Eichler-Shimura theory for general congruence subgroups

Soumyadip Sahu
School of Mathematics, Tata Institute of Fundamental Research,
1 Homi Bhabha Road, Mumbai, 400005, Maharashtra, India,
Email: [email protected]

Abstract

This article aims to extend the Eichler-Shimura isomorphism theorem for cusp forms on a congruence subgroup of $\text{SL}_{2}(\mathbb{Z})$ to the whole space of modular forms. To regularize the Eichler-Shimura integral at infinity we reconstruct the standard formalism of the Eichler-Shimura theory starting from an abstract differential definition of the integral. The article also provides new proof of Hida’s twisted Eichler-Shimura isomorphism for the modular forms with nebentypus using the isomorphism theorem mentioned above. We compute the Eichler-Shimura homomorphism for a family of Eisenstein series on $\Gamma(N)$ and prove algebraicity results for the cohomology classes attached to a basis of the space of Eisenstein series on a general congruence subgroup.

MSC2020: Primary 11F67; Secondary 11F75

Keywords: Eichler-Shimura isomorphism, congruence subgroups

1 Introduction

Let $k\geq 2$ be an integer and $\mathcal{G}$ be a finite index subgroup of $\text{SL}_{2}(\mathbb{Z})$ . Suppose that $\mathcal{M}_{k}(\mathcal{G})$ , resp. $\mathcal{S}_{k}(\mathcal{G})$ , is the $\mathbb{C}$ -vector space of modular forms, resp. cusp forms, of weight $k$ on $\mathcal{G}$ . Set $\mathcal{S}^{a}_{k}(\mathcal{G})=\{\bar{f}\mid f\in\mathcal{S}_{k}(\mathcal{G})\}$ . Let $V_{k-2,\mathbb{C}}\subseteq\mathbb{C}[X]$ be the $\mathbb{C}$ -subspace of polynomials having degree $\leq k-2$ . The classical Eichler-Shimura theory constructs a $\mathbb{C}$ -linear Hecke equivariant homomorphism $\mathcal{S}_{k}(\mathcal{G})\to H^{1}(\mathcal{G},V_{k-2,\mathbb{C}})$ that formally extends to a $\mathbb{C}$ -linear Hecke equivariant map

(1.1)

\mathcal{S}_{k}(\mathcal{G})\oplus\mathcal{S}^{a}_{k}(\mathcal{G})\to H^{1}(\mathcal{G},V_{k-2,\mathbb{C}}).

The Eichler-Shimura isomorphism theorem for the cusp forms [16] asserts that the map (1.1) is injective and its image equals the parabolic subspace of $H^{1}(\mathcal{G},V_{k-2,\mathbb{C}})$ . Hida [10] employed the Eisenstein series with nebentypus to establish a version of the Eichler-Shimura isomorphism for the spaces of modular forms with nebentypus. In general, his investigations suggest that for $\Gamma_{1}(N)$ and $\Gamma_{0}(N)$ the injection (1.1) should extend to a Hecke equivariant isomorphism

(1.2)

\mathcal{M}_{k}(\mathcal{G})\oplus\mathcal{S}^{a}_{k}(\mathcal{G})\xrightarrow{\cong}H^{1}(\mathcal{G},V_{k-2,\mathbb{C}}).

One can also use Zagier’s approach towards the period polynomials of Eisenstein series [19] to formulate and prove a holomorphic variant of (1.2) on these subgroups [14]. The modular symbols provide a homological approach to this problem which allows us to establish an analogue of these results for $\Gamma_{1}(N)$ [3]. The main result of the article is as follows:

Theorem 1.1.

Let $k\geq 2$ be an integer and $\mathcal{G}$ be an arbitrary congruence subgroup of $\emph{SL}_{2}(\mathbb{Z})$ . Then the Eichler-Shimura homomorphism for $\mathcal{S}_{k}(\mathcal{G})$ admits a natural Hecke equivariant extension

\mathcal{M}_{k}(\mathcal{G})\to H^{1}(\mathcal{G},V_{k-2,\mathbb{C}})

so that (1.1) extends to an isomorphism (1.2).

The statement above furnishes a modular form avatar of the general theorems on the cohomology of arithmetic groups that describe the cohomology as a sum of the cuspidal and Eisenstein subspaces [9]. Our work provides complete proof of the result in the spirit of the Eichler-Shimura theory of cusp forms. A usual treatment of the isomorphism theorem for cusp forms involves Eichler-Shimura integrals with endpoints at the boundary of the upper half plane and its convergence requires the cuspidal decay condition. Thus a naive attempt to generalize the constructions to the whole space of modular forms runs into issues regarding convergence. To circumvent this problem we develop our formalism in terms of a differential variant of the classical Eichler-Shimura integral that appears in literature in the context of weakly holomorphic modular forms [4].

Let $\mathbb{H}$ be the upper half-plane. Suppose that $\text{Hol}(\mathbb{H})$ is the $\mathbb{C}$ -algebra of the holomorphic functions on $\mathbb{H}$ . Now suppose $k$ is an integer $\geq 2$ and $f\in\text{Hol}(\mathbb{H})$ . A weight $k$ Eichler-Shimura integral of $f$ is a polynomial $\mathbb{I}_{k}(\tau,f,X)$ of degree $k-2$ with coefficients in $\text{Hol}(\mathbb{H})$ so that

(1.3)

\partial_{\tau}\big{(}\mathbb{I}_{k}(\tau,f,X)\big{)}=f(\tau)(X-\tau)^{k-2}d\tau.

The differential equation above always has a solution determined up to an element in $V_{k-2,\mathbb{C}}$ . The familiar Eichler-Shimura integral

(1.4)

\int^{\tau}_{\tau_{0}}f(\xi)(X-\xi)^{k-2}d\xi\hskip 14.22636pt(\begin{subarray}{c}\tau_{0}\in\mathbb{H}\end{subarray})

provides a canonical solution of (1.3). One can develop a working Eichler-Shimura theory by considering Eichler-Shimura integrals only of the form (1.4); see [10], [15]. The advantage of the differential definition is that it allows us to talk about Eichler-Shimura integrals with the base point at infinity whenever $f$ lies in a suitable growth class containing the weakly holomorphic modular forms. Note that the definition of the Eichler-Shimura integral does not require any transformation property for the test function $f$ . However, the action of $\text{SL}_{2}(\mathbb{Z})$ on the right-hand side of (1.3) gives rise to a transformation formula for the Eichler-Shimura integral on the left-hand side of the equation. This approach leads to a new equivariant formulation of the period functions where the group has a nontrivial action on both the test function and module of coefficients. Our formalism allows us to bypass the subtle convergence issues involved in the change of variables of integrals having endpoints at the boundary.

Let $\mathcal{G}$ be a finite index subgroup of $\text{SL}_{2}(\mathbb{Z})$ and $\mathcal{X}$ be a subset of $\text{Hol}(\mathbb{H})$ such that $\mathcal{X}$ is closed under the $\lvert_{k}$ action of $\mathcal{G}$ on $\text{Hol}(\mathbb{H})$ . Given a choice of Eichler-Shimura integrals $\{\mathbb{I}_{k}(\tau,f,X)\mid f\in\mathcal{X}\}$ one defines a period function

(1.5)

\mathsf{p}_{k,\mathcal{X}}:\mathcal{G}\to\text{Fun}(\mathcal{X},V_{k-2,\mathbb{C}})

where $\text{Fun}(\cdot,\cdot)$ is the collection of all set-theoretic maps. Now $\text{Fun}(\mathcal{X},V_{k-2,\mathbb{C}})$ has a natural right $\mathbb{C}[\mathcal{G}]$ module structure stemming from $\mathcal{G}$ -actions on $\mathcal{X}$ and $V_{k-2,\mathbb{C}}$ ; see (1.7). The period function $\mathsf{p}_{k,\mathcal{X}}$ is a cocycle on $\mathcal{G}$ with values in $\text{Fun}(\mathcal{X},V_{k-2,\mathbb{C}})$ . Moreover the image of $\mathsf{p}_{k,\mathcal{X}}$ in $H^{1}\big{(}\mathcal{G},\text{Fun}(\mathcal{X},V_{k-2,\mathbb{C}})\big{)}$ , denoted $[\mathsf{p}_{k}]_{\mathcal{G},\mathcal{X}}$ , is independent of the choice of Eichler-Shimura integrals. We call $[\mathsf{p}_{k}]_{\mathcal{G},\mathcal{X}}$ the Eichler-Shimura class of weight $k$ associated with $\mathcal{G}$ and $\mathcal{X}$ (Section 2.2). The Eichler-Shimura classes are compatible with the restriction of the subgroup as well as the subset of test functions. In particular, every Eichler-Shimura class arises as the restriction of the global class

[\mathsf{p}_{k}]_{\text{SL}_{2}(\mathbb{Z}),\text{Hol}(\mathbb{H})}.

Now suppose $\mathcal{X}$ is a trivial $\mathcal{G}$ -set, i.e., $\mathcal{X}$ consists of $\mathcal{G}$ -invariant functions. Then there is a natural isomorphism

H^{1}\big{(}\mathcal{G},\text{Fun}(\mathcal{X},V_{k-2,\mathbb{C}})\big{)}\xrightarrow{\cong}\text{Fun}\big{(}\mathcal{X},H^{1}(\mathcal{G},V_{k-2,\mathbb{C}})\big{)};

see Section 4.1. We define the Eichler-Shimura homomorphism

[\text{ES}_{k}]_{\mathcal{G},\mathcal{X}}:\mathcal{X}\to H^{1}(\mathcal{G},V_{k-2,\mathbb{C}})

to be the image of $[\mathsf{p}_{k}]_{\mathcal{G},\mathcal{X}}$ under the isomorphism mentioned above. This construction furnishes the desired extension of the Eichler-Shimura map promised in Theorem 1.1. The properties of the equivariant period function (1.5) yield a direct proof of the Hecke-equivariance of this map. We also distill another simple yet neglected equivariance property of $[\text{ES}_{k}]_{\mathcal{G},\mathcal{X}}$ that plays a crucial role in the proof of the isomorphism theorem. Let the notation be as in the discussion above. Moreover, suppose that $\mathcal{H}$ is a subgroup of $\text{SL}_{2}(\mathbb{Z})$ that contains $\mathcal{G}$ as a normal subgroup and $\mathcal{X}$ is closed under $\lvert_{k}$ action of $\mathcal{H}$ . Note that $\mathcal{H}$ acts on $H^{1}(\mathcal{G},V_{k-2,\mathbb{C}})$ by conjugation.

Proposition 1.2.

(Proposition 4.7) The Eichler-Shimura map $[\emph{ES}_{k}]_{\mathcal{G},\mathcal{X}}$ is compatible with the action of $\mathcal{H}$ on its domain and codomain.

To perform effective computations with the period polynomials one needs to work with a choice of Eichler-Shimura integral that amounts to (1.4) with $\tau_{0}$ replaced by $\infty$ . We treat this problem for a class of holomorphic functions that admit decomposition of the form $f=f_{\infty}+f_{c}$ where $f_{\infty}$ is an exponential polynomial and $f_{c}$ is a function with cuspidal decay at $\infty$ (Section 3). Given such a function $f$ one can write down a solution of (1.3) as follows:

(1.6)

\int^{\tau}_{0}f_{\infty}(\xi)(X-\xi)^{k-2}d\xi-\int^{\infty}_{\tau}f_{c}(\xi)(X-\xi)^{k-2}d\xi.

Let $\mathcal{X}$ be a $\mathcal{G}$ -stable subset consisting of functions with a suitable growth rate. Then our general formalism ensures that the choice of Eichler-Shimura integrals (1.6) recovers the same Eichler-Shimura homomorphism. However to determine the period function attached to (1.6) one needs to discuss the transformation of the integrals having an endpoint on the boundary of the upper-half plane. We handle the convergence problem at the boundary by introducing a family of punctured neighborhoods around boundary points that resemble the Satake topology for the Baily-Borel compactification and transform well under the action of $\text{SL}_{2}(\mathbb{Z})$ . This strategy to regularize the Eichler-Shimura integral provides an upper half-plane analog of Brown’s construction of period polynomials where he interprets (1.6) in the light of a tangential basepoint [5, Part I]. Readers may also note that our choice of compactification differs from the standard automorphic method [9] that prefers the Borel-Serre compactification over the Baily-Borel approach.

We end the introduction with a glimpse into the isomorphism theorem and its consequences. Let the notation be as in the statement of Theorem 1.1. Recall that $\mathcal{M}_{k}(\mathcal{G})=\mathcal{E}_{k}(\mathcal{G})\oplus\mathcal{S}_{k}(\mathcal{G})$ where $\mathcal{E}_{k}(\mathcal{G})$ is the space of Eisenstein series. To prove the theorem it suffices to check that the Eichler-Shimura map attached to $\mathcal{E}_{k}(\mathcal{G})$ is injective and the image of $\mathcal{E}_{k}(\mathcal{G})$ in $H^{1}(\mathcal{G},V_{k-2,\mathbb{C}})$ is a linear complement of the parabolic subspace. The main ingredient of the argument is an explicit basis for the space of Eisenstein series that is parameterized by the cusps of $\mathcal{G}$ and satisfies an appropriate nonvanishing property at the cusps (Section 5.3). Our treatment makes free use of the abstract group cohomological techniques developed in the books of Shimura [16] and Hida [10], but, we completely bypass the Haberland pairing between cocycles. There is a perspective towards the Eichler-Shimura isomorphism that appeals to Shapiro’s lemma to transform the problem into an assertion regarding the cohomology of the full modular group $\Gamma$ with coefficients in an induced module [14]. Our version of the Eichler-Shimura theorem directly implies the corresponding isomorphism in the induced picture.

Theorem 1.3.

(Theorem 7.5) Let $\mathcal{G}$ be a congruence subgroup of $\Gamma$ . There exists an induced Eichler-Shimura homomorphism

\mathcal{M}_{k}(\mathcal{G})\to H^{1}\big{(}\Gamma,\emph{Fun}(\mathcal{G}\backslash\Gamma,V_{k-2,\mathbb{C}})\big{)}

extending the map for the cusp forms so that the extended map gives rise to a Hecke equivariant isomorphism $\mathcal{M}_{k}(\mathcal{G})\oplus\mathcal{S}^{a}_{k}(\mathcal{G})\xrightarrow{\cong}H^{1}\big{(}\Gamma,\emph{Fun}(\mathcal{G}\backslash\Gamma,V_{k-2,\mathbb{C}})\big{)}$ .

Hida developed a theory of twisted Eichler-Shimura isomorphism [10, 6.3] in the context of modular forms with nebentypus. The equivariant formalism proposed in this article provides a natural framework for the twisted Eichler-Shimura theory (Section 7.2). Let $\chi$ be a Dirichlet character modulo $N$ . Suppose that $\mathcal{M}_{k}(N,\chi)$ , resp. $\mathcal{S}_{k}(N,\chi)$ , is the space of all modular forms, resp. cusp forms, on $\Gamma_{1}(N)$ that transform according to the nebentypus character $\chi$ under $\Gamma_{0}(N)$ . As before set $\mathcal{S}_{k}^{a}(N,\chi):=\{\bar{f}\mid f\in\mathcal{S}_{k}(N,\bar{\chi})\}$ . Here one needs to replace the module of coefficients by its twisted counterpart, denoted $V_{k-2,\mathbb{C}}^{\chi}$ , on which $\Gamma_{0}(N)$ action is twisted by the character $\chi^{-1}$ . We use the equivariant period function (1.5) to extend the Eichler-Shimura isomorphism for cusp forms and provide the following isomorphism statement strengthening Hida’s earlier result for primitive nebentypus character and prime power level (loc.cit., Theorem 3). Our proof of the result only requires Theorem 1.1 and we do not appeal to the existence Eisenstein series with nebentypus contrary to Hida’s approach; cf. [15].

Theorem 1.4.

(Theorem 7.8) Let the notation be as above. The twisted Eichler-Shimura map for cusp forms extends to a Hecke equivariant isomorphism

\mathcal{M}_{k}(N,\chi)\oplus\mathcal{S}^{a}_{k}(N,\chi)\xrightarrow{\cong}H^{1}\big{(}\Gamma_{0}(N),V_{k-2,\mathbb{C}}^{\chi}\big{)}.

The final topic of the article is an explicit computation of the period polynomials attached to the Eisenstein series (Section 8). Recall that if $\mathbb{K}$ is a subfield of $\mathbb{C}$ and $\mathcal{G}$ is a finite index subgroup then the natural inclusion $V_{k-2,\mathbb{K}}\hookrightarrow V_{k-2,\mathbb{C}}$ induces an isomorphism $H^{1}(\mathcal{G},V_{k-2,\mathbb{K}})\otimes_{\mathbb{K}}\mathbb{C}\cong H^{1}(\mathcal{G},V_{k-2,\mathbb{C}})$ ; see (5.4). A class $\mathcal{\alpha}\in H^{1}(\mathcal{G},V_{k-2,\mathbb{C}})$ is definable over the subfield $\mathbb{K}$ if $\alpha$ lies in the canonical $\mathbb{K}$ -form $H^{1}(\mathcal{G},V_{k-2,\mathbb{K}})$ . Now suppose $\mathcal{G}$ is a congruence subgroup of $\text{SL}_{2}(\mathbb{Z})$ . Let $\mathcal{B}_{\mathcal{G},k}$ denote the basis of the space of Eisenstein series $\mathcal{E}_{k}(\mathcal{G})$ described in Section 5.3.

Theorem 1.5.

(Theorem 8.8) Let $\mathcal{G}$ be a congruence subgroup containing $\Gamma(N)$ and $f\in\mathcal{B}_{\mathcal{G},k}$ . Then the image of $f$ under the Eichler-Shimura map for $\mathcal{G}$ is definable over $\mathbb{Q}(\mathbf{e}(\frac{1}{N}))$ .

The algebraicity theorem above apparently contradicts the fact that the coefficients of the period polynomial of an Eisenstein series involve transcendental expressions closely related to the zeta values; see [19] and Lemma 8.5. However, an observation of Gangl and Zagier suggests that one can modify the period polynomial of the Eisenstein series on $\text{SL}_{2}(\mathbb{Z})$ by an appropriate coboundary so that the modified cocycle has coefficients in $\mathbb{Q}$ [20, 7]. We first establish a generalization of their observation for $\Gamma(N)$ and then use the equivariance property in Proposition 1.2 to deduce the general statement. We also prove a similar rationality result (Theorem 8.11) for the twisted Eichler-Shimura isomorphism in Theorem 1.4.

1.1 Notation and conventions

Throughout the article $\xi$ and $\tau$ are variables with values in the upper half-plane $\mathbb{H}$ . Let $i$ denote the imaginary unit. For $\tau\in\mathbb{H}$ one writes $\tau=u(\tau)+iv(\tau)$ where $u(\tau)$ and $v(\tau)$ are real numbers with $v(\tau)>0$ . We normalize the complex exponential function as $\textbf{e}(z):=\exp(2\pi iz)$ . One uses $\mathbb{K}$ to refer to a field of characteristic $0$ which in some places is assumed to be a subfield of $\mathbb{C}$ . For a positive integer $N$ set

\mathbb{Q}_{N}:=\mathbb{Q}(\mathbf{e}(\frac{1}{N})).

Let $\text{Hol}(\mathbb{H})$ , resp. $\Omega^{1}_{\text{hol}}(\mathbb{H})$ , be the space of all $\mathbb{C}$ -valued holomorphic functions, resp. holomorphic differential $1$ -forms, on $\mathbb{H}$ . The constant functions define a natural inclusion $\mathbb{C}\hookrightarrow\text{Hol}(\mathbb{H})$ which turns $\text{Hol}(\mathbb{H})$ into a commutative $\mathbb{C}$ -algebra. We have a $\mathbb{C}$ -linear differential operator

\partial_{\tau}:\text{Hol}(\mathbb{H})\to\Omega^{1}_{\text{hol}}(\mathbb{H}),\hskip 8.5359ptf\to\frac{\partial f}{\partial\tau}d\tau

whose kernel equals the subspace of constant functions in $\text{Hol}(\mathbb{H})$ . During the calculations, one considers integrals along paths on $\mathbb{H}$ . All paths are piecewise smooth unless otherwise stated.

