ddagger}$

3.3.3. Definitions of polynomials $P^{(1)}_{\nu,k,j}$ and $P^{(0)}_{\nu,k,j}$

3.3.4. Definition of the cycle $\widetilde{T_{p}^{m}(C_{\nu}(\tau))}$

4.1. $\langle\mathrm{Eis}_{n},\widetilde{C}_{\nu,k,j}\left(\tau_{0},\tau_{1}\right)\rangle$

4.2. Summation over $j$

4.3. Summation over $k$ and the $p$ -adic limits

4.4. Rationality of the Eisenstein class $\mathrm{Eis}_{n}$