We abbreviate $\text{SL}_{2}(\mathbb{Z})$ as $\Gamma$ and write

T:=\begin{pmatrix}1&1\\ 0&1\end{pmatrix},\hskip 8.5359ptS:=\begin{pmatrix}0&-1\\ 1&0\end{pmatrix}

for the generators of this group. Sometimes one may refer to the subgroups of $\text{GL}_{2}(\mathbb{R})$ . For a subring $R$ of $\mathbb{R}$ set $\text{GL}^{+}_{2}(R):=\{\gamma\in\text{GL}_{2}(R)\mid\text{det}(\gamma)>0\}$ . The letter $\gamma$ stands for an arbitrary $2\times 2$ matrix and we put $\gamma=\begin{pmatrix}a&b\\ c&d\end{pmatrix}$ where $a,b,c,d$ take values in an appropriate ring. One uses the notation $\mathcal{C}(\cdot)$ , resp. $\mathcal{C}_{\infty}(\cdot)$ , to denote the set of cusps, resp. regular cusps, attached to a finite index subgroup of $\Gamma$ . Our treatment also makes use of the standard congruence subgroups namely $\Gamma(N)$ , $\Gamma_{1}(N)$ , and $\Gamma_{0}(N)$ [7, 1.2]. We call $\Gamma(N)$ the principal congruence subgroup of level $N$ . For a subgroup $\mathcal{G}$ of $\text{SL}_{2}(\mathbb{Z})$ let

\text{P}\mathcal{G}:=\mathcal{G}/\mathcal{G}\cap\{\pm\text{Id}\}

denote the projectivization of $\mathcal{G}$ .

The group actions relevant to us are usually right actions. In the context of group cohomology $Z^{1}$ , resp. $B^{1}$ , denotes the collection of $1$ -cocycles, $1$ -coboundaries. If $\mathcal{X},\mathcal{Y}$ are two sets then $\text{Fun}(\mathcal{X},\mathcal{Y})$ is the set of all set-theoretic functions from $\mathcal{X}$ to $\mathcal{Y}$ . Suppose that $\mathcal{Y}$ is a $\mathbb{F}$ -vector space for some field $\mathbb{F}$ . Then $\text{Fun}(\mathcal{X},\mathcal{Y})$ possesses a $\mathbb{F}$ -linear structure defined by pointwise addition and scalar multiplication. Now suppose $G$ is an abstract group and $\mathcal{X}$ , $\mathcal{Y}$ are right $G$ -sets. The set $\text{Fun}(\mathcal{X},\mathcal{Y})$ admits a right $G$ -action given by

(1.7)

(F.g)(x)=F(xg^{-1})g.\hskip 28.45274pt\big{(}\begin{subarray}{c}F\in\text{Fun}(\mathcal{X},\mathcal{Y}),x\in\mathcal{X},\,g\in G\end{subarray}\big{)}

If $\mathcal{Y}$ is a right $\mathbb{F}[G]$ -module then the resulting $\mathbb{F}$ -linear structure and $G$ -structure turn $\text{Fun}(\mathcal{X},\mathcal{Y})$ into a right $\mathbb{F}[G]$ -module.

2 Construction of period cocycle

We begin by recalling some background relevant to the Eichler-Shimura theory. The following subsection generalizes the notion of the period polynomial to not necessarily invariant functions. This abstract formulation leads us to the concept of the Eichler-Shimura class that effectively captures our equivariant period function.

2.1 Preliminaries

Group action on $\mathbb{H}$ .

The group $\text{GL}_{2}^{+}(\mathbb{R})$ acts on $\mathbb{H}$ from left by holomorphic automorphisms via fractional linear transformations;

(2.1)

(\gamma,\tau)\to\gamma\tau:=\frac{a\tau+b}{c\tau+d},\hskip 8.5359pt\gamma=\begin{pmatrix}a&b\\ c&d\end{pmatrix}\in\text{GL}_{2}^{+}(\mathbb{R}).

The restriction of this action to $\Gamma$ is our primary interest. The $\Gamma$ -action on $\mathbb{H}$ descends to a $\text{P}\Gamma$ -action on $\mathbb{H}$ , i.e, $-\text{Id}\in\Gamma$ acts trivially on each point. Note that the fractional linear transformations (2.1) define a holomorphic left $\Gamma$ -action on the Riemann sphere $\widehat{\mathbb{C}}=\mathbb{C}\cup\{\infty\}$ . The boundary of $\mathbb{H}$ in $\widehat{\mathbb{C}}$ equals $\partial\mathbb{H}=\{\infty\}\cup\mathbb{R}=\mathbb{P}^{1}(\mathbb{R})$ and the $\Gamma$ -action on $\widehat{\mathbb{C}}$ preserves this boundary. We define the extended upper-half plane as

\mathbb{H}^{\ast}:=\mathbb{H}\cup\mathbb{P}^{1}(\mathbb{Q}).

For $x\in\mathbb{P}^{1}(\mathbb{Q})$ let $\Gamma_{x}$ denote the stabilizer of $x$ in $\Gamma$ . With this notation $\Gamma_{\infty}=\{\pm T^{n}\mid n\in\mathbb{Z}\}$ . If $\mathcal{G}$ is a subgroup of $\Gamma$ then set $\mathcal{G}_{x}:=\mathcal{G}\cap\Gamma_{x}$ . Now suppose $\mathcal{G}$ is a finite index subgroup. Then a cusp of $\mathcal{G}$ refers to a $\mathcal{G}$ -orbit in $\mathbb{P}^{1}(\mathbb{Q})$ .

Group action on function spaces.

Let $k\in\mathbb{Z}$ . We define the $\lvert_{k}$ right action of $\text{GL}_{2}^{+}(\mathbb{R})$ on $\text{Hol}(\mathbb{H})$ by

(2.2)

f\lvert_{k,\gamma}(\tau):=\text{det}(\gamma)^{k-1}(c\tau+d)^{-k}f(\gamma\tau).\hskip 14.22636pt\big{(}\begin{subarray}{c}f\in\text{Hol}(\mathbb{H}),\gamma\in\text{GL}_{2}^{+}(\mathbb{R})\end{subarray}\big{)}

As before one primarily uses restriction of this action to $\Gamma$ . If $k$ remains fixed throughout the discussion then we drop it from the notation.

Representations of $\text{GL}_{2}(\mathbb{Z})$ .

Let $n$ be a nonnegative integer. Set $V_{n}:=\{P\in\mathbb{Z}[X]\mid\text{deg}(P)\leq n\}$ . The group $\text{GL}_{2}(\mathbb{Z})$ acts on $V_{n}$ from right by

(2.3)

P(X)\lvert_{-n,\gamma}:=(cX+d)^{n}P\Big{(}\frac{aX+b}{cX+d}\Big{)}.\hskip 8.5359pt\big{(}\begin{subarray}{c}P(X)\in V_{n},\gamma\in\text{GL}_{2}(\mathbb{Z})\end{subarray}\big{)}

We omit the index of weight from the subscript if it is clear from the context. More generally, let $R$ be a commutative ring and $C$ be a commutative $R$ -algebra. Write $V_{n,C}=C\otimes_{\mathbb{Z}}V_{n}$ . Then (2.3) defines a right $C[\text{GL}_{2}(R)]$ action on $V_{n,C}$ . The pair $(\mathbb{Z},C)$ yields a $C[\text{GL}_{2}(\mathbb{Z})]$ module structure on $V_{n,C}$ . In this article, $C$ is typically a field of characteristic zero and we consider the cohomology vector spaces $H^{\ast}(\mathcal{G},V_{k-2,C})$ for the subgroups $\mathcal{G}$ of $\Gamma$ .

Differential operators.

The action of $\partial_{\tau}$ on the coefficients gives rise to a $\mathbb{C}$ -linear differential operator

(2.4)

\partial_{\tau}:\text{Hol}(\mathbb{H})[X]_{n}\to\Omega_{\text{hol}}^{1}(\mathbb{H})\otimes_{\mathbb{C}}V_{n,\mathbb{C}}\hskip 28.45274pt(\begin{subarray}{c}n\in\mathbb{Z}_{\geq 0}\end{subarray})

where $\text{Hol}(\mathbb{H})[X]_{n}=\{P\in\text{Hol}(\mathbb{H})[X]\mid\text{deg}(P)\leq n\}$ . The kernel of (2.4) is equal to $V_{n,\mathbb{C}}\subseteq\text{Hol}(\mathbb{H})[X]_{n}$ .

2.2 Period cocycle and Eichler-Shimura class

Definition 2.1.

Let $k$ be an integer $\geq 2$ and $f\in\text{Hol}(\mathbb{H})$ . A weight $k$ Eichler-Shimura integral of $f$ is a polynomial $\mathbb{I}_{k}(\tau,f,X)\in\text{Hol}(\mathbb{H})[X]_{k-2}$ that satisfies $\partial_{\tau}\big{(}\mathbb{I}_{k}(\tau,f,X)\big{)}=f(\tau)(X-\tau)^{k-2}d\tau$ .

Now suppose $\mathcal{X}$ is a collection of holomorphic functions on $\mathbb{H}$ . A choice of weight $k$ Eichler-Shimura integrals for $\mathcal{X}$ is a set-theoretic function

\begin{gathered}\mathbb{I}_{k,\mathcal{X}}:\mathcal{X}\to\text{Hol}(\mathbb{H})[X]_{k-2},\hskip 8.5359ptf\to\mathbb{I}_{k,\mathcal{X}}(\tau,f,X);\\ \partial_{\tau}\big{(}\mathbb{I}_{k,\mathcal{X}}(\tau,f,X)\big{)}=f(\tau)(X-\tau)^{k-2}d\tau.\end{gathered}

Note that two choice functions, say $\mathbb{I}^{1}_{k,\mathcal{X}}$ and $\mathbb{I}^{2}_{k,\mathcal{X}}$ , give rise to a set map

\big{(}\mathbb{I}^{1}_{k,\mathcal{X}}-\mathbb{I}^{2}_{k,\mathcal{X}}\big{)}:\mathcal{X}\to V_{k-2,\mathbb{C}},\hskip 8.5359ptf\to\mathbb{I}^{1}_{k,\mathcal{X}}(\tau,f,X)-\mathbb{I}^{2}_{k,\mathcal{X}}(\tau,f,X).

Example 2.2.

Let $\tau_{0}\in\mathbb{H}$ . The canonical choice for $\mathcal{X}$ with respect to $\tau_{0}$ is

\mathbb{I}_{k,\mathcal{X}}^{\tau_{0}}:\mathcal{X}\to\text{Hol}(\mathbb{H})[X]_{k-2},\hskip 8.5359ptf\to\int_{\tau_{0}}^{\tau}f(\xi)(X-\xi)^{k-2}d\xi.

This choice function plays a key role throughout the paper; cf. (1.4).

Let $\mathcal{G}$ be a finite index subgroup of $\Gamma$ . Suppose that $\mathcal{X}$ is a subset of $\text{Hol}(\mathbb{H})$ which is stable under the $\lvert_{k}$ action of $\mathcal{G}$ . The $\mathcal{G}$ -actions on $\mathcal{X}$ and $V_{k-2,\mathbb{C}}$ induce a right $\mathbb{C}[\mathcal{G}]$ module structure on $\text{Fun}(\mathcal{X},V_{k-2,\mathbb{C}})$ according to the recipe (1.7). Let $\mathbb{I}_{k,\mathcal{X}}$ be a choice of weight $k$ Eichler-Shimura integrals for $\mathcal{X}$ . For $\gamma\in\mathcal{G}$ and $f\in\mathcal{X}$ put

(2.5)

\mathsf{p}_{k,\mathcal{X}}(\gamma,f,X):=(cX+d)^{k-2}\mathbb{I}_{k,\mathcal{X}}(\gamma\tau,f\lvert_{\gamma^{-1}},\frac{aX+b}{cX+d})-\mathbb{I}_{k,\mathcal{X}}(\tau,f,X).

We now have the following result:

Lemma 2.3.

$\mathsf{p}_{k,\mathcal{X}}(\gamma,f,X)\in V_{k-2,\mathbb{C}}$ .

Proof.

One first verifies the assertion for the canonical choice with respect to $\tau_{0}\in\mathbb{H}$ . A direct calculation using the change of variable formula demonstrates that

(2.6)			$\displaystyle(cX+d)^{k-2}\mathbb{I}^{\tau_{0}}_{k,\mathcal{X}}(\gamma\tau,f\lvert_{\gamma^{-1}},\frac{aX+b}{cX+d})-\mathbb{I}^{\tau_{0}}_{k,\mathcal{X}}(\tau,f,X)\hskip 14.22636pt\big{(}\begin{subarray}{c}f\in\mathcal{X},\gamma\in\mathcal{G}\end{subarray}\big{)}$
(2.6)			$\displaystyle=\int_{\gamma^{-1}\tau_{0}}^{\tau_{0}}f(\xi)(X-\xi)^{k-2}d\xi\in V_{k-2,\mathbb{C}}.$

Therefore the statement holds for $\mathbb{I}_{k,\mathcal{X}}^{\tau_{0}}$ . Note that

(2.7)			$\displaystyle(cX+d)^{k-2}\mathbb{I}_{k,\mathcal{X}}(\gamma\tau,f\lvert_{\gamma^{-1}},\frac{aX+b}{cX+d})\hskip 85.35826pt\big{(}\begin{subarray}{c}f\in\mathcal{X},\gamma\in\mathcal{G}\end{subarray}\big{)}$
			$\displaystyle\hskip 85.35826pt-(cX+d)^{k-2}\mathbb{I}^{\tau_{0}}_{k,\mathcal{X}}(\gamma\tau,f\lvert_{\gamma^{-1}},\frac{aX+b}{cX+d})$
			$\displaystyle=(\mathbb{I}_{k,\mathcal{X}}-\mathbb{I}^{\tau_{0}}_{k,\mathcal{X}})(f\lvert_{\gamma^{-1}})\bigl{\lvert}_{\gamma}\in V_{k-2,\mathbb{C}}.$

The identities (2.6) and (2.7) together imply

(2.8)		$\displaystyle\mathsf{p}_{k,\mathcal{X}}(\gamma,f,X)$	$\displaystyle=\int^{\tau_{0}}_{\gamma^{-1}\tau_{0}}f(\xi)(X-\xi)^{k-2}d\xi+(\mathbb{I}_{k,\mathcal{X}}-\mathbb{I}^{\tau_{0}}_{k,\mathcal{X}})(f\lvert_{\gamma^{-1}})\bigl{\lvert}_{\gamma}$
(2.8)			$\displaystyle\hskip 113.81102pt-(\mathbb{I}_{k,\mathcal{X}}-\mathbb{I}^{\tau_{0}}_{k,\mathcal{X}})(f)\in V_{k-2,\mathbb{C}}.$

∎

Definition 2.4.

Let the notation be as above. The period function of the choice of Eichler-Shimura integrals $\mathbb{I}_{k,\mathcal{X}}$ is a function on $\mathcal{G}$ given by

\mathsf{p}_{k,\mathcal{X}}:\mathcal{G}\to\text{Fun}(\mathcal{X},V_{k-2,\mathbb{C}}),\hskip 5.69046pt\gamma\to\mathsf{p}_{k,\mathcal{X}}(\gamma);\hskip 8.5359pt\mathsf{p}_{k,\mathcal{X}}(\gamma)(f):=\mathsf{p}_{k,\mathcal{X}}(\gamma,f,X).

Example 2.5.

Let $\mathsf{p}^{\tau_{0}}_{k,\mathcal{X}}$ be the period function of $\mathbb{I}_{k,\mathcal{X}}^{\tau_{0}}$ . Then (2.6) amounts to the identity

(2.9)

\mathsf{p}^{\tau_{0}}_{k,\mathcal{X}}(\gamma,f,X)=\int_{\gamma^{-1}\tau_{0}}^{\tau_{0}}f(\xi)(X-\xi)^{k-2}d\xi.\hskip 28.45274pt\big{(}\begin{subarray}{c}f\in\mathcal{X},\gamma\in\mathcal{G}\end{subarray}\big{)}

The following lemma checks that the period function is a cocycle.

Lemma 2.6.

Let $\mathbb{I}_{k,\mathcal{X}}$ be a choice of Eichler-Shimura integrals for $\mathcal{X}$ and $\mathsf{p}_{k,\mathcal{X}}$ be the period function of $\mathbb{I}_{k,\mathcal{X}}$ . Then $\mathsf{p}_{k,\mathcal{X}}$ is a cocycle on $\mathcal{G}$ with values in $\emph{Fun}(\mathcal{X},V_{k-2,\mathbb{C}})$ , i.e.,

(2.10)

\mathsf{p}_{k,\mathcal{X}}(\gamma_{1}\gamma_{2})=\mathsf{p}_{k,\mathcal{X}}(\gamma_{1})\lvert_{\gamma_{2}}+\mathsf{p}_{k,\mathcal{X}}(\gamma_{2}).\hskip 28.45274pt(\begin{subarray}{c}\gamma_{1},\gamma_{2}\in\mathcal{G}\end{subarray})

Proof.

We start by proving the statement for $\mathbb{I}_{k,\mathcal{X}}^{\tau_{0}}$ . Let $\gamma_{1},\gamma_{2}\in\mathcal{G}$ and $f\in\mathcal{X}$ . One employs (2.9) and the change of variable formula to discover that

\mathsf{p}_{k,\mathcal{X}}^{\tau_{0}}(\gamma_{1})\lvert_{\gamma_{2}}(f)=\int^{\gamma_{2}^{-1}\tau_{0}}_{\gamma_{2}^{-1}\gamma_{1}^{-1}\tau_{0}}f(\xi)(X-\xi)^{k-2}d\xi.

The identity above together with (2.9) implies that

(2.11)

\mathsf{p}_{k,\mathcal{X}}^{\tau_{0}}(\gamma_{1}\gamma_{2})(f)=\mathsf{p}_{k,\mathcal{X}}^{\tau_{0}}(\gamma_{1})\lvert_{\gamma_{2}}(f)+\mathsf{p}_{k,\mathcal{X}}^{\tau_{0}}(\gamma_{2})(f).

Therefore (2.10) holds for $\mathbb{I}_{k,\mathcal{X}}^{\tau_{0}}$ . Now the equation (2.8) translates into

(2.12)

\mathsf{p}_{k,\mathcal{X}}(\gamma)=\mathsf{p}_{k,\mathcal{X}}^{\tau_{0}}(\gamma)+\big{(}\mathbb{I}_{k,\mathcal{X}}-\mathbb{I}_{k,\mathcal{X}}^{\tau_{0}}\big{)}\bigl{\lvert}_{\gamma}-\big{(}\mathbb{I}_{k,\mathcal{X}}-\mathbb{I}_{k,\mathcal{X}}^{\tau_{0}}\big{)}.\hskip 28.45274pt(\begin{subarray}{c}\gamma\in\mathcal{G}\end{subarray})

The assertion is a direct consequence of (2.11) and (2.12). ∎

We next study the effect of changing the choice function.

Proposition 2.7.

The image of $\mathsf{p}_{k,\mathcal{X}}$ in $H^{1}\big{(}\mathcal{G},\emph{Fun}(\mathcal{X},V_{k-2,\mathbb{C}})\big{)}$ is independent of the choice function $\mathbb{I}_{k,\mathcal{X}}$ .

Proof.

Let $\tau_{0}\in\mathbb{H}$ be a fixed base point and $\mathbb{I}_{k,\mathcal{X}}$ be an arbitrary choice function. The identity (2.12) implies that $\mathsf{p}_{k,\mathcal{X}}$ and $\mathsf{p}_{k,\mathcal{X}}^{\tau_{0}}$ differ by the coboundary defined by $(\mathbb{I}_{k,\mathcal{X}}-\mathbb{I}^{\tau_{0}}_{k,\mathcal{X}})$ . Therefore the images of $\mathsf{p}_{k,\mathcal{X}}$ and $\mathsf{p}_{k,\mathcal{X}}^{\tau_{0}}$ in cohomology are equal. ∎

Proposition 2.7 proves that the pair $(\mathcal{G},\mathcal{X})$ determines the image of $\mathsf{p}_{k,\mathcal{X}}$ .

Definition 2.8.

The Eichler-Shimura class of weight $k$ attached to $(\mathcal{G},\mathcal{X})$ , denoted $[\mathsf{p}_{k}]_{\mathcal{G},\mathcal{X}}$ , is the image of $\mathsf{p}_{k,\mathcal{X}}$ in $H^{1}\big{(}\mathcal{G},\text{Fun}(\mathcal{X},V_{k-2,\mathbb{C}})\big{)}$ .

The Eichler-Shimura class exhibits good functorial properties concerning restriction of the corresponding set of functions and subgroup; cf. [18, 3]. Let $\mathcal{X}_{2}\subseteq\mathcal{X}_{1}$ be two $\mathcal{G}$ -stable subsets of $\text{Hol}(\mathbb{H})$ . Now restriction induces a $\mathbb{C}[\mathcal{G}]$ -linear homomorphism $\text{Res}^{\mathcal{X}_{1}}_{\mathcal{X}_{2}}:\text{Fun}(\mathcal{X}_{1},V_{k-2,\mathbb{C}})\to\text{Fun}(\mathcal{X}_{2},V_{k-2,\mathbb{C}})$ that gives rise to a $\mathbb{C}$ -linear map

\text{R}^{\mathcal{X}_{1}}_{\mathcal{X}_{2}}:H^{1}\big{(}\mathcal{G},\text{Fun}(\mathcal{X}_{1},V_{k-2,\mathbb{C}})\big{)}\to H^{1}\big{(}\mathcal{G},\text{Fun}(\mathcal{X}_{2},V_{k-2,\mathbb{C}})\big{)}.

Let $\mathbb{I}_{k,\mathcal{X}_{1}}$ be a choice of weight $k$ Eichler-Shimura integrals for $\mathcal{X}_{1}$ . Set $\mathbb{I}_{k,\mathcal{X}_{2}}:=\mathbb{I}_{k,\mathcal{X}_{1}}\bigl{\lvert}_{\mathcal{X}_{2}}$ . One computes the corresponding Eichler-Shimura classes with these compatible choice functions to discover that

(2.13)

[\mathsf{p}_{k}]_{\mathcal{G},\mathcal{X}_{2}}=\text{R}^{\mathcal{X}_{1}}_{\mathcal{X}_{2}}\big{(}[\mathsf{p}_{k}]_{\mathcal{G},\mathcal{X}_{1}}\big{)}.

Now suppose $\mathcal{G}_{2}\subseteq\mathcal{G}_{1}$ are two finite index subgroups of $\Gamma$ . Let $\mathcal{X}$ be a $\mathcal{G}_{1}$ -stable subset of $\text{Hol}(\mathbb{H})$ . The restriction $\text{Res}^{\mathcal{G}_{1}}_{\mathcal{G}_{2}}:\text{Fun}\big{(}\mathcal{G}_{1},\text{Fun}(\mathcal{X},V_{k-2,\mathbb{C}})\big{)}\to\text{Fun}\big{(}\mathcal{G}_{2},\text{Fun}(\mathcal{X},V_{k-2,\mathbb{C}})\big{)}$ yields change of group map

\text{R}^{\mathcal{G}_{1}}_{\mathcal{G}_{2}}:H^{1}\big{(}\mathcal{G}_{1},\text{Fun}(\mathcal{X},V_{k-2,\mathbb{C}})\big{)}\to H^{1}\big{(}\mathcal{G}_{2},\text{Fun}(\mathcal{X},V_{k-2,\mathbb{C}})\big{)}.

As before one calculates the Eichler-Shimura classes using compatible choice functions to find

(2.14)

[\mathsf{p}_{k}]_{\mathcal{G}_{2},\mathcal{X}}=\text{R}^{\mathcal{G}_{1}}_{\mathcal{G}_{2}}([\mathsf{p}_{k}]_{\mathcal{G}_{1},\mathcal{X}}).

Global Eichler-Shimura class.

Let $[\mathsf{p}_{k}]_{\Gamma,\text{Hol}(\mathbb{H})}$ be the Eichler-Shimura class attached to the $\Gamma$ -stable set $\text{Hol}(\mathbb{H})$ . Then, using (2.13) and (2.14), we conclude that

[\mathsf{p}_{k}]_{\mathcal{G},\mathcal{X}}=\text{R}^{\text{Hol}(\mathbb{H})}_{\mathcal{X}}\circ\text{R}^{\Gamma}_{\mathcal{G}}\big{(}[\mathsf{p}_{k}]_{\Gamma,\text{Hol}(\mathbb{H})}\big{)}.

The identity above shows that each Eichler-Shimura class arises as a restriction of the global class $[\mathsf{p}_{k}]_{\Gamma,\text{Hol}(\mathbb{H})}$ .

2.3 Variations of formalism

2.3.1 General subgroups of $\text{GL}_{2}^{+}(\mathbb{R})$

The constructions of this section are completely algebraic and continue to work for an arbitrary subgroup of $\text{GL}_{2}^{+}(\mathbb{R})$ . Let $\mathcal{G}$ be a subgroup of $\text{GL}_{2}^{+}(\mathbb{R})$ and $\mathcal{X}\subseteq\text{Hol}(\mathbb{H})$ be stable under $\lvert_{k}$ action of $\mathcal{G}$ . Note that (2.3) defines a right $\mathbb{C}[\text{GL}_{2}(\mathbb{C})]$ -module structure on $V_{k-2,\mathbb{C}}$ . Therefore $\text{Fun}(\mathcal{X},V_{k-2,\mathbb{C}})$ has a natural right $\mathbb{C}[\mathcal{G}]$ -module structure. A straightforward calculation using (2.2) demonstrates that (2.6) and Lemma 2.3 hold in this setup. As a consequence, one can define the period function using (2.5). The rest of the treatment goes through without any change and we obtain a well-defined Eichler-Shimura class $[\mathsf{p}_{k}]_{\mathcal{G},\mathcal{X}}$ in $H^{1}\big{(}\mathcal{G},\text{Fun}(\mathcal{X},V_{k-2,\mathbb{C}})\big{)}$ . The general setup described above yields a formalism for Fuchsian groups, cf. [16, 8]. The subgroups of $\text{GL}_{2}^{+}(\mathbb{Q})$ appear in the context of the action of double coset operators. The cocycle formula for the subgroups of $\text{GL}_{2}^{+}(\mathbb{Q})$ is relevant to our treatment of Hecke theory.

2.3.2 Linear cocycles and projectivization

Our formalism of period function allows the choice of Eichler-Shimura integrals to be a set-theoretic function. However, the standard definitions of the Eichler-Shimura integral in literature force $f\to\mathbb{I}_{k}(\tau,f,X)$ to be linear. We now sketch a variant of our construction that incorporates the linearity condition. Let the notation be as in Section 2.2. Suppose that $\mathbb{K}$ is a subfield of $\mathbb{C}$ and $\mathcal{X}$ is a $\mathbb{K}[\mathcal{G}]$ -submodule of $\text{Hol}(\mathbb{H})$ . A choice of Eichler-Shimura integrals for $\mathcal{X}$ , say $\mathbb{I}_{k,\mathcal{X}}$ , is $\mathbb{K}$ -linear if $\mathbb{I}_{k,\mathcal{X}}$ is a $\mathbb{K}$ -linear map.

Example 2.9.

The canonical choice $\mathbb{I}_{k,\mathcal{X}}^{\tau_{0}}$ is a $\mathbb{K}$ -linear choice of Eichler-Shimura integrals.

If $\mathbb{I}_{k,\mathcal{X}}^{1}$ and $\mathbb{I}_{k,\mathcal{X}}^{2}$ are two $\mathbb{K}$ -linear choices for $\mathcal{X}$ then $\big{(}\mathbb{I}_{k,\mathcal{X}}^{1}-\mathbb{I}_{k,\mathcal{X}}^{2}\big{)}:\mathcal{X}\to V_{k-2,\mathbb{C}}$ is a $\mathbb{K}$ -linear map. Let $\mathbb{I}_{k,\mathcal{X}}$ be a $\mathbb{K}$ -linear choice of Eichler-Shimura integrals and $\mathsf{p}_{k,\mathcal{X}}$ be the period function of $\mathbb{I}_{k,\mathcal{X}}$ . It is clear that

\mathsf{p}_{k,\mathcal{X}}(\gamma)\in\text{Hom}_{\mathbb{K}}(\mathcal{X},V_{k-2,\mathbb{C}})

for each

\gamma\in\mathcal{G}

Note that $\text{Hom}_{\mathbb{K}}(\mathcal{X},V_{k-2,\mathbb{C}})$ is a $\mathbb{C}[\mathcal{G}]$ -submodule of $\text{Fun}(\mathcal{X},V_{k-2,\mathbb{C}})$ . Hence $\mathsf{p}_{k,\mathcal{X}}$ is a cocycle with values in $\text{Hom}_{\mathbb{K}}(\mathcal{X},V_{k-2,\mathbb{C}})$ . The image of $\mathsf{p}_{k,\mathcal{X}}$ in $H^{1}\big{(}\mathcal{G},\text{Hom}_{\mathbb{K}}(\mathcal{X},V_{k-2,\mathbb{C}})\big{)}$ is independent of the $\mathbb{K}$ -linear choice function $\mathbb{I}_{k,\mathcal{X}}$ and determines the $\mathbb{K}$ -linear Eichler-Shimura class $[\mathsf{p}_{k}]_{\mathcal{G},\mathcal{X}}^{\mathbb{K}}$ . The natural homomorphism in cohomology induced by the inclusion of $\text{Hom}_{\mathbb{K}}(\mathcal{X},V_{k-2,\mathbb{C}})$ into $\text{Fun}(\mathcal{X},V_{k-2,\mathbb{C}})$ maps $[\mathsf{p}_{k}]_{\mathcal{G},\mathcal{X}}^{\mathbb{K}}$ to $[\mathsf{p}_{k}]_{\mathcal{G},\mathcal{X}}$ .

For the classical Eichler-Shimura theory of modular forms the linear formalism adds no new structure; see Proposition 4.2. The advantage of linear theory is that it allows us to construct the equivariant period cocycle on the projectivization of $\mathcal{G}$ . If $k$ is odd then the action of $\mathcal{G}$ on $\text{Fun}\big{(}\mathcal{X},V_{k-2,\mathbb{C}}\big{)}$ does not descend to $\text{P}\mathcal{G}$ . On the other hand the action of $\mathcal{G}$ on $\text{Hom}_{\mathbb{K}}\big{(}\mathcal{X},V_{k-2,\mathbb{C}}\big{)}$ always descends to $\text{P}\mathcal{G}$ . Moreover, we have the following lemma:

Lemma 2.10.

Assume that $-\emph{Id}\in\mathcal{G}$ . Let $\mathbb{I}_{k,\mathcal{X}}$ be a $\mathbb{K}$ -linear choice function for $\mathcal{X}$ and $\mathsf{p}_{k,\mathcal{X}}$ be the period function of $\mathbb{I}_{k,\mathcal{X}}$ . Then

\mathsf{p}_{k,\mathcal{X}}(\gamma)=\mathsf{p}_{k,\mathcal{X}}(-\gamma).\hskip 8.5359pt(\begin{subarray}{c}\gamma\in\mathcal{G}\end{subarray})

Proof.

The canonical choice function $\mathbb{I}_{k,\mathcal{X}}^{\tau_{0}}$ is $\mathbb{K}$ -linear and $(\mathbb{I}_{k,\mathcal{X}}-\mathbb{I}_{k,\mathcal{X}}^{\tau_{0}})\in\text{Hom}_{\mathbb{K}}(\mathcal{X},V_{k-2,\mathbb{C}})$ . But (2.9) implies that the identity holds for $\mathsf{p}_{k,\mathcal{X}}^{\tau_{0}}$ . Now the assertion is a consequence of (2.12). ∎

If $-\text{Id}\notin\mathcal{G}$ then the natural projection map $\mathcal{G}\twoheadrightarrow\text{P}\mathcal{G}$ identifies the group with its projectivization. On the other hand, if $-\text{Id}\in\mathcal{G}$ then Lemma 2.10 demonstrates that the linear cocycles on $\mathcal{G}$ gives rise to well-defined cocycles on $\text{P}\mathcal{G}$ . Thus, in both the cases, one obtains a projective Eichler-Shimura class

[\mathsf{p}_{k}]_{\text{P}\mathcal{G},\mathcal{X}}^{\mathbb{K}}\in H^{1}\big{(}\text{P}\mathcal{G},\text{Hom}_{\mathbb{K}}(\mathcal{X},V_{k-2,\mathbb{C}})\big{)}

so that its pullback under $\mathcal{G}\twoheadrightarrow\text{P}\mathcal{G}$ equals $[\mathsf{p}_{k}]^{\mathbb{K}}_{\mathcal{G},\mathcal{X}}$ .

3 Base point at the boundary

The current section constructs Eichler-Shimura integrals with the base point at infinity for a class of holomorphic functions that satisfies an appropriate growth rate at the boundary of the upper half-plane.

3.1 Growth conditions at the boundary

The closure of $\mathbb{H}$ in the Riemann sphere equals $\overline{\mathbb{H}}:=\mathbb{H}\cup\mathbb{R}\cup\{\infty\}$ . We endow the extended upper half-plane $\mathbb{H}^{\ast}\subseteq\overline{\mathbb{H}}$ with the subspace topology arising from $\widehat{\mathbb{C}}$ because it is favorable for discussing path integrals. In the theory of Baily-Borel compactification, one extends the topology of $\mathbb{H}$ to $\mathbb{H}^{\ast}$ by prescribing special types of neighborhoods for the points of $\mathbb{P}^{1}(\mathbb{Q})$ that define the so-called Satake topology [8, 2.1]. We next introduce the notion of triangular neighborhoods of the points of $\mathbb{P}^{1}(\mathbb{Q})$ which capture the punctured neighborhoods for the Satake topology. Our definition of the growth classes relies on the behavior of test functions on these subsets.

An admissible triangle based at $x\in\mathbb{P}^{1}(\mathbb{Q})$ refers to a geodesic triangle on $\mathbb{H}^{\ast}$ whose one vertex is at $x$ and two other vertices lie in $\mathbb{H}$ .

Definition 3.1.

A triangular neighborhood of $x\in\mathbb{P}^{1}(\mathbb{Q})$ is the intersection of an open neighborhood of $x$ in $\mathbb{H}^{\ast}$ with the interior of an admissible triangle based at $x$ .

Note that if $\mathcal{T}$ is an admissible triangle based at $x$ then the interior of $\mathcal{T}$ , denoted $\text{int}(\mathcal{T})$ , is a triangular neighborhood of $x$ . The action of $\gamma\in\Gamma$ maps a triangular neighborhood of $x$ onto a triangular neighborhood of $\gamma x$ .

Example 3.2.

This example describes the triangular neighborhoods of $\infty$ .

(i)

Let $t_{1},t_{2},h\in\mathbb{R}$ so that $t_{1}<t_{2}$ and $h>0$ . Set

$\mathcal{N}^{h}_{t_{1},t_{2}}=\{\tau\in\mathbb{H}\mid v(\tau)>h,t_{1}<u(\tau)<t_{2}\}.$

If $h>\frac{t_{2}-t_{1}}{2}$ then $\mathcal{N}^{h}_{t_{1},t_{2}}$ is a triangular neighborhood of $\infty$ (Figure 1).

Figure 1: The triangular neighborhood $\mathcal{N}^{h}_{t_{1},t_{2}}$
(ii)

Let $\tau_{1},\tau_{2}\in\mathbb{H}$ and consider the geodesic triangle $\mathcal{T}$ with vertices at $\tau_{1},\tau_{2},\infty$ . Write $t_{1}=u(\tau_{1}),t_{2}=u(\tau_{2})$ . If $t_{1}=t_{2}$ then $\mathcal{T}$ is degenerate and $\text{int}(\mathcal{T})=\emptyset$ . Without loss of generality assume that $t_{1}<t_{2}$ . The geodesic arc joining $\tau_{1}$ and $\tau_{2}$ is unique. Therefore there exists $h_{0}\geq\frac{t_{2}-t_{1}}{2}$ such that $\text{int}(\mathcal{T})=\mathcal{N}_{t_{1},t_{2}}^{h}\cup\mathcal{C}_{h}$ for each $h>h_{0}$ where $\mathcal{C}_{h}$ is a relatively compact subset of $\mathbb{H}$ , i.e., the closure of $\mathcal{C}_{h}$ in $\mathbb{H}$ is compact; cf. Figure 1. Now suppose $W$ is an open neighborhood $\infty$ in $\mathbb{H}^{\ast}$ . Choose $h>h_{0}$ with $\mathcal{N}_{t_{1},t_{2}}^{h}\subseteq W$ . It follows that $\text{int}(\mathcal{T})\cap W=\mathcal{N}_{t_{1},t_{2}}^{h}\cup(\mathcal{C}_{h}\cap W)$ . Thus each nonempty triangular neighborhood of $\infty$ is a union of $\mathcal{N}_{t_{1},t_{2}}^{h}$ and a relatively compact subset of $\mathbb{H}$ for suitable $t_{1},t_{2},h\in\mathbb{R}$ with $h>\frac{t_{2}-t_{1}}{2}>0$ .

Growth rate at infinity.

We first introduce a scale for the growth of a function in the triangular neighborhoods of $\infty$ . Let $\alpha$ be a fixed real number. A function $f\in\text{Hol}(\mathbb{H})$ decays with exponent $\alpha$ at $\infty$ if $\lvert f(\tau)\rvert\lvert\tau\rvert^{\alpha}$ is bounded on each triangular neighborhood of $\infty$ . Note that the definition does not require the bound to be uniform. Write

\mathfrak{F}_{\alpha}:=\{f\in\text{Hol}(\mathbb{H})\mid\text{$f$ decays with exponent $\alpha$ at $\infty$}\}.

It is clear that $\mathfrak{F}_{\alpha}$ is a $\mathbb{C}$ -subspace of $\text{Hol}(\mathbb{H})$ . Next, we provide an equivalent version of the decay condition at $\infty$ that is useful for practical applications.

Lemma 3.3.

The following are equivalent:

(i)

$f$ decays with exponent $\alpha$ at $\infty$ .
(ii)

$\lvert f(\tau)\rvert v(\tau)^{\alpha}$ is bounded on each of $\big{\{}\mathcal{N}^{h}_{t_{1},t_{2}}\mid\text{$t_{1}<t_{2}$, $h>h_{0}$}\big{\}}$ for some fixed $h_{0}\geq 0$ .

Proof.

Given $t_{1},t_{2},h\in\mathbb{R}$ with $t_{1}<t_{2}$ and $h>0$ there exists a positive constant $C=C(t_{1},t_{2},h)>1$ so that $v(\tau)\leq\lvert\tau\rvert\leq Cv(\tau)$ on $\mathcal{N}^{h}_{t_{1},t_{2}}$ . Since continuous functions are bounded on relatively compact subsets the assertion is a consequence of Example 3.2 and the definition above. ∎

One employs the equivalence in Lemma 3.3 with $h_{0}=1$ to discover that $\mathfrak{F}_{\alpha_{2}}\subseteq\mathfrak{F}_{\alpha_{1}}$ whenever $\alpha_{1}\leq\alpha_{2}$ . Set

\mathfrak{F}_{\infty}:=\bigcap_{\alpha\in\mathbb{R}}\mathfrak{F}_{\alpha}.

Growth near the real line.

The boundary points in $\mathbb{P}^{1}(\mathbb{Q})-\{\infty\}$ naturally play a role in the theory due to the action of $\Gamma$ on the extended upper half plane. Let $\alpha$ be a real number and $x_{0}\in\mathbb{P}^{1}(\mathbb{Q})-\{\infty\}$ . A function $f\in\text{Hol}(\mathbb{H})$ grows with exponent $\alpha$ at $x_{0}$ if $\lvert f(\tau)\rvert\lvert\tau-x_{0}\rvert^{\alpha}$ is bounded on each triangular neighborhood of $x_{0}$ . Define

\mathfrak{F}_{\alpha}^{x_{0}}:=\{f\in\text{Hol}(\mathbb{H})\mid\text{$f$ grows with exponent $\alpha$ at $x_{0}$}\}.

It is clear that $\mathfrak{F}^{x_{0}}_{\alpha}$ is a $\mathbb{C}$ -subspace of $\text{Hol}(\mathbb{H})$ . Note that $\lvert\tau-x_{0}\rvert\leq 1$ in a small enough neighborhood of $x_{0}$ . Therefore $\mathfrak{F}_{\alpha_{1}}^{x_{0}}\subseteq\mathfrak{F}_{\alpha_{2}}^{x_{0}}$ if $\alpha_{1}\leq\alpha_{2}$ . We write

\mathfrak{F}_{-\infty}^{x_{0}}:=\bigcap_{\alpha\in\mathbb{R}}\mathfrak{F}_{\alpha}^{x_{0}}.

Exponential polynomials.

The Fourier expansion of a weakly holomorphic modular form at $\infty$ involves a principal part of the Laurent expansion along with a constant term (Section 3.3). Let

\mathcal{J}_{n}:=\{(\kappa_{1},\cdots,\kappa_{n})\in\mathbb{R}_{\geq 0}^{n}\mid\kappa_{1}<\cdots<\kappa_{n}\}.\hskip 8.5359pt(\begin{subarray}{c}n\geq 1\end{subarray})

For $\kappa=(\kappa_{1},\cdots,\kappa_{n})\in\mathcal{J}_{n}$ define $\mathsf{EP}(\kappa)$ to be the $\mathbb{C}$ -subspace of $\text{Hol}(\mathbb{H})$ spanned by $\big{\{}\textbf{e}(-\kappa_{r}\tau)\mid 1\leq r\leq n\big{\}}$ . An exponential polynomial at $\infty$ of index $\kappa$ is an element of $\mathsf{EP}(\kappa)$ . The subspace $\mathsf{EP}(0)$ consists of the constant functions. Set

\mathsf{EP}:=\bigcup_{\begin{subarray}{c}\kappa\in\mathcal{J}_{n},\\ n\geq 1\end{subarray}}\mathsf{EP}(\kappa).

Since $\mathbb{R}_{\geq 0}$ is totally ordered $\mathsf{EP}$ is a $\mathbb{C}$ -subspace of $\text{Hol}(\mathbb{H})$ . Note that the polynomials in $\mathsf{EP}$ have unique holomorphic extensions to $\mathbb{C}$ .

Lemma 3.4.

With notation above

\mathsf{EP}\cap\mathfrak{F}_{\alpha}=\{0\}.\hskip 28.45274pt(\begin{subarray}{c}\alpha\in(0,\infty]\end{subarray})

Proof.

Suppose $f\in\mathsf{EP}\cap\mathfrak{F}_{\alpha}$ . Then $f(iv)\to 0$ as $v\to\infty$ . It follows that $f$ must be the zero exponential polynomial. Therefore $\mathsf{EP}\cap\mathfrak{F}_{\alpha}=\{0\}$ . ∎

Let $\alpha\in(0,\infty]$ . Define

\mathsf{EP}\mathfrak{F}_{\alpha}:=\mathsf{EP}+\mathfrak{F}_{\alpha}\subseteq\text{Hol}(\mathbb{H}).

Lemma 3.4 shows that $\mathsf{EP}\mathfrak{F}_{\alpha}=\mathsf{EP}\oplus\mathfrak{F}_{\alpha}$ . Suppose that $\pi_{\infty}:\mathsf{EP}\mathfrak{F}_{\alpha}\to\mathsf{EP}$ and $\pi_{c}:\mathsf{EP}\mathfrak{F}_{\alpha}\to\mathfrak{F}_{\alpha}$ are the projection maps. For brevity one writes $f_{\infty}:=\pi_{\infty}(f)$ and $f_{c}:=\pi_{c}(f)$ .

The final topic of the current subsection is the action of conformal automorphisms on growth families. With $(\lambda,\mu)\in\mathbb{R}_{>0}\times\mathbb{R}$ we attach a conformal automorphism

\phi_{\lambda,\mu}:\mathbb{H}\to\mathbb{H},\hskip 8.5359pt\tau\to\lambda\tau+\mu.

Lemma 3.3 implies that $f\circ\phi_{\lambda,\mu}\in\mathfrak{F}_{\alpha}$ whenever $f\in\mathfrak{F}_{\alpha}$ for some $\alpha\in(-\infty,\infty]$ . It is clear that if $f\in\mathsf{EP}(\kappa)$ then $f\circ\phi_{\lambda,\mu}\in\mathsf{EP}(\lambda\kappa)$ . Therefore $\mathsf{EP}$ is also closed under $\phi_{\lambda,\mu}$ .

Lemma 3.5.

Let $k\in\mathbb{Z}$ , $\alpha\in(0,\infty]$ , and $\gamma\in\Gamma$ .

(i)

If $\gamma\in\Gamma_{\infty}$ then both $\mathsf{EP}$ and $\mathfrak{F}_{\alpha}$ are stable under $\lvert_{k,\gamma}$ . Moreover

(3.1)

\pi_{\infty}(f\lvert_{k,\gamma})=\pi_{\infty}(f)\lvert_{k,\gamma},\hskip 8.5359pt\pi_{c}(f\lvert_{k,\gamma})=\pi_{c}(f)\lvert_{k,\gamma}.\hskip 14.22636pt(\begin{subarray}{c}\gamma\in\Gamma_{\infty},f\in\mathsf{EP}\mathfrak{F}_{\alpha}\end{subarray})

(ii)

Now suppose $\gamma\in\Gamma-\Gamma_{\infty}$ and $f\in\mathfrak{F}_{\alpha}$ . Then $f\lvert_{k,\gamma}\in\mathfrak{F}^{-\frac{d}{c}}_{k-\alpha}$ .

Proof.

(i)

Let $\gamma\in\Gamma_{\infty}$ and $f\in\text{Hol}(\mathbb{H})$ . Then $f\lvert_{k,\gamma}(\tau)=d^{-k}f\circ\phi_{1,\frac{b}{d}}(\tau)$ . We conclude that both $\mathsf{EP}$ and $\mathfrak{F}_{\alpha}$ are stable under $\lvert_{k,\gamma}$ . Now (3.1) is a consequence of Lemma 3.4.
(ii)

Suppose $\gamma\in\Gamma-\Gamma_{\infty}$ and $f\in\mathfrak{F}_{\alpha}$ . Here $c\neq 0$ and $\gamma^{-1}(\infty)=-\frac{d}{c}\in\mathbb{Q}$ . Assume that $\alpha\in\mathbb{R}_{>0}$ . Let $\mathcal{N}$ be a triangular neighborhood of $-\frac{d}{c}$ . Then $\gamma\mathcal{N}$ is a triangular neighborhood of $\infty$ . Note that $\lvert a\tau+b\rvert$ is bounded away from zero on $\mathcal{N}$ . We use the decay estimates for $\gamma\mathcal{N}$ to obtain $C>0$ so that $\lvert f\lvert_{k,\gamma}(\tau)\rvert\lvert\tau+\frac{d}{c}\rvert^{k-\alpha}\leq C$ for each $\tau\in\mathcal{N}$ . Since the statement holds for each $\alpha\in\mathbb{R}_{>0}$ it is true for $\alpha=\infty$ also.

∎

3.2 Eichler-Shimura integrals with base point at $\infty$

Our discussion primarily concerns the following classes of continuous paths on the extended upper half-plane:

•

Let $\tau_{1},\tau_{2}\in\mathbb{H}^{\ast}-\{\infty\}$ . A good path between $\tau_{1}$ and $\tau_{2}$ refers to a piecewise smooth path on $\mathbb{H}^{\ast}$ that maps $(0,1)$ inside $\mathbb{H}$ .
•

The unique good path joining $\tau\in\mathbb{H}^{\ast}-\{\infty\}$ and $\infty$ is the vertical ray originating at $\tau$ .

In the sequel the path integrals are along good paths joining the endpoints.

Lemma 3.6.

Let $\tau\in\mathbb{H}$ and $x\in\mathbb{P}^{1}(\mathbb{Q})-\{\infty\}$ . Suppose that $f\in\mathfrak{F}^{x}_{0}$ , i.e., $f$ is bounded on each triangular neighborhood of $x$ . Then

\int^{\tau}_{x}f(\xi)d\xi

does not depend on the choice of good path joining the endpoints.

Proof.

Let $\gamma_{1},\gamma_{2}$ be two good paths joining $\tau$ and $x$ . We use piecewise smoothness to obtain an admissible triangle $\mathcal{T}$ at $x$ and $0<\epsilon_{0}\ll 1$ so that $\gamma_{1}\big{(}[1-\epsilon,1)\big{)},\gamma_{2}\big{(}[1-\epsilon,1)\big{)}\subseteq\text{int}(\mathcal{T})$ for each $0<\epsilon\leq\epsilon_{0}$ . Note that $\text{int}(\mathcal{T})$ is geodesically convex. Set

\begin{gathered}\ell_{1}(\epsilon)=\text{length of $\gamma_{1}\bigl{\lvert}_{[1-\epsilon,1]}$},\;\ell_{2}(\epsilon)=\text{length of $\gamma_{2}\bigl{\lvert}_{[1-\epsilon,1]}$},\\ \ell_{12}(\epsilon)=\text{length of the geodesic segment joining $\gamma_{1}(1-\epsilon)$ and $\gamma_{2}(1-\epsilon)$}\end{gathered}

where length is measured with respect to the underlying metric on $\mathbb{C}$ . Suppose that $\lvert f(\tau)\rvert\leq M$ for all $\tau\in\text{int}(\mathcal{T})$ . Then

\bigl{\lvert}\int_{\gamma_{1}}f(\xi)d\xi-\int_{\gamma_{2}}f(\xi)d\xi\bigl{\rvert}\leq M\big{(}\ell_{1}(\epsilon)+\ell_{2}(\epsilon)+\ell_{12}(\epsilon)\big{)}.\hskip 8.5359pt(\begin{subarray}{c}0<\epsilon\leq\epsilon_{0}\end{subarray})

We let $\epsilon\to 0$ and conclude that the above inequality’s LHS is zero. ∎

Corollary 3.7.

Let $x_{1},x_{2}\in\mathbb{P}^{1}(\mathbb{Q})-\{\infty\}$ . Suppose that $f\in\mathfrak{F}^{x_{1}}_{0}\cap\mathfrak{F}^{x_{2}}_{0}$ . Then

\int^{x_{2}}_{x_{1}}f(\xi)d\xi

does not depend on the choice of good path joining the endpoints.

Proof.

The assertion is a simple consequence of Lemma 3.6. ∎

We next record an observation that is useful for the subsequent discussion.

Lemma 3.8.

Let $f\in\mathfrak{F}_{2}$ . Then $\int_{\tau}^{\infty}f(\xi)d\xi$ converges absolutely to define a holomorphic function on $\mathbb{H}$ and

\frac{d}{d\tau}\int_{\tau}^{\infty}f(\xi)d\xi=-f(\tau).

Proof.

The hypothesis on $f$ amounts to the fact that $\lvert f(\xi)\rvert\leq\frac{C}{v(\xi)^{2}}$ on every vertical strip bounded below. Now the assertion is an easy exercise in standard techniques of complex analysis. ∎

Let $k\geq 2$ and $\alpha\in[k,\infty]$ . Suppose $f\in\mathsf{EP}\mathfrak{F}_{\alpha}$ . Write $f=f_{\infty}+f_{c}$ . Set

\mathbb{I}^{\infty}_{k}(\tau,f,X):=\int^{\tau}_{0}f_{\infty}(\xi)(X-\xi)^{k-2}d\xi\in\text{Hol}(\mathbb{H})[X]_{k-2}.

It is clear that $\partial_{\tau}\big{(}\mathbb{I}^{\infty}_{k}(\tau,f,X)\big{)}=f_{\infty}(\tau)(X-\tau)^{k-2}d\tau$ . Also, we have $f(\tau)\tau^{r}\in\mathfrak{F}_{2}$ for each $0\leq r\leq k-2$ . One employs Lemma 3.8 to conclude that

\mathbb{I}^{c}_{k}(\tau,f,X):=-\int^{\infty}_{\tau}f_{c}(\xi)(X-\xi)^{k-2}d\xi\in\text{Hol}(\mathbb{H})[X]_{k-2}

and satisfies $\partial_{\tau}\big{(}\mathbb{I}^{c}_{k}(\tau,f,X)\big{)}=f_{c}(\tau)(X-\tau)^{k-2}d\tau$ . Put

(3.2)

\mathbb{I}^{*}_{k}(\tau,f,X)=\mathbb{I}^{\infty}_{k}(\tau,f,X)+\mathbb{I}^{c}_{k}(\tau,f,X)\in\text{Hol}(\mathbb{H})[X]_{k-2}.

The differential relations for $\mathbb{I}^{\infty}_{k}$ and $\mathbb{I}^{c}_{k}$ together imply that $\mathbb{I}^{*}_{k}(\tau,f,X)$ is a weight $k$ Eichler-Shimura integral of $f$ . We refer to $\mathbb{I}^{*}_{k}(\tau,f,X)$ as the Eichler-Shimura integral of $f$ with base point at $\infty$ .

Now suppose $\mathcal{G}$ is a finite index subgroup of $\Gamma$ and $\mathcal{X}\subseteq\mathsf{EP}\mathfrak{F}_{\alpha}$ is stable under the $\lvert_{k}$ action of $\mathcal{G}$ . Let

\mathbb{I}_{k,\mathcal{X}}^{*}:\mathcal{X}\to\text{Hol}(\mathbb{H})[X]_{k-2}

be the choice function attached to (3.2) and $\mathsf{p}^{*}_{k,\mathcal{X}}$ is the period function of $\mathbb{I}^{*}_{k,\mathcal{X}}$ . The Eichler-Shimura integrals with base point at $\infty$ behave like a canonical choice to some extent. In particular $\mathbb{I}_{k,\mathcal{X}}^{*}$ is linear whenever $\mathcal{X}$ is a linear subspace of $\text{Hol}(\mathbb{H})$ . Next, we provide an explicit formula for the period function. For convenience write

f_{\infty}(\gamma):=\pi_{\infty}(f\lvert_{\gamma}),\hskip 5.69046ptf_{c}(\gamma):=\pi_{c}(f\lvert_{\gamma}).\hskip 8.5359pt(\begin{subarray}{c}f\in\mathcal{X},\gamma\in\mathcal{G}\end{subarray})

Proposition 3.9.

Let $f\in\mathcal{X}$ and $\gamma\in\mathcal{G}$ . Then

(3.3)			$\displaystyle\mathsf{p}_{k,\mathcal{X}}^{*}(\gamma,f,X)$
			$\displaystyle=\int_{\gamma^{-1}0}^{\tau_{0}}f_{\infty}(\gamma^{-1})\lvert_{\gamma}(\xi)(X-\xi)^{k-2}d\xi-\int_{0}^{\tau_{0}}f_{\infty}(\xi)(X-\xi)^{k-2}d\xi$
			$\displaystyle-\int_{\tau_{0}}^{\gamma^{-1}\infty}f_{c}(\gamma^{-1})\lvert_{\gamma}(\xi)(X-\xi)^{k-2}d\xi+\int_{\tau_{0}}^{\infty}f_{c}(\xi)(X-\xi)^{k-2}d\xi$

for each $\tau_{0}\in\mathbb{H}$ .

Proof.

Let $\tau_{0}\in\mathbb{H}$ . One uses (3.2) with $\tau=\tau_{0}$ to conclude that

(3.4)			$\displaystyle\mathsf{p}_{k,\mathcal{X}}^{*}(\gamma,f,X)$
			$\displaystyle=(cX+d)^{k-2}\int^{\gamma\tau_{0}}_{0}f_{\infty}(\gamma^{-1})(\xi)(\frac{aX+b}{cX+d}-\xi)^{k-2}d\xi-\mathbb{I}^{\infty}_{k}(\tau_{0},f,X)$
			$\displaystyle-(cX+d)^{k-2}\int_{\gamma\tau_{0}}^{\infty}f_{c}(\gamma^{-1})(\xi)(\frac{aX+b}{cX+d}-\xi)^{k-2}d\xi-\mathbb{I}^{c}_{k}(\tau_{0},f,X).$

To deduce (3.3) from (3.4) one must show that the change of variable $\xi\to\gamma\xi$ transforms the first, resp. third, term in RHS of (3.4) into the first, resp. third, term in RHS of (3.3). We assume that the path of integration in the first integral of the RHS is the geodesic arc joining $0$ and $\gamma\tau_{0}$ which ensures that the transformed path is a good path between the endpoints whenever $\gamma^{-1}0=\infty$ . Now a routine computation using the definition of growth classes verifies that the boundary contribution in each of the improper integrals is negligible and the change of variable goes through without any problem. ∎

The following specialization of Proposition 3.9 is particularly useful to us.

Corollary 3.10.

Let $\gamma\in\mathcal{G}_{\infty}$ and $f\in\mathcal{X}$ . Then

(3.5)

\mathsf{p}_{k,\mathcal{X}}^{*}(\gamma,f,X)=\int_{\gamma^{-1}0}^{0}f_{\infty}(\xi)(X-\xi)^{k-2}d\xi.

Proof.

We have $\gamma^{-1}\infty=\infty$ . Moreover $a\neq 0$ and $\gamma^{-1}0=-\frac{b}{a}\in\mathbb{Q}$ . Lemma 3.5(i) implies that $f_{\infty}(\gamma^{-1})\lvert_{\gamma}=f_{\infty}$ and $f_{c}(\gamma^{-1})\lvert_{\gamma}=f_{c}$ . Therefore the assertion is a direct consequence of (3.3). ∎

3.3 Connections with the theory of modular forms

Let $k\geq 2$ be an integer and $\mathcal{G}$ be a finite index subgroup $\Gamma$ . Suppose that $\mathcal{M}_{k}^{!}(\mathcal{G})$ , resp. $\mathcal{M}_{k}(\mathcal{G})$ , is the space of weight $k$ weakly holomorphic, resp. holomorphic, modular forms on $\mathcal{G}$ . We choose a positive integer $N$ so that $T^{N}=\begin{pmatrix}1&N\\ 0&1\end{pmatrix}\in\mathcal{G}$ . A weakly holomorphic modular form $f\in\mathcal{M}_{k}^{!}(\mathcal{G})$ admits a $q$ -series expansion

(3.6)

f(\tau)=\sum_{n=-\infty}^{\infty}a(n)q_{N}^{n},\hskip 14.22636ptq_{N}=\textbf{e}(\tau/N),

where $\big{(}a(n)\big{)}_{n\in\mathbb{Z}}$ is a sequence of complex numbers so that $a(n)=0$ for $n<<0$ and $\sum_{n\geq 1}a(n)z^{n}$ has radius of convergence $1$ on the open unit disc. If $f\in\mathcal{M}_{k}(\mathcal{G})$ then $a(n)=0$ for each $n<0$ .

Lemma 3.11.

$\mathcal{M}_{k}^{!}(\mathcal{G})\subseteq\mathsf{EP}\mathfrak{F}_{\infty}$ .

Proof.

Let $f\in\mathcal{M}_{k}^{!}(\mathcal{G})$ and consider the Fourier expansion (3.6). The condition on the radius of convergence yields positive constants $C,M$ with $M>1$ so that $\lvert a(n)\rvert\leq CM^{n}$ for each $n\geq 1$ . Set $h_{0}=\frac{N\log M}{2\pi}$ and let $t_{1},t_{2},h\in\mathbb{R}$ with $t_{1}<t_{2}$ and $h>h_{0}$ . Then

\sum_{n\geq 1}\lvert a(n)q_{N}^{n}\rvert\leq\frac{CM}{1-\frac{M}{\exp{\frac{2\pi h}{N}}}}\exp{(-\frac{2\pi v(\tau)}{N})}.\hskip 14.22636pt(\begin{subarray}{c}\tau\in\mathcal{N}_{t_{1},t_{2}}^{h}\end{subarray})

Thus $\lvert\sum_{n\geq 1}a(n)q_{N}^{n}\rvert v(\tau)^{\alpha}$ is bounded on $\mathcal{N}_{t_{1},t_{2}}^{h}$ for each $\alpha>0$ . This calculation shows that $\sum_{n\geq 1}a(n)q_{N}^{n}\in\mathfrak{F}_{\infty}$ (Lemma 3.3). Therefore $f\in\mathsf{EP}\mathfrak{F}_{\infty}$ . ∎

Observe that the space of constants $\mathbb{C}$ is contained in $\mathsf{EP}$ . Set

\mathbb{C}\mathfrak{F}_{\infty}:=\mathbb{C}+\mathfrak{F}_{\infty}\subseteq\mathsf{EP}\mathfrak{F}_{\infty}.

The following result is an immediate consequence of the lemma above.

Corollary 3.12.

$\mathcal{M}_{k}(\mathcal{G})\subseteq\mathbb{C}\mathfrak{F}_{\infty}$ .

The formula (3.5) simplifies for the class of functions $\mathbb{C}\mathfrak{F}_{\infty}$ . Let the notation be as in Corollary 3.10. Moreover assume that $\mathcal{X}\subseteq\mathbb{C}\mathfrak{F}_{\infty}$ . Write $\gamma=\epsilon T^{r}$ where $\epsilon\in\{\pm 1\}$ and $r\in\mathbb{Z}$ .

Lemma 3.13.

$\mathsf{p}_{k,\mathcal{X}}^{*}(\gamma,f,X)=\pi_{\infty}(f)\frac{(X+r)^{k-1}-X^{k-1}}{k-1}$ .

Proof.

Note that $\gamma^{-1}0=-r$ . Moreover $f_{\infty}(\xi)$ is a constant function. Thus the desired identity follows from the integral formula (3.5). ∎

Our equivariant period function is particularly appropriate for the induced period polynomial perspective towards Eichler-Shimura theory since here one can always evaluate the period function at $\gamma=S$ even if $f$ is not $S$ -invariant; see Proposition 7.6. The results above also verify that the formalism of the current section applies to the weakly holomorphic modular forms. Thus the theory developed in this article offers a cohomological vista of the recent developments on weakly holomorphic modular forms [4].

4 Eichler-Shimura homomorphism

This section aims to retrieve the setup of Eichler-Shimura theory for invariant functions from the equivariant formalism developed in Section 2.

4.1 Definition and basic properties

We start with a simple algebraic observation that allows us to translate our theory of period functions into the classical language. Let $\mathcal{G}$ be an abstract group and $\mathcal{X}$ be a trivial right $\mathcal{G}$ -set. Suppose that $\mathbb{K}$ is a field and $V$ is a right $\mathbb{K}[\mathcal{G}]$ -module. There is a canonical $\mathbb{K}$ -linear isomorphism

(4.1)			$\displaystyle\Phi:\text{Fun}\big{(}\mathcal{G},\text{Fun}(\mathcal{X},V)\big{)}\xrightarrow{\cong}\text{Fun}\big{(}\mathcal{X},\text{Fun}(\mathcal{G},V)\big{)};$
(4.1)			$\displaystyle F\to\Phi(F),\hskip 8.5359pt\big{(}\Phi(F)(f)\big{)}(\gamma):=F(\gamma)(f).\hskip 14.22636pt(\begin{subarray}{c}f\in\mathcal{X},\gamma\in\mathcal{G}\end{subarray})$

Since $\mathcal{G}$ acts trivially on $\mathcal{X}$ the map $\Phi$ gives rise to $\mathbb{K}$ -linear isomorphisms at the level of cocyles and cohomology:

		$\displaystyle\Phi_{\text{cy}}:Z^{1}\big{(}\mathcal{G},\text{Fun}(\mathcal{X},V)\big{)}\xrightarrow{\cong}\text{Fun}\big{(}\mathcal{X},Z^{1}(\mathcal{G},V)\big{)},$
		$\displaystyle\overline{\Phi}:H^{1}\big{(}\mathcal{G},\text{Fun}(\mathcal{X},V)\big{)}\xrightarrow{\cong}\text{Fun}\big{(}\mathcal{X},H^{1}(\mathcal{G},V)\big{)}.$

Let the notation be as in Section 2.2 and assume that $\mathcal{X}\subseteq\text{Hol}(\mathbb{H})^{\mathcal{G}}$ . Now suppose $\mathbb{I}_{k,\mathcal{X}}$ is a choice function for $\mathcal{X}$ . With $\mathbb{I}_{k,\mathcal{X}}$ one associates an Eichler-Shimura cocycle map given by

(4.2)

\text{ES}_{k,\mathcal{X}}:\mathcal{X}\to Z^{1}\big{(}\mathcal{G},V_{k-2,\mathbb{C}}\big{)},\hskip 8.5359pt\text{ES}_{k,\mathcal{X}}:=\Phi_{\text{cy}}(\mathsf{p}_{k,\mathcal{X}})

where $\mathsf{p}_{k,\mathcal{X}}$ is the period function of $\mathbb{I}_{k,\mathcal{X}}$ . This cocycle map naturally descends to a map into the cohomology module.

Definition 4.1.

The Eichler-Shimura homomorphism attached to $(\mathcal{G},\mathcal{X})$ is the image of the Eichler-Shimura class $[\mathsf{p}_{k}]_{\mathcal{G},\mathcal{X}}$ under $\overline{\Phi}$ , i.e.,

[\text{ES}_{k}]_{\mathcal{G},\mathcal{X}}:\mathcal{X}\to H^{1}\big{(}\mathcal{G},V_{k-2,\mathbb{C}}\big{)},\hskip 8.5359pt[\text{ES}_{k}]_{\mathcal{G},\mathcal{X}}:=\overline{\Phi}([\mathsf{p}_{k}]_{\mathcal{G},\mathcal{X}}).

Proposition 2.7 ensures that $[\text{ES}_{k}]_{\mathcal{G},\mathcal{X}}$ is determined by $(\mathcal{G},\mathcal{X})$ and does not depend on the choice of the Eichler-Shimura cocycle map. If $(\mathcal{G},\mathcal{X})$ is clear from context then we drop $\mathcal{G}$ and $\mathcal{X}$ from the notation. If $\mathcal{X}$ consists of functions lying in suitable growth classes, say $\mathcal{X}\subseteq\mathbb{C}\mathfrak{F}_{\infty}$ , then the cocycle function (4.2) determined by base point at $\infty$ retrieves the period cocycle used by Zagier and subsequent authors [14]. However, at the level of cohomology, the corresponding Eichler-Shimura map coincides with the map defined by a canonical choice $\mathbb{I}_{k,\mathcal{X}}^{\tau_{0}}$ .

The following result records the basic properties of the Eichler-Shimura map.

Proposition 4.2.

Let $\mathcal{X}\subseteq\emph{Hol}(\mathbb{H})$ be a collection of $\mathcal{G}$ -invariant functions.

(i)

The class $[\mathsf{p}_{k}]_{\mathcal{G},\mathcal{X}}$ vanishes if and only if $[\emph{ES}_{k}]$ is the zero map.
(ii)

Let $\mathcal{X}$ be a $\mathbb{K}$ -subspace of $\emph{Hol}(\mathbb{H})$ for some subfield $\mathbb{K}$ of $\mathbb{C}$ . Then $[\emph{ES}_{k}]$ is automatically $\mathbb{K}$ -linear.

Proof.

Part (i) follows from the observation that $\overline{\Phi}$ is an isomorphism. To prove (ii) let $\mathbb{I}_{k,\mathcal{X}}^{\tau_{0}}$ be a canonical choice function for $\mathcal{X}$ . We know that $\mathbb{I}_{k,\mathcal{X}}^{\tau_{0}}$ is a $\mathbb{K}$ -linear choice function, i.e.,

\text{Im}\big{(}\mathsf{p}^{\tau_{0}}_{k,\mathcal{X}}\big{)}\subseteq\text{Hom}_{\mathbb{K}}(\mathcal{X},V_{k-2,\mathbb{C}})\subseteq\text{Fun}(\mathcal{X},V_{k-2,\mathbb{C}}).

One now uses the definition of $\Phi$ in (4.1) to discover that the cocycle map attached to $\mathbb{I}_{k,\mathcal{X}}^{\tau_{0}}$ is also $\mathbb{K}$ -linear. But $\text{ES}_{k,\mathcal{X}}$ descends to the map $[\text{ES}_{k}]$ . Therefore $[\text{ES}_{k}]$ is $\mathbb{K}$ -linear. ∎

4.2 The action of double coset operators

Next, we verify the Hecke equivariance of the Eichler-Shimura homomorphism defined in the previous subsection. Recall that the double coset operators generate the Hecke algebra of a congruence subgroup. In this article, the term Hecke operator refers to a general double coset operator unless otherwise specified.

The following paragraphs digress from our convention and deal with the action of subgroups of $\text{GL}_{2}^{+}(\mathbb{Q})$ on function spaces and module of coefficients (Section 2.3.1). Let $\alpha\in\text{GL}_{2}^{+}(\mathbb{Q})$ and $\mathcal{G}_{1}$ , $\mathcal{G}_{2}$ be two finite index subgroups of $\Gamma$ . Suppose that $\{\alpha_{1},\ldots,\alpha_{n}\}$ is a set of representatives for the orbits of the left $\mathcal{G}_{1}$ -action on $\mathcal{G}_{1}\alpha\mathcal{G}_{2}$ . For each $k\in\mathbb{Z}$ we have a $\mathbb{C}$ -linear double coset operator on functions given by

[\mathcal{G}_{1}\alpha\mathcal{G}_{2}]_{k}:\text{Hol}(\mathbb{H})^{\mathcal{G}_{1}}\to\text{Hol}(\mathbb{H})^{\mathcal{G}_{2}};\hskip 8.5359ptf\lvert_{[\mathcal{G}_{1}\alpha\mathcal{G}_{2}]_{k}}:=\sum_{j=1}^{n}f|_{k,\alpha_{j}}.

The operator $[\mathcal{G}_{1}\alpha\mathcal{G}_{2}]_{k}$ is independent of the choice of $\alpha$ -s. There is also a notion of double coset operators for group cohomology. Let $\mathcal{G}$ denote the subgroup of $\text{GL}_{2}^{+}(\mathbb{Q})$ generated by $\mathcal{G}_{1}$ , $\mathcal{G}_{2}$ , and $\alpha$ . Suppose that $V$ is a right $\mathbb{K}[\mathcal{G}]$ -module where $\mathbb{K}$ is a field of characteristic $0$ . Given $\gamma\in\mathcal{G}_{2}$ and $1\leq j\leq n$ we obtain unique $\gamma_{j}\in\mathcal{G}_{1}$ and $j(\gamma)\in\{1,\ldots,n\}$ so that $\alpha_{j}\gamma=\gamma_{j}\alpha_{j(\gamma)}$ . Define

(4.3)

\begin{gathered}{}_{V}:\text{Fun}(\mathcal{G}_{1},V)\to\text{Fun}(\mathcal{G}_{2},V),\\ P\bigl{\lvert}_{[\mathcal{G}_{1}\alpha\mathcal{G}_{2}]_{V}}(\gamma):=\sum_{j=1}^{n}P(\gamma_{j})\alpha_{j(\gamma)}.\end{gathered}

The operator $[\mathcal{G}_{1}\alpha\mathcal{G}_{2}]_{V}$ sends cocycles to cocycles and descends to a $\mathbb{K}$ -linear map in cohomology. We denote the map in cohomology by the same notation and often drop $V$ from the subscript. The map at the level of cohomology does not depend on the choice of representatives for $\mathcal{G}_{1}$ -orbits. Readers may like to notice that in our convention the double coset operators act on the right of the cohomology module exactly like function spaces.

Proposition 4.3.

(cf. [16], Proposition 8.5) Let $\mathcal{X}_{1}\subseteq\emph{Hol}(\mathbb{H})^{\mathcal{G}_{1}}$ and $\mathcal{X}_{2}\subseteq\emph{Hol}(\mathbb{H})^{\mathcal{G}_{2}}$ be two subsets of functions so that $[\mathcal{G}_{1}\alpha\mathcal{G}_{2}]_{k}(\mathcal{X}_{1})\subseteq\mathcal{X}_{2}$ . Then the Eichler-Shimura maps commute with the action of double coset operators, i.e., $[\mathcal{G}_{1}\alpha\mathcal{G}_{2}]\circ[\emph{ES}_{k}]_{\mathcal{G}_{1},\mathcal{X}_{1}}=[\emph{ES}_{k}]_{\mathcal{G}_{2},\mathcal{X}_{2}}\circ[\mathcal{G}_{1}\alpha\mathcal{G}_{2}]_{k}$ .

Proof.

Let $\mathcal{G}$ be the subgroup of $\text{GL}_{2}^{+}(\mathbb{Q})$ generated by $\mathcal{G}_{1}\cup\mathcal{G}_{2}\cup\{\alpha\}$ . Suppose that $\mathcal{X}$ is a $\mathcal{G}$ -stable subset of $\text{Hol}(\mathbb{H})$ containing $\mathcal{X}_{1}\cup\mathcal{X}_{2}$ . We fix a choice function $\mathbb{I}_{k,\mathcal{X}}$ and consider the period function $\mathsf{p}_{k,\mathcal{X}}$ . Restriction yields choice functions for $\mathcal{X}_{1}$ and $\mathcal{X}_{2}$ . Let $f\in\mathcal{X}_{1}$ and $\gamma\in\mathcal{G}_{2}$ . Then

		$\displaystyle\mathsf{p}_{k,\mathcal{X}}(\gamma_{j},f,X)\lvert_{\alpha_{j(\gamma)}}=\mathsf{p}_{k,\mathcal{X}}(\gamma_{j}\alpha_{j(\gamma)},f\lvert_{\alpha_{j(\gamma)}},X)-\mathsf{p}_{k,\mathcal{X}}(\alpha_{j(\gamma)},f\lvert_{\alpha_{j(\gamma)}},X)\hskip 5.69046pt(\begin{subarray}{c}1\leq j\leq n\end{subarray})$
		$\displaystyle=\mathsf{p}_{k,\mathcal{X}}(\gamma,f\lvert_{\alpha_{j(\gamma)}},X)+\mathsf{p}_{k,\mathcal{X}}(\alpha_{j},f\lvert_{\alpha_{j}},X)\bigl{\lvert}_{\gamma}-\mathsf{p}_{k,\mathcal{X}}(\alpha_{j(\gamma)},f\lvert_{\alpha_{j(\gamma)}},X).$

Summing over $j$ one finds that

[\mathcal{G}_{1}\alpha\mathcal{G}_{2}]\circ\text{ES}_{k,\mathcal{X}_{1}}(f)-\text{ES}_{k,\mathcal{X}_{2}}\circ[\mathcal{G}_{1}\alpha\mathcal{G}_{2}]_{k}(f)

equals a coboundary on $\mathcal{G}_{2}$ . ∎

4.3 Equivariance under group action

We begin with a recap of the conjugation action on the group cohomology [6, III.8]. Let $\mathcal{H}$ be an abstract group and $\mathcal{G}$ be a subgroup of $\mathcal{H}$ . Suppose that $\mathbb{K}$ is a field of characteristic $0$ and $V$ is a right $\mathbb{K}[\mathcal{H}]$ -module. For each $h\in\mathcal{H}$ conjugation gives rise to an isomorphism of pairs

c(h):(\mathcal{G},V)\to(h^{-1}\mathcal{G}h,V);\hskip 8.5359pt(g\mapsto h^{-1}gh,m\mapsto mh)

that yields a pullback isomorphism $c(h)^{*}:H^{*}(h^{-1}\mathcal{G}h,V)\to H^{*}(\mathcal{G},V)$ . For $\alpha\in H^{*}(\mathcal{G},V)$ set

(4.4)

\alpha\lvert_{h}:=\big{(}c(h)^{*}\big{)}^{-1}(\alpha)\in H^{*}(h^{-1}\mathcal{G}h,V).

Suppose that $\mathcal{G}$ is a normal subgroup of $\mathcal{H}$ . Then (4.4) defines a right $\mathbb{K}[\mathcal{H}]$ -module structure on $H^{*}(\mathcal{G},V)$ . The subgroup $\mathcal{G}$ acts trivially on $H^{*}(\mathcal{G},V)$ and the $\mathcal{H}$ -action descends to an action of $\mathcal{H}/\mathcal{G}$ .

Lemma 4.4.

Let the notation be as above. Suppose that $\mathcal{G}$ is a finite index normal subgroup of $\mathcal{H}$ . Then the restriction homomorphism

\emph{res}^{\mathcal{H}}_{\mathcal{G}}:H^{*}(\mathcal{H},V)\to H^{*}(\mathcal{G},V)

maps $H^{*}(\mathcal{H},V)$ isomorphically onto $H^{*}(\mathcal{G},V)^{\mathcal{H}}$ .

Proof.

See Proposition 10.4 in [6, III.10]. ∎

We next explicate the action of $\mathcal{H}$ at the level of $1$ -cocycles; cf. [6, p.79]. Assume that $\mathcal{G}$ is a normal subgroup of $\mathcal{H}$ . The (right) conjugation action of $\mathcal{H}$ on $\mathcal{G}$ turns $\text{Fun}(\mathcal{G},V)$ into a $\mathbb{K}[\mathcal{H}]$ -module as follows:

(4.5)

(P,h)\mapsto P\lvert_{h};\hskip 8.5359ptP\lvert_{h}(g)=P(hgh^{-1})h.

Note that $Z^{1}(\mathcal{G},V)$ and $B^{1}(\mathcal{G},V)$ are $\mathbb{K}[\mathcal{H}]$ -submodules of $\text{Fun}(\mathcal{G},V)$ . Thus $H^{1}(\mathcal{G},V)$ inherits a $\mathcal{H}$ -action that coincides with (4.4).

Example 4.5.

Let $\mathcal{G}$ be a subgroup of $\Gamma$ containing $-\text{Id}$ and assume that $-\text{Id}$ acts on $V$ as $-\text{Id}_{V}$ . Then, by (4.5), the conjugation action of $-\text{Id}$ on $H^{1}(\mathcal{G},V)$ equals $-\text{Id}_{H^{1}(\mathcal{G},V)}$ . But each element of $\mathcal{G}$ must act trivially on the cohomology. It follows that $H^{1}(\mathcal{G},V)$ must be zero. In particular, if $k$ is odd and $-\text{Id}\in\mathcal{G}$ then $H^{1}(\mathcal{G},V_{k-2,\mathbb{K}})$ is zero.

Remark 4.6.

Let the notation be as in Lemma 4.4 but we drop the assumption that $\mathcal{G}$ is normal in $\mathcal{H}$ . By general theory

\text{cores}^{\mathcal{H}}_{\mathcal{G}}\circ\text{res}^{\mathcal{H}}_{\mathcal{G}}=\text{multiplication by $[\mathcal{H}:\mathcal{G}]$}

where $\text{res}^{\mathcal{H}}_{\mathcal{G}}$ is the restriction map and $\text{cores}^{\mathcal{H}}_{\mathcal{G}}:H^{*}(\mathcal{G},V)\to H^{*}(\mathcal{H},V)$ is the corestriction map [6, III.9]. Therefore the restriction map is injective.

We put ourselves in the setting of Section 4.1 and suppose that $\mathcal{H}$ is a subgroup of $\Gamma$ that contains $\mathcal{G}$ as a normal subgroup. Moreover assume that $\mathcal{X}$ is stable under the $\lvert_{k}$ -action of $\mathcal{H}$ . The discussion of this subsection yields a canonical right $\mathbb{C}[\mathcal{H}]$ -module structure on $H^{1}(\mathcal{G},V_{k-2,\mathbb{C}})$ .

Proposition 4.7.

The Eichler-Shimura homomorphism

[\emph{ES}_{k}]:\mathcal{X}\to H^{1}(\mathcal{G},V_{k-2,\mathbb{C}})

is a morphism of $\mathcal{H}$ -sets.

Proof.

We fix a choice function for $\mathcal{X}$ and consider the period function and the cocycle map attached to that choice. Let $f\in\mathcal{X}$ and $\gamma_{0}\in\mathcal{H}$ . Then

	$\displaystyle\text{ES}_{k,\mathcal{X}}(f)\lvert_{\gamma_{0}}(\gamma)$	$\displaystyle=\mathsf{p}_{k,\mathcal{X}}(\gamma_{0}\gamma\gamma_{0}^{-1},f,X)\lvert_{\gamma_{0}}\hskip 14.22636pt(\begin{subarray}{c}\gamma\in\mathcal{G}\end{subarray})$
		$\displaystyle=\mathsf{p}_{k,\mathcal{X}}(\gamma_{0},f\lvert_{\gamma_{0}\gamma^{-1}},X)\lvert_{\gamma}+\mathsf{p}_{k,\mathcal{X}}(\gamma,f\lvert_{\gamma_{0}},X)+\mathsf{p}_{k,\mathcal{X}}(\gamma_{0}^{-1},f,X)\lvert_{\gamma_{0}}$
		$\displaystyle=\text{ES}_{k,\mathcal{X}}(f\lvert_{\gamma_{0}})(\gamma)+\mathsf{p}_{k,\mathcal{X}}(\gamma_{0},f\lvert_{\gamma_{0}},X)\lvert_{\gamma}-\mathsf{p}_{k,\mathcal{X}}(\gamma_{0},f\lvert_{\gamma_{0}},X).$

Therefore $[\text{ES}_{k}](f)\lvert_{\gamma_{0}}=[\text{ES}_{k}](f\lvert_{\gamma_{0}})$ in $H^{1}(\mathcal{G},V_{k-2,\mathbb{C}})$ as desired. ∎

Remark 4.8.

With the notation above let $\mathcal{G}_{1}$ be a subgroup of $\mathcal{H}$ containing $\mathcal{G}$ . Set $\mathcal{X}_{1}:=\mathcal{X}^{\mathcal{G}_{1}}$ . Then using compatible choice functions for $\mathcal{X}$ and $\mathcal{X}_{1}$ (cf. (2.13) and (2.14)) one discovers that the following diagram commutes:

Here the top arrow is inclusion and the bottom arrow is the restriction map in group cohomology. Note that the bottom arrow maps $H^{1}(\mathcal{G}_{1},V_{k-2,\mathbb{C}})$ isomorphically onto the $\mathcal{G}_{1}$ -invariant subspace of $H^{1}(\mathcal{G},V_{k-2,\mathbb{C}})$ .

5 Preparations for the main theorem

5.1 Parabolic cohomology

We follow the treatment given in the book of Shimura [16] and first introduce the notion of parabolic cohomology for the subgroups of the projective modular group $\text{P}\Gamma$ . As per standard convention, we denote the cusp of a finite index subgroup of $\Gamma$ or $\text{P}\Gamma$ by choice of representative for the equivalence class. In particular, the stabilizer of a cusp refers to the stabilizer of a fixed choice of representative for the cusp.

Let $G$ be a finite index subgroup of $\text{P}\Gamma$ . Suppose that $\mathcal{C}(G)$ is the set of cusps attached to $G$ . For $x\in\mathcal{C}(G)$ let $G_{x}$ denote the stablizer of $x$ in $G$ . Write $G_{x}=\langle\pi_{x}\rangle$ . We also fix a set of representatives $\{\varepsilon_{1},\ldots,\varepsilon_{r}\}$ of the elliptic elements in $G$ . Let $V$ be a right $\mathbb{K}[G]$ -module where $\mathbb{K}$ is a field of characteristic $0$ . For a subset $Q$ of $G$ one defines the cohomology groups vanishing on $Q$ , denoted $H^{\ast}_{Q}(G,V)$ , by imposing constraints on $1$ -cochains [16, p.224]. Suppose that $\mathsf{P}$ is the collection of all parabolic elements in $G$ . We refer to $H^{\ast}_{\mathsf{P}}(G,V)$ as the parabolic cohomology of $V$ . However, for practical purposes, it is more convenient to work with $Q=\{\pi_{x}\mid x\in\mathcal{C}(G)\}\subseteq\mathsf{P}$ . The space of $1$ -cocycles vanishing on $Q$ and $\mathsf{P}$ coincide and there is a canonical identification $H^{1}_{\mathsf{P}}(G,V)\cong H^{1}_{Q}(G,V)$ . This description of $H^{1}_{\mathsf{P}}(G,V)$ provides an exact sequence of $\mathbb{K}$ -vector spaces

(5.1)

0\to H^{1}_{\mathsf{P}}(G,V)\to H^{1}(G,V)\to\oplus_{x\in\mathcal{C}(G)}H^{1}(G_{x},V)

where the first map is inclusion and the second map is componentwise restriction. The parabolic subspace is compatible with the restriction map.

Lemma 5.1.

Let $H\subseteq G$ be two finite index subgroups of $\emph{P}\Gamma$ and $V$ be a $\mathbb{K}[G]$ -module. Suppose that $\alpha\in H^{1}(G,V)$ . Then

$\alpha\in H^{1}_{\mathsf{P}}(G,V)$ if and only if $\emph{res}^{G}_{H}(\alpha)\in H^{1}_{\mathsf{P}}(H,V)$ .

Proof.

Let $x\in\mathbb{P}^{1}(\mathbb{Q})$ . The natural inclusions give rise to a commutative diagram

where all the arrows are restriction maps. By Remark 4.6 the bottom arrow in the diagram above is injective. Therefore the image of $\alpha$ in $H^{1}(G_{x},V)$ is zero if and only if the image of $\text{res}^{G}_{H}(\alpha)$ in $H^{1}(H_{x},V)$ is zero. Since this assertion holds for each $x\in\mathbb{P}^{1}(\mathbb{Q})$ the statement follows from (5.1). ∎

We can also compute $H^{*}(G,V)$ in terms of a $2$ -dimensional simplicial complex $\mathfrak{K}$ that contains the fixed points of $\varepsilon_{j}$ -s as $0$ -simplices and admits a simplicial $G$ -action [16, p.225]. With $\mathfrak{K}$ one associates a complex of finitely presented $\mathbb{K}[G]$ -modules $(A_{\ast},\partial)$ so that the complex

A^{\ast}(V):=\text{Hom}_{\mathbb{K}[G]}(A_{\ast},V)

computes $H^{\ast}(G,V)$ . Readers may like to note that Shimura defines $A_{\ast}$ as a complex of left modules which we endow with the canonical right action $x\gamma:=\gamma^{-1}x$ . Moreover $\mathfrak{K}$ also provides an alternate description of $H^{\ast}_{Q}(G,V)$ . In particular, there is an isomorphism

(5.2)

H^{2}_{Q}(G,V)\cong\frac{V}{\{v(\gamma-1)\mid v\in V,\gamma\in G\}}.

Lemma 5.2.

Let $\mathbb{L}$ be a field containing $\mathbb{K}$ . Then the natural change of scalar map induces isomorphisms

H^{1}(G,V)\otimes_{\mathbb{K}}\mathbb{L}\cong H^{1}(G,V\otimes_{\mathbb{K}}\mathbb{L}),\;\;H^{1}_{\mathsf{P}}(G,V)\otimes_{\mathbb{K}}\mathbb{L}\cong H^{1}_{\mathsf{P}}(G,V\otimes_{\mathbb{K}}\mathbb{L}).

Proof.

Follows from the construction above and the sequence (5.1). For details see [10, p. 168]. ∎

Now suppose $V$ is a finite-dimensional $\mathbb{K}$ -vector space. Write

\begin{gathered}n_{0}=\text{dim}_{\mathbb{K}}V^{G},\hskip 8.5359ptn_{1}=\text{dim}_{\mathbb{K}}H^{2}_{Q}(G,V),\\ n_{j}^{\text{ell}}=\text{dim}_{\mathbb{K}}\big{\{}v\in V\mid v\varepsilon_{j}=v\big{\}},\hskip 11.38092pt(\begin{subarray}{c}1\leq j\leq r\end{subarray})\\ n_{x}^{\text{par}}=\text{dim}_{\mathbb{K}}\big{\{}v(\pi_{x}-1)\mid v\in V\big{\}}.\hskip 11.38092pt(\begin{subarray}{c}x\in\mathcal{C}(G)\end{subarray})\end{gathered}

Proposition 5.3.

Let the notation be as above and $g$ denote the genus of the canonical compact Riemann surface attached to $G\backslash\mathbb{H}$ . Then

(i)

$\sum_{j=0}^{2}(-1)^{j}\emph{dim}_{\mathbb{K}}H^{j}(G,V)=\big{(}2-2g-\lvert\mathcal{C}(G)\rvert\big{)}\emph{dim}_{\mathbb{K}}V$
$\hskip 213.39566pt-\sum_{j=1}^{r}(\emph{dim}_{\mathbb{K}}V-n_{j}^{\emph{ell}})$ ,
(ii)

$\emph{dim}_{\mathbb{K}}H^{1}_{\mathsf{P}}(G,V)=(2g-2)\emph{dim}_{\mathbb{K}}V+n_{0}+n_{1}$
$\hskip 142.26378pt+\sum_{x\in\mathcal{C}(G)}n_{x}^{\emph{par}}+\sum_{j=1}^{r}(\emph{dim}_{\mathbb{K}}V-n^{\emph{ell}}_{j})$ .

Proof.

The second identity is standard and we refer the readers to [16, p. 229] for a demonstration. The first identity is a consequence of the arguments employed to establish the other identity. Let $m_{j}$ be the number of $G$ -inequivalent $j$ -simplices of $\mathfrak{K}$ . Here $A_{1}$ , resp. $A_{2}$ , is a free $\mathbb{K}[G]$ -module of rank $m_{1}$ , resp. $m_{2}$ but the elliptic fixed points may prevent $A_{0}$ from being a free module. A straightforward analysis demonstrates that

\begin{gathered}\text{dim}_{\mathbb{K}}A^{0}(V)=m_{0}\text{dim}_{\mathbb{K}}V-\sum_{j=1}^{r}(\text{dim}_{\mathbb{K}}V-n_{j}^{\text{ell}}),\\ \text{dim}_{\mathbb{K}}A^{1}(V)=m_{1}\text{dim}_{\mathbb{K}}V,\hskip 28.45274pt\text{dim}_{\mathbb{K}}A^{2}(V)=m_{2}\text{dim}_{\mathbb{K}}V.\end{gathered}

But

\sum_{j=0}^{2}(-1)^{j}\text{dim}_{\mathbb{K}}H^{j}(G,V)=\sum_{j=0}^{2}(-1)^{j}\text{dim}_{\mathbb{K}}A^{j}(V)

and $m_{0}-m_{1}+m_{2}=2-2g-\lvert\mathcal{C}(G)\rvert$ . We now substitute the values of $\text{dim}_{\mathbb{K}}A^{j}(V)$ in the identity above to arrive at (i). ∎

Lemma 5.4.

Let $G$ be a finite index subgroup of $\emph{P}\Gamma$ and $V$ be a right $\mathbb{K}[G]$ -module. Then $H^{2}(G,V)=0$ .

Proof.

Let $H$ be a finite index subgroup contained in $G$ so that $H$ is torsion-free. Then the arguments given in Proposition 1 of [10, 6.1] demonstrates that $H^{2}(H,V)=0$ . But the natural restriction map from $H^{2}(G,V)$ to $H^{2}(H,V)$ is injective (Remark 4.6). Therefore $H^{2}(G,V)$ is also zero. Note that Hida originally proves the result for congruence subgroups but his proof goes through for all torsion-free finite index subgroups. ∎

Next, we introduce a few conventions to facilitate the transition between the subgroups of $\Gamma$ and $\text{P}\Gamma$ . Let $\mathcal{G}$ be a subgroup of $\Gamma$ and $V$ be a right $\mathbb{K}[\mathcal{G}]$ module. If $\mathcal{G}$ is a finite index subgroup and $\mathsf{P}$ is the subset of all parabolic elements of $\mathcal{G}$ then one can introduce the notion of parabolic subspace, denoted $H^{1}_{\mathsf{P}}(\mathcal{G},V)$ , using the same formalism mentioned before. The subspace $H^{1}_{\mathsf{P}}(\mathcal{G},V)$ of $H^{1}(\mathcal{G},V)$ is stable under the action of double coset operators on cohomology [16, 8.3]. We call $(\mathcal{G},V)$ a descent module if either $-\text{Id}\notin\mathcal{G}$ or $-\text{Id}\in\mathcal{G}$ acts trivially on $V$ . For example, the familiar pair $(\mathcal{G},V_{k-2,\mathbb{K}})$ is a descent module if either $k$ is even or $-\text{Id}\notin\mathcal{G}$ . Let $(\mathcal{G},V)$ be a descent module. Then there exists a natural $\mathbb{K}[\text{P}\mathcal{G}]$ -structure on $V$ so that the $\mathbb{K}[\mathcal{G}]$ -structure on $V$ arises from the projection $\mathcal{G}\twoheadrightarrow\text{P}\mathcal{G}$ . Moreover, the pullback of the projection map identifies $H^{1}(\text{P}\mathcal{G},V)$ and $H^{1}(\mathcal{G},V)$ [16, 8.2]. Now suppose $\mathcal{G}$ is a finite index subgroup of $\Gamma$ . Note that the image of $\mathsf{P}$ in $\text{P}\mathcal{G}$ , denoted $\bar{\mathsf{P}}$ , equals the collection of parabolic elements of $\text{P}\mathcal{G}$ and there is an identification $H^{1}_{\mathsf{P}}(\mathcal{G},V)\cong H^{1}_{\bar{\mathsf{P}}}(\text{P}\mathcal{G},V)$ . The procedure described above allows us to rewrite (5.1) as

(5.3)

0\to H^{1}_{\mathsf{P}}(\mathcal{G},V)\to H^{1}(\mathcal{G},V)\to\oplus_{x\in\mathcal{C}(\mathcal{G})}H^{1}(\mathcal{G}_{x},V).

And furthermore, the change of scalar property of Lemma 5.2 lifts to $\mathcal{G}$ in the following manner

(5.4)

H^{1}(\mathcal{G},V)\otimes_{\mathbb{K}}\mathbb{L}\cong H^{1}(\mathcal{G},V\otimes_{\mathbb{K}}\mathbb{L}),\;H^{1}_{\mathsf{P}}(\mathcal{G},V)\otimes_{\mathbb{K}}\mathbb{L}\cong H^{1}_{\mathsf{P}}(\mathcal{G},V\otimes_{\mathbb{K}}\mathbb{L})

whenever $\mathbb{L}$ is a field extension of $\mathbb{K}$ .

Remark 5.5.

Let $\mathcal{G}$ be a finite index subgroup so that $-\text{Id}\in\mathcal{G}$ and $-\text{Id}$ acts on $V$ by $-\text{Id}_{V}$ . Then $H^{1}(\mathcal{G},V)$ is zero (Example 4.5) and the change of coefficients isomorphisms (5.4) trivially hold for such pairs. We use this observation for the pair $(\mathcal{G},V_{k-2,\mathbb{K}})$ where $k$ is odd and $-\text{Id}\in\mathcal{G}$ to obtain change of scalars isomorphisms for all $(\mathcal{G},V_{k-2,\mathbb{K}})$ .

The final topic of this subsection is the third term appearing in (5.3). We first consider the subgroup $\langle\langle T^{h}\rangle\rangle\subseteq\Gamma$ for some positive integer $h$ where $\langle\langle\cdot\rangle\rangle$ refers to the subgroup generated by the element. Since $\langle\langle T^{h}\rangle\rangle\subseteq\Gamma$ is an infinite cyclic group it follows that

(5.5)

H^{1}(\langle\langle T^{h}\rangle\rangle,V_{k-2,\mathbb{K}})\cong\frac{V_{k-2,\mathbb{K}}}{V_{k-2,\mathbb{K}}(T^{h}-1)}\cong\mathbb{K}.

Here the last isomorphism stems from a linear functional $V_{k-2,\mathbb{K}}\to\mathbb{K}$ described as $P\mapsto$ the coefficient of $X^{k-2}$ in $P(T^{h})$ . Let $\mathcal{G}$ be a finite index subgroup $\Gamma$ and $x\in\mathcal{C}(\mathcal{G})$ . Choose $\gamma\in\Gamma$ so that $x=\mathcal{G}(\gamma\infty)$ . Then $\gamma^{-1}\mathcal{G}_{x}\gamma=\gamma^{-1}\mathcal{G}\gamma\cap\Gamma_{\infty}$ . In more detail, there exists a positive integer $h$ so that

\gamma^{-1}\mathcal{G}_{x}\gamma=\begin{cases}\langle\langle-\text{Id},T^{h}\rangle\rangle\text{ or }\langle\langle T^{h}\rangle\rangle,&\text{if $x\in\mathcal{C}_{\infty}(\mathcal{G})$;}\\ \langle\langle-T^{h}\rangle\rangle,&\text{if $x\notin\mathcal{C}_{\infty}(\mathcal{G})$}\end{cases}

where $\mathcal{C}_{\infty}(\mathcal{G})$ is the collection of regular cusps. The following result is certainly well-known among the experts; cf. [10, 6.3].

Proposition 5.6.

Let $\mathcal{G}$ be as above and assume that $-\emph{Id}\notin\mathcal{G}$ if $k$ is odd. Then

H^{1}(\mathcal{G}_{x},V_{k-2,\mathbb{K}})\cong\begin{cases}\mathbb{K},&\text{if $k$ is even;}\\ \mathbb{K},&\text{if $k$ is odd and $x\in\mathcal{C}_{\infty}(\mathcal{G})$;}\\ 0,&\text{if $k$ is odd and $x\notin\mathcal{C}_{\infty}(\mathcal{G})$.}\end{cases}

Proof.

We use the conjugation by $\gamma$ isomorphism (Section 4.3) to obtain

H^{1}(\mathcal{G}_{x},V_{k-2,\mathbb{K}})\cong H^{1}(\gamma^{-1}\mathcal{G}_{x}\gamma,V_{k-2,\mathbb{K}}).

If $\gamma^{-1}\mathcal{G}_{x}\gamma=\langle\langle T^{h}\rangle\rangle$ then $H^{1}(\gamma^{-1}\mathcal{G}_{x}\gamma,V_{k-2,\mathbb{K}})\cong\mathbb{K}$ . Now suppose $\gamma^{-1}\mathcal{G}_{x}\gamma=\langle\langle-\text{Id},T^{h}\rangle\rangle$ . Observe that

H^{1}(\langle\langle-\text{Id},T^{h}\rangle\rangle,V_{k-2,\mathbb{K}})\cong H^{1}(\langle\langle T^{h}\rangle\rangle,V_{k-2,\mathbb{K}})^{\langle\langle-\text{Id},T^{h}\rangle\rangle}.

Here $k$ is even and $-\text{Id}$ acts trivially on $H^{1}(\langle\langle T^{h}\rangle\rangle,V_{k-2,\mathbb{K}})$ ; see (4.5). As a consequence $H^{1}(\langle\langle-\text{Id},T^{h}\rangle\rangle,V_{k-2,\mathbb{K}})\cong\mathbb{K}$ . It remains to consider the case $\gamma^{-1}\mathcal{G}_{x}\gamma=\langle\langle-T^{h}\rangle\rangle$ . In this situation, there is an isomorphism

H^{1}(\langle\langle-T^{h}\rangle\rangle,V_{k-2,\mathbb{K}})\cong H^{1}(\langle\langle T^{2h}\rangle\rangle,V_{k-2,\mathbb{K}})^{\langle\langle-T^{h}\rangle\rangle}.

One again employs (4.5) to conclude that $-T^{h}$ acts on $H^{1}(\langle\langle T^{2h}\rangle\rangle,V_{k-2,\mathbb{K}})$ by $(-1)^{k-2}$ . Hence $H^{1}(\langle\langle-T^{h}\rangle\rangle,V_{k-2,\mathbb{K}})\cong\mathbb{K}$ if $k$ is even and equals zero if $k$ is odd. ∎

5.2 Isomorphism theorem for cusp forms

We first describe the action of complex conjugation on our cohomology spaces. Let $k$ be an integer $\geq 2$ and $\mathcal{G}$ be a finite index subgroup of $\Gamma$ . The complex conjugation on $\mathbb{C}$ gives rise to an conjugate linear automorphism of $Z^{1}\big{(}\mathcal{G},V_{k-2,\mathbb{C}}\big{)}$ described as $P\mapsto\#P$ where $\#P(\gamma)=\overline{P(\gamma)}$ . This automorphism descends to a complex conjugation on $H^{1}\big{(}\mathcal{G},V_{k-2,\mathbb{C}}\big{)}$ . Now suppose $\mathcal{S}_{k}(\mathcal{G})$ is the space of all weight $k$ cusp form on $\mathcal{G}$ and $\mathcal{S}^{a}_{k}(\mathcal{G})=\{\bar{f}\mid f\in\mathcal{S}_{k}(\mathcal{G})\}$ is the space of antiholomorphic cusp forms on $\mathcal{G}$ . Let

[\text{ES}_{k}]_{\text{c},\mathcal{G}}:\mathcal{S}_{k}(\mathcal{G})\to H^{1}(\mathcal{G},V_{k-2,\mathbb{C}})

be the Eichler-Shimura homomorphism constructed in Section 4. One defines an antiholomorphic avatar of $[\text{ES}_{k}]_{\text{c},\mathcal{G}}$ by

[\text{ES}_{k}]^{a}_{\text{c},\mathcal{G}}:\mathcal{S}^{a}_{k}(\mathcal{G})\xrightarrow{}H^{1}(\mathcal{G},V_{k-2,\mathbb{C}}),\hskip 8.5359ptf\to\#[\text{ES}_{k}]_{\text{c},\mathcal{G}}(\bar{f}).

It is clear that $[\text{ES}_{k}]^{a}_{\text{c},\mathcal{G}}$ is $\mathbb{C}$ -linear. We extend the action of a double coset operator $[\mathcal{G}\alpha\mathcal{G}]_{k}$ to the space of antiholomorphic cusp form using the formula $f\to\#[\mathcal{G}\alpha\mathcal{G}]_{k}(\bar{f})$ . Since the action of $[\mathcal{G}\alpha\mathcal{G}]_{k}$ on a cocycle preserves the decomposition into real and imaginary parts it follows that $[\text{ES}_{k}]^{a}_{\text{c},\mathcal{G}}$ is also Hecke equivariant. Hence $[\text{ES}_{k}]_{\text{c},\mathcal{G}}$ and $[\text{ES}_{k}]^{a}_{\text{c},\mathcal{G}}$ together provide a Hecke-equivariant $\mathbb{C}$ -linear map

[\text{ES}_{k}]_{\text{c},\mathcal{G}}\oplus[\text{ES}_{k}]^{a}_{\text{c},\mathcal{G}}:\mathcal{S}_{k}(\mathcal{G})\oplus\mathcal{S}^{a}_{k}(\mathcal{G})\to H^{1}(\mathcal{G},V_{k-2,\mathbb{C}}).

The Eichler-Shimura isomorphism theorem for the cusp forms is as follows:

Theorem 5.7.

[10, p.171] The map $[\emph{ES}_{k}]_{\emph{c},\mathcal{G}}\oplus[\emph{ES}_{k}]^{a}_{\emph{c},\mathcal{G}}$ is an injection whose image equals the parabolic subspace of $H^{1}(\mathcal{G},V_{k-2,\mathbb{C}})$ .

Proof.

For $[P]\in H^{1}(\mathcal{G},V_{k-2,\mathbb{C}})$ write $\mathfrak{R}([P])=\frac{[P]+\#[P]}{2}$ and $\mathfrak{I}([P])=\frac{[P]-\#[P]}{2i}$ . The $\mathbb{R}$ -linear version of the isomorphism theorem given in [16, 8.2] asserts that the assignment $\mathcal{S}_{k}(\mathcal{G})\to H^{1}_{\mathsf{P}}(\mathcal{G},V_{k-2,\mathbb{R}})$ , $f\mapsto\mathfrak{R}\big{(}[\text{ES}_{k}]_{c,\mathcal{G}}(f)\big{)}$ , is an isomorphism of real vector spaces. One twists the map above by the $\mathbb{R}$ -linear automorphism $f\to if$ of $\mathcal{S}_{k}(\mathcal{G})$ to deduce that the imaginary part map $f\mapsto\mathfrak{I}\big{(}[\text{ES}_{k}]_{c,\mathcal{G}}(f)\big{)}$ also defines a real linear isomorphism between $\mathcal{S}_{k}(\mathcal{G})$ and $H^{1}_{\mathsf{P}}(\mathcal{G},V_{k-2,\mathbb{R}})$ . Now the assertion is a consequence of (5.4). ∎

The injectivity statement of the theorem combined with the functorial properties of the Eichler-Shimura class leads to the following result.

Corollary 5.8.

Let $k$ be an integer $\geq 2$ and $\mathcal{G}$ be a finite index subgroup of $\Gamma$ . Then $[\mathsf{p}_{k}]_{\mathcal{G},\emph{Hol}(\mathbb{H})}$ is nonzero.

Proof.

Let $\mathcal{G}^{\prime}\subseteq\mathcal{G}$ be a finite index subgroup of $\Gamma$ with $[\mathcal{G}:\mathcal{G}^{\prime}]$ so large that $\mathcal{S}_{k}(\mathcal{G}^{\prime})$ is nonzero. One uses (2.13) and (2.14) to discover that

[\mathsf{p}_{k}]_{\mathcal{G}^{\prime},\mathcal{S}_{k}(\mathcal{G}^{\prime})}=\text{R}^{\text{Hol}(\mathbb{H})}_{\mathcal{S}_{k}(\mathcal{G}^{\prime})}\text{R}^{\mathcal{G}}_{\mathcal{G}^{\prime}}[\mathsf{p}_{k}]_{\mathcal{G},\text{Hol}(\mathbb{H})}.

But $[\text{ES}_{k}]_{\mathcal{G}^{\prime},\mathcal{S}_{k}(\mathcal{G}^{\prime})}$ is a nonzero injective map. Therefore by Proposition 4.2 the class $[\mathsf{p}_{k}]_{\mathcal{G}^{\prime},\mathcal{S}_{k}(\mathcal{G}^{\prime})}$ is nonzero. Thus $[\mathsf{p}_{k}]_{\mathcal{G},\text{Hol}(\mathbb{H})}$ cannot be zero. ∎

5.3 The space of Eisenstein series

Let $k$ be an integer $\geq 2$ and $\mathcal{G}$ be a congruence subgroup of $\Gamma$ . Suppose that $\mathcal{M}_{k}(\mathcal{G})$ is the space of weight $k$ modular forms on $\mathcal{G}$ . Then the space of Eisenstein series on $\mathcal{G}$ , denoted $\mathcal{E}_{k}(\mathcal{G})$ , is the unique linear complement of $\mathcal{S}_{k}(\mathcal{G})$ in $\mathcal{M}_{k}(\mathcal{G})$ that is orthogonal to $\mathcal{S}_{k}(\mathcal{G})$ with respect to the Petersson inner product [7, 5.11]. The equivariance properties of the inner product imply that the decomposition $\mathcal{M}_{k}(\mathcal{G})=\mathcal{E}_{k}(\mathcal{G})\oplus\mathcal{S}_{k}(\mathcal{G})$ is stable under the action of a general double coset operator. Next, we explain how to use the spectral Eisenstein series on $\mathcal{G}$ to construct a basis for the space of Eisenstein series following the author’s recent account of the theory [15].

Let $x\in\mathcal{C}(\mathcal{G})$ and $\sigma_{x}\in\Gamma$ be a scaling matrix for $x$ , i.e., $x=\sigma_{x}\infty$ . If $k$ is odd then additionally assume that $-\text{Id}\notin\mathcal{G}$ and $x$ is a regular cusp. The spectral Eisenstein series of weight $k$ attached to $x$ and the scaling matrix $\sigma_{x}$ is

E_{k,x}(\tau;s):=\sum_{\gamma\in\mathcal{G}_{x}\backslash\mathcal{G}}\mathbf{j}(\sigma_{x}^{-1}\gamma,\tau)^{-k}(\text{Im }\sigma_{x}^{-1}\gamma\tau)^{s}\hskip 8.5359pt\begin{subarray}{c}(\text{Re}(k+2s)>2)\end{subarray}

where $\mathbf{j}(\gamma,\tau)=c\tau+d$ for $\gamma=\begin{pmatrix}a&b\\ c&d\end{pmatrix}\in\Gamma$ . If $k$ is odd then our Eisenstein series may depend on the choice of scaling matrix. The series above admits analytic continuation to the larger domain $\text{Re}(k+2s)>1$ allowing us to specialize at $(2,0)$ . For simplicity we abbreviate $E_{k,x}(\tau;0)$ as $E_{k,x}(\tau)$ . An explicit calculation with the Fourier series shows that

E_{k,x}(\tau)\in\begin{cases}\text{Hol}(\mathbb{H}),&\text{$k\geq 3$,}\\ \frac{\alpha_{x}}{\pi\text{Im}(\tau)}+\text{Hol}(\mathbb{H}),&\text{$k=2$}\end{cases}

where $\alpha_{x}$ is a nonzero rational number. Moreover, we have

(5.6)

\pi_{\infty}(E_{k,x}\lvert_{\gamma})=\begin{cases}(\pm 1)^{k},&\text{if $\gamma\infty=x$ in $\mathcal{C}(\mathcal{G})$;}\\ 0,&\text{otherwise.}\end{cases}\hskip 8.5359pt(\begin{subarray}{c}\forall\gamma\in\Gamma\end{subarray})

Set

\mathcal{B}_{\mathcal{G},k}=\begin{cases}\{E_{k,x}\mid x\in\mathcal{C}(\mathcal{G})\},&\begin{subarray}{c}\text{if $k$ is even and $\geq 4$;}\end{subarray}\\ \{E_{k,x}\mid x\in\mathcal{C}_{\infty}(\mathcal{G})\},&\begin{subarray}{c}\text{if $k$ is odd and $-\text{Id}\notin\mathcal{G}$;}\end{subarray}\\ \{E_{k,x}-\frac{\alpha_{x}}{\alpha_{\infty}}E_{k,\infty}\mid x\in\mathcal{C}(\mathcal{G})-\{\infty\}\},&\begin{subarray}{c}\text{if $k=2$;}\end{subarray}\\ \emptyset,&\begin{subarray}{c}\text{otherwise}\end{subarray}\end{cases}

where in the second case we write the series for a fixed choice of scaling matrix for each regular cusp. Then $\mathcal{B}_{\mathcal{G},k}$ is a basis for the space of Eisenstein series $\mathcal{E}_{k}(\mathcal{G})$ . For a subfield $\mathbb{K}$ of $\mathbb{C}$ we write $\mathcal{E}_{k}(\mathcal{G},\mathbb{K})$ for the $\mathbb{K}$ span of $\mathcal{B}_{\mathcal{G},k}$ . This $\mathbb{K}$ -structure on the space of Eisenstein series is compatible with restriction to a principal level. In more detail, if $\Gamma(N)\subseteq\mathcal{G}$ then

(5.7)

\mathcal{E}_{k}(\mathcal{G},\mathbb{K})=\mathcal{E}_{k}\big{(}\Gamma(N),\mathbb{K}\big{)}^{\mathcal{G}}.

To write down uniform statements for all weights $\geq 2$ we introduce an extended space of Eisenstein series

\mathcal{E}^{\ast}_{k}(\mathcal{G},\mathbb{K})=\begin{cases}\text{$\mathbb{K}$-span of $\{E_{k,x}\mid x\in\mathcal{C}(\mathcal{G})\}$,}&\text{if $k=2$;}\\ \mathcal{E}_{k}(\mathcal{G},\mathbb{K}),&\text{if $k\geq 3$.}\end{cases}

It is clear that $\mathcal{E}^{\ast}_{2}(\mathcal{G},\mathbb{K})\cap\text{Hol}(\mathbb{H})=\mathcal{E}_{2}(\mathcal{G},\mathbb{K})$ and $\mathcal{E}_{2}(\mathcal{G},\mathbb{K})$ is a subspace of codimension $1$ inside $\mathcal{E}^{\ast}_{2}(\mathcal{G},\mathbb{K})$ . The $\mathbb{K}$ -rational structure on the extended space of weight $2$ is also compatible with restriction to a principal level in the sense described above.

Remark 5.9.

The author expects that there are spectral Eisenstein series of weight $k$ for every finite index subgroup so that their specializations at $s=0$ give rise to a basis of the space of Eisenstein series. Such a construction would directly lead to a generalization of Theorem 1.1 to finite index subgroups. However, in this article, we restrict our attention to congruence subgroups with a view towards arithmetic application.

6 Proof of Theorem 1.1

Let $k$ be an integer $\geq 2$ and $\mathcal{G}$ be a congruence subgroup of $\Gamma$ . Recall the decomposition $\mathcal{M}_{k}(\mathcal{G})=\mathcal{E}_{k}(\mathcal{G})\oplus\mathcal{S}_{k}(\mathcal{G})$ from Section 5.3. Suppose that

[\text{ES}_{k}]_{\text{m},\mathcal{G}}:\mathcal{M}_{k}(\mathcal{G})\xrightarrow{}H^{1}(\mathcal{G},V_{k-2,\mathbb{C}})

is the Eichler-Shimura homomorphism attached to the space of modular forms on $\mathcal{G}$ . Then $[\text{ES}_{k}]_{\text{m},\mathcal{G}}$ is a Hecke equivariant extension of the map $[\text{ES}_{k}]_{\text{c},\mathcal{G}}$ on the space of cusp forms. We need to prove that

[\text{ES}_{k}]_{\text{m},\mathcal{G}}\oplus[\text{ES}_{k}]^{a}_{\text{c},\mathcal{G}}:\mathcal{M}_{k}(\mathcal{G})\oplus\mathcal{S}^{a}_{k}(\mathcal{G})\to H^{1}(\mathcal{G},V_{k-2,\mathbb{C}})

is an isomorphism. If $k$ is odd and $-\text{Id}\in\mathcal{G}$ then the domain and codomain of the map above are zero (Example 4.5). Assume that $-\text{Id}\notin\mathcal{G}$ whenever $k$ is odd. Let $[\text{ES}_{k}]_{\text{E},\mathcal{G}}$ denote the restriction of $[\text{ES}_{k}]_{\text{m},\mathcal{G}}$ to the space of Eisenstein series on $\mathcal{G}$ . In the light of Theorem 5.7 it suffices to verify that $[\text{ES}_{k}]_{\text{E},\mathcal{G}}$ is injective and the image of $[\text{ES}_{k}]_{\text{E},\mathcal{G}}$ is a linear complement of $H^{1}_{\mathsf{P}}(\mathcal{G},V_{k-2,\mathbb{C}})$ in $H^{1}(\mathcal{G},V_{k-2,\mathbb{C}})$ .

6.1 Restriction to the cusps

With each $x\in\mathcal{C}(\mathcal{G})$ one can associate a cuspidal Eichler-Shimura map

\Theta_{x}:\mathcal{M}_{k}(\mathcal{G})\to H^{1}(\mathcal{G}_{x},V_{k-2,\mathbb{C}}),\hskip 8.5359pt\Theta_{x}=\text{Res}_{x}\circ[\text{ES}_{k}]_{\text{m},\mathcal{G}}

where $\text{Res}_{x}$ is the restriction map from $H^{1}(\mathcal{G},V_{k-2,\mathbb{C}})$ to $H^{1}(\mathcal{G}_{x},V_{k-2,\mathbb{C}})$ . Proposition 5.6 furnishes a complete description of the target of $\Theta_{x}$ . The subsequent discussion aims to find the kernel of this map.

Let $x\in\mathcal{C}(\mathcal{G})$ and choose $\gamma_{0}\in\Gamma$ so that $x=\gamma_{0}\infty$ . Now suppose $\Gamma(N)\subseteq\mathcal{G}$ with $N\geq 3$ . Note that the restriction maps fit into a commutative diagram

(6.1)

exactly as in the proof of Lemma 5.1. Here $\Gamma(N)$ is a normal subgroup of $\Gamma$ and $\Gamma$ acts by conjugation on $H^{1}\big{(}\Gamma(N),V_{k-2,\mathbb{C}}\big{)}$ (Section 4.3). The conjugation action of $\gamma_{0}$ yields another commutative diagram described by

(6.2)

where the vertical arrows are restriction maps. Next, we use our equivariant machinery to verify the key property of the map $\Theta_{x}$ .

Proposition 6.1.

Suppose that $f\in\mathcal{M}_{k}(\mathcal{G})$ . Then $\Theta_{x}(f)=0$ if and only if $\pi_{\infty}(f\lvert_{\gamma_{0}})=0$ .

Proof.

We identify $\mathcal{M}_{k}(\mathcal{G})$ , resp. $H^{1}(\mathcal{G},V_{k-2,\mathbb{C}})$ , with the $\mathcal{G}$ -invariant subspace of $\mathcal{M}_{k}(\Gamma(N))$ , resp. $H^{1}\big{(}\Gamma(N),V_{k-2,\mathbb{C}}\big{)}$ , and utilize

[\text{ES}_{k}]_{\text{m},\Gamma(N)}:\mathcal{M}_{k}(\Gamma(N))\xrightarrow{}H^{1}\big{(}\Gamma(N),V_{k-2,\mathbb{C}}\big{)}

to describe $[\text{ES}_{k}]_{\text{m},\mathcal{G}}$ (Remark 4.8). First, one invokes Remark 4.6 to conclude that the bottom arrow of (6.1) is injective. Hence $\Theta_{x}(f)=0$ if and only if the restriction of $[\text{ES}_{k}]_{\text{m},\Gamma(N)}(f)$ to $H^{1}\big{(}\Gamma(N)_{x},V_{k-2,\mathbb{C}}\big{)}$ is zero. But $[\text{ES}_{k}]_{\text{m},\Gamma(N)}$ is $\Gamma$ -equivariant (Proposition 4.7). Therefore $[\text{ES}_{k}]_{\text{m},\Gamma(N)}(f)\lvert_{\gamma_{0}}=[\text{ES}_{k}]_{\text{m},\Gamma(N)}(f\lvert_{\gamma_{0}})$ . One now employs (6.2) to conclude that $\Theta_{x}(f)=0$ if and only if the restriction of $[\text{ES}_{k}]_{\text{m},\Gamma(N)}(f\lvert_{\gamma_{0}})$ to $H^{1}\big{(}\Gamma(N)_{\infty},V_{k-2,\mathbb{C}}\big{)}$ is zero. Note that $\Gamma(N)_{\infty}=\langle\langle T^{N}\rangle\rangle$ . From Lemma 3.13 it follows that

\mathsf{p}^{\ast}_{k}(T^{N},f\lvert_{\gamma_{0}},X)=\pi_{\infty}(f\lvert_{\gamma_{0}})\frac{(X+N)^{k-1}-X^{k-1}}{k-1}.

Therefore, by (5.5), $\Theta_{x}(f)=0$ if and only if $\pi_{\infty}(f\lvert_{\gamma_{0}})=0$ . ∎

One uses the theorem above to determine the cuspidal behavior of the cohomology classes attached to the basis $\mathcal{B}_{\mathcal{G},k}$ for the space of Eisenstein series. Let the notation be as in the discussion around Section 5.3. For convenience write $\mathcal{C}_{k}(\mathcal{G})=\mathcal{C}(\mathcal{G})$ if $k$ is even and $\mathcal{C}_{\infty}(\mathcal{G})$ if $k$ is odd.

Corollary 6.2.

Let $k\geq 3$ and assume that $x,y\in\mathcal{C}_{k}(\mathcal{G})$ . Then $\Theta_{x}(E_{k,y})\neq 0$ if and only if $x=y$ . Moreover, let $k=2$ and $x,y\in\mathcal{C}(\mathcal{G})-\{\infty\}$ . Then

\Theta_{x}\big{(}E_{2,y}-\frac{\alpha_{y}}{\alpha_{\infty}}E_{2,\infty}\big{)}\neq 0

if and only if $x=y$ .

Proof.

The assertion is an easy consequence of Proposition 6.1 and (5.6). ∎

6.2 Completion of the proof

The cuspidal Eichler-Shimura maps together give rise to a homomorphism

\Theta:\mathcal{E}_{k}(\mathcal{G})\to\bigoplus_{x\in\mathcal{C}(\mathcal{G})}H^{1}(\mathcal{G}_{x},V_{k-2,\mathbb{C}});\hskip 8.5359pt\Theta:=(\Theta_{x})_{x\in\mathcal{C}(\mathcal{G})}.

We use this homomorphism to extract information about the Eichler-Shimura map attached to the space of Eisenstein series.

Case I: $k\geq 3$ .

Proposition 5.6 shows that $H^{1}(\mathcal{G}_{x},V_{k-2,\mathbb{C}})\neq 0$ if and only if $x\in\mathcal{C}_{k}(\mathcal{G})$ . Therefore $\Theta$ maps our basis of $\mathcal{E}_{k}(\mathcal{G})$ to a basis of $\oplus_{x\in\mathcal{C}(\mathcal{G})}H^{1}(\mathcal{G}_{x},V_{k-2,\mathbb{C}})$ (Corollary 6.2). In particular, the map $\Theta$ is injective. It follows that $[\text{ES}_{k}]_{\text{E},\mathcal{G}}$ is also injective. Moreover $\text{im}([\text{ES}_{k}]_{\text{E},\mathcal{G}})\cap H^{1}_{\mathsf{P}}(\mathcal{G},V_{k-2,\mathbb{C}})=\{0\}$ where $\text{im}(\cdot)$ refers to the image a map. The surjectivity of $\Theta$ demonstrates that (5.3) is exact at the right end. Therefore $\text{im}([\text{ES}_{k}]_{\text{E},\mathcal{G}})$ is a linear complement of $H^{1}_{\mathsf{P}}(\mathcal{G},V_{k-2,\mathbb{C}})$ as desired.

Case II: $k=2$ .

One again uses Corollary 6.2 to conclude that $\Theta$ maps the given basis of $\mathcal{E}_{k}(\mathcal{G})$ to a linearly independent subset of $\oplus_{x\in\mathcal{C}(\mathcal{G})}H^{1}\big{(}\mathcal{G}_{x},V_{k-2,\mathbb{C}}\big{)}$ . Hence $[\text{ES}_{k}]_{\text{E},\mathcal{G}}$ is injective and $\text{im}([\text{ES}_{k}]_{\text{E},\mathcal{G}})\cap H^{1}_{\mathsf{P}}(\mathcal{G},V_{k-2,\mathbb{C}})=\{0\}$ . Here $V_{k-2,\mathbb{C}}$ equals the trivial $\mathcal{G}$ -module $\mathbb{C}$ . We combine (5.2), Proposition 5.3, and Lemma 5.4 to discover that

\text{dim}_{\mathbb{C}}H^{1}(\mathcal{G},V_{k-2,\mathbb{C}})=\text{dim}_{\mathbb{C}}H^{1}_{\mathsf{P}}(\mathcal{G},V_{k-2,\mathbb{C}})+\lvert\mathcal{C}(\mathcal{G})\rvert-1.

But $\text{dim}_{\mathbb{C}}\text{im}\big{(}[\text{ES}_{k}]_{\text{E},\mathcal{G}}\big{)}=\text{dim}_{\mathbb{C}}\mathcal{E}_{k}(\mathcal{G})=\lvert\mathcal{C}(\mathcal{G})\rvert-1$ . Thus $\text{im}([\text{ES}_{k}]_{\text{E},\mathcal{G}})$ is a linear complement of $H^{1}_{\mathsf{P}}(\mathcal{G},V_{k-2,\mathbb{C}})$ in $H^{1}(\mathcal{G},V_{k-2,\mathbb{C}})$ .

7 Versions of the isomorphism theorem

7.1 The induced picture

We begin with a convenient description of Shapiro’s lemma [12, p.62]. Let $\mathcal{H}$ be an abstract group and $\mathcal{G}$ be a subgroup of $\mathcal{H}$ . Suppose that $V$ is a right $\mathbb{K}[\mathcal{G}]$ -module where $\mathbb{K}$ is a field of characteristics zero. Write

\text{Ind}^{\mathcal{G}}_{\mathcal{H}}(V):=\text{Hom}_{\mathbb{K}[\mathcal{G}]}(\mathbb{K}[\mathcal{H}],V)

and endow it with the right $\mathcal{H}$ -action $(fh)(x)=f(hx)$ . If the $\mathbb{K}[\mathcal{G}]$ -module structure on $V$ extends to a $\mathbb{K}[\mathcal{H}]$ -structure then $\text{Ind}^{\mathcal{G}}_{\mathcal{H}}(V)\cong\text{Fun}(\mathcal{G}\backslash\mathcal{H},V)$ as $\mathbb{K}[\mathcal{H}]$ -module via the assignment $f\to\tilde{f}$ where $\tilde{f}(\mathcal{G}x):=f(x^{-1})x$ . Shapiro’s lemma states that the canonical homomorphism of $\mathcal{G}$ -modules $\text{Ind}^{\mathcal{G}}_{\mathcal{H}}(V)\to V$ described by $f\mapsto f(\text{Id})$ gives rise to a $\mathbb{K}$ -linear isomorphism

\text{sh}^{*}:H^{\ast}\big{(}\mathcal{H},\text{Ind}^{\mathcal{G}}_{\mathcal{H}}(V)\big{)}\xrightarrow{\cong}H^{\ast}(\mathcal{G},V).

Let $\mathcal{G}$ be a finite index subgroup of $\Gamma$ . Suppose that $V$ is a right $\mathbb{K}[\Gamma]$ -module. We consider the Shapiro isomorphism

\text{sh}^{1}:H^{1}\big{(}\Gamma,\text{Fun}(\mathcal{G}\backslash\Gamma,V)\big{)}\xrightarrow{\cong}H^{1}(\mathcal{G},V)

explicitly described by

(7.1)

[P]\to[\psi(P)];\hskip 8.5359pt\psi(P)(\gamma):=P(\gamma)(\mathcal{G}\text{Id}).\hskip 8.5359pt(\begin{subarray}{c}\gamma\in\mathcal{G}\end{subarray})

In practice working with the cocycles on $\Gamma$ is more convenient since each cocycle is determined by its value on $\{T,S\}$ . One pulls back the action of the double coset operators on $H^{1}(\mathcal{G},V)$ to obtain a $\mathcal{G}$ -Hecke action on $H^{1}(\Gamma,\text{Fun}(\mathcal{G}\backslash\Gamma,V))$ ; cf. [14, 5]. The $\Gamma$ -action on $\text{Fun}(\mathcal{G}\backslash\Gamma,V)$ need not descend to an action of $\text{P}\Gamma$ . Define

\text{Fun}^{\pm}(\mathcal{G}\backslash\Gamma,V)=\big{\{}F\lvert_{-\text{Id}}=\pm F\mid F\in\text{Fun}(\mathcal{G}\backslash\Gamma,V)\big{\}}.

It is clear that $\text{Fun}(\mathcal{G}\backslash\Gamma,V)=\text{Fun}^{+}(\mathcal{G}\backslash\Gamma,V)\oplus\text{Fun}^{-}(\mathcal{G}\backslash\Gamma,V)$ as $\mathbb{K}[\Gamma]$ -modules. There is a natural projection

\begin{gathered}Z^{1}\big{(}\Gamma,\text{Fun}(\mathcal{G}\backslash\Gamma,V)\big{)}\to Z^{1}\big{(}\Gamma,\text{Fun}^{+}(\mathcal{G}\backslash\Gamma,V)\big{)};\\ F\to F^{+},\hskip 14.22636ptF^{+}(\gamma):=\frac{F(\gamma)+F(\gamma)\lvert_{-\text{Id}}}{2}.\end{gathered}

Note that $(F-F^{+})(\gamma)=\frac{F(-\text{Id})(1-\gamma)}{2}$ for each $\gamma\in\Gamma$ . Therefore the natural inclusion

H^{1}\big{(}\Gamma,\text{Fun}^{+}(\mathcal{G}\backslash\Gamma,V)\big{)}\hookrightarrow H^{1}\big{(}\Gamma,\text{Fun}(\mathcal{G}\backslash\Gamma,V)\big{)}

is an isomorphism. The action of $\Gamma$ on $\text{Fun}^{+}(\mathcal{G}\backslash\Gamma,V)$ descends to a $\text{P}\Gamma$ -action and one identifies $H^{1}\big{(}\Gamma,\text{Fun}^{+}(\mathcal{G}\backslash\Gamma,V)\big{)}$ and $H^{1}\big{(}\text{P}\Gamma,\text{Fun}^{+}(\mathcal{G}\backslash\Gamma,V)\big{)}$ as in Section 5.1. Thus one can use the isomorphism above to deduce that the induced module satisfies the change of scalar properties given in (5.4). The experts are presumably familiar with the following result.

Proposition 7.1.

[14, 2] Let $(\mathcal{G},V)$ be a descent module. Then the Shapiro map $\emph{sh}^{1}$ induces an isomorphism $H^{1}_{\mathsf{P}}\big{(}\Gamma,\emph{Fun}(\mathcal{G}\backslash\Gamma,V)\big{)}\cong H^{1}_{\mathsf{P}}(\mathcal{G},V)$ . As a consequence the parabolic subspace $H^{1}_{\mathsf{P}}\big{(}\Gamma,\emph{Fun}(\mathcal{G}\backslash\Gamma,V)\big{)}$ is stable under the $\mathcal{G}$ -Hecke action.

For a finite index subgroup $\mathcal{G}$ the normal core of $\mathcal{G}$ , denoted $\mathcal{G}_{\text{nc}}$ , is

\mathcal{G}_{\text{nc}}:=\bigcap_{\gamma\in\Gamma}\gamma\mathcal{G}\gamma^{-1}.

This subgroup is the largest finite index normal subgroup of $\Gamma$ that is contained in $\mathcal{G}$ .

Proof.

Let $N$ be a positive integer so that $T^{N}\in\mathcal{G}_{\text{nc}}$ . Then $\gamma T^{N}\gamma^{-1}\in\mathcal{G}$ for each $\gamma\in\Gamma$ . Suppose that $F$ is a cocycle on $\Gamma$ with values in $\text{Fun}^{+}(\mathcal{G}\backslash\Gamma,V)$ . A straightforward calculation yields

F(\gamma T^{N}\gamma^{-1})(\mathcal{G}\text{Id})\in F(T^{N})(\mathcal{G}\gamma)\gamma^{-1}+V(\gamma T^{N}\gamma^{-1}-1).\hskip 8.5359pt(\begin{subarray}{c}\gamma\in\Gamma\end{subarray})

Therefore $F(T^{N})(\mathcal{G}\gamma)\in V(T^{N}-1)$ if and only if $F(\gamma T^{N}\gamma^{-1})(\mathcal{G}\text{Id})\in V(\gamma T^{N}\gamma^{-1}-1)$ . It follows that

		$\displaystyle F(T^{N})\in\text{Fun}^{+}(\mathcal{G}\backslash\Gamma,V)(T^{N}-1)$
		$\displaystyle\iff F(\gamma T^{N}\gamma^{-1})(\mathcal{G}\text{Id})\in V(\gamma T^{N}\gamma^{-1}-1),\;\forall\gamma\in\Gamma.$

But $\langle\langle\gamma T^{N}\gamma^{-1}\rangle\rangle$ , resp. $\langle\langle T^{N}\rangle\rangle$ , is a finite index subgroup of the abelian group $\mathcal{G}_{\gamma\infty}$ , resp. $\Gamma_{\infty}$ . The equivalence above along with the injectivity of restriction argument and (5.3) implies that $[F]\in H^{1}_{\mathsf{P}}\big{(}\Gamma,\text{Fun}^{+}(\mathcal{G}\backslash\Gamma,V)\big{)}$ if and only if $\text{sh}^{1}([F])\in H^{1}_{\mathsf{P}}(\mathcal{G},V)$ where $[\cdot]$ denotes the image in cohomology. Hence $\text{sh}^{1}$ induces an isomorphism between $H^{1}_{\mathsf{P}}\big{(}\Gamma,\text{Fun}(\mathcal{G}\backslash\Gamma,V)\big{)}$ and $H^{1}_{\mathsf{P}}\big{(}\mathcal{G},V\big{)}$ . The second part of the assertion is clear from the definition of $\mathcal{G}$ -Hecke action. ∎

We next explicitly describe the preimage under $\text{sh}^{1}$ of the cohomology classes attached to modular forms. Write $\mathcal{X}_{\Gamma}=\text{the $\Gamma$-span of $\mathcal{M}_{k}(\mathcal{G})$ in $\text{Hol}(\mathbb{H})$}$ . Note that $\mathcal{X}_{\Gamma}\subseteq\mathcal{M}_{k}(\mathcal{G}_{\text{nc}})$ . In particular $\mathcal{X}_{\Gamma}\subseteq\mathbb{C}\mathfrak{F}_{\infty}$ and one can invoke the theory of Eichler-Shimura integrals with base-point at infinity for this collection of functions. Let

\text{ES}_{k,\mathcal{G}}^{*}:\mathcal{M}_{k}(\mathcal{G})\to Z^{1}(\mathcal{G},V_{k-2,\mathbb{C}})

be the Eichler-Shimura cocycle map attached to $(\mathcal{G},\mathcal{M}_{k}(\mathcal{G}))$ for the choice of base point at $\infty$ . Define the induced Eichler-Shimura cocycle map by

(7.2)		$\displaystyle\text{Ind-ES}_{k,\mathcal{G}}^{*}$	$\displaystyle:\mathcal{M}_{k}(\mathcal{G})\to Z^{1}\big{(}\Gamma,\text{Fun}(\mathcal{G}\backslash\Gamma,V_{k-2,\mathbb{C}})\big{)},$
(7.2)			$\displaystyle\text{Ind-ES}_{k,\mathcal{G}}^{}(f)(\gamma)(\mathcal{G}\sigma):=\mathsf{p}_{k}^{}(\gamma,f\lvert_{\sigma},X).$

Our theory of equivariant period functions for the pair $(\Gamma,\mathcal{X}_{\Gamma})$ directly verifies that $\text{Ind-ES}_{k,\mathcal{G}}^{*}(f)$ is indeed a cocycle on $\Gamma$ . Moreover, there is a commutative diagram

(7.3)

where $\psi$ is as in (7.1). Note that the cocycle $\text{Ind-ES}_{k,\mathcal{G}}^{*}(f)$ takes values in $\text{Fun}^{+}(\mathcal{G}\backslash\Gamma,V_{k-2,\mathbb{C}})$ . The induced Eichler-Shimura homomorphism attached to $\mathcal{M}_{k}(\mathcal{G})$ is

\begin{gathered}\text{Ind-}[\text{ES}_{k}]_{\text{m},\mathcal{G}}:\mathcal{M}_{k}(\mathcal{G})\to H^{1}\big{(}\Gamma,\text{Fun}(\mathcal{G}\backslash\Gamma,V_{k-2,\mathbb{C}})\big{)};\\ \text{Ind-}[\text{ES}_{k}]_{\text{m},\mathcal{G}}(f):=[\text{Ind-}\text{ES}_{k,\mathcal{G}}^{*}(f)].\end{gathered}

We denote the restriction of $\text{Ind-}[\text{ES}_{k}]_{\text{m},\mathcal{G}}$ to $\mathcal{S}_{k}(\mathcal{G})$ by $\text{Ind-}[\text{ES}_{k}]_{\text{c},\mathcal{G}}$ .

Lemma 7.2.

$\emph{sh}^{1}\big{(}\emph{Ind-}[\emph{ES}_{k}]_{\emph{m},\mathcal{G}}(f)\big{)}=[\emph{ES}_{k}]_{\emph{m},\mathcal{G}}(f),\;\forall f\in\mathcal{M}_{k}(\mathcal{G})$ .

Proof.

The assertion is an easy consequence of (7.3). ∎

Remark 7.3.

In principle, one can start with any choice of Eichler-Shimura integrals for $\mathcal{X}_{\Gamma}$ to obtain a lift of the corresponding Eichler-Shimura cocycle map as in (7.2). Lemma 7.2 ensures that all the lifts give rise to the same map at the level of cohomology.

The $\Gamma$ -modules $V_{k-2,\mathbb{C}}$ and $\text{Fun}(\mathcal{G}\backslash\Gamma,V_{k-2,\mathbb{C}})$ inherit real structures from $V_{k-2,\mathbb{R}}$ and $\psi$ in (7.3) is compatible with the associated real structures. As before the real structures descend to the corresponding cohomology modules and $\text{sh}^{1}$ is compatible with complex conjugations in the source and target. Let

\text{Ind-}[\text{ES}_{k}]^{a}_{\text{c},\mathcal{G}}:\mathcal{S}^{a}_{k}(\mathcal{G})\to H^{1}\big{(}\Gamma,\text{Fun}(\mathcal{G}\backslash\Gamma,V_{k-2,\mathbb{C}})\big{)};\hskip 5.69046ptf\mapsto\#\text{Ind-}[\text{ES}_{k}]_{\text{c},\mathcal{G}}(\bar{f})

be the induced antiholomorphic Eichler-Shimura map where $\#$ is the conjugation in cohomology.

Corollary 7.4.

$\emph{sh}^{1}\big{(}\emph{Ind-}[\emph{ES}_{k}]^{a}_{\emph{c},\mathcal{G}}(f)\big{)}=[\emph{ES}_{k}]^{a}_{\emph{c},\mathcal{G}}(f);\;\forall f\in\mathcal{S}^{a}_{k}(\mathcal{G})$ .

Proof.

Follows from Lemma 7.2 and the discussion above. ∎

Lemma 7.2 and Corollary 7.4 culminate into the isomorphism theorem in the induced picture.

Theorem 7.5.

Let $\mathcal{G}$ be a congruence subgroup of $\Gamma$ . The map

\emph{Ind-}[\emph{ES}_{k}]_{\emph{m},\mathcal{G}}\oplus\emph{Ind-}[\emph{ES}_{k}]^{a}_{\emph{c},\mathcal{G}}:\mathcal{M}_{k}(\mathcal{G})\oplus\mathcal{S}^{a}_{k}(\mathcal{G})\to H^{1}\big{(}\Gamma,\emph{Fun}(\mathcal{G}\backslash\Gamma,V_{k-2,\mathbb{C}})\big{)}

is a $\mathcal{G}$ -Hecke equivariant isomorphism. Moreover, this isomorphism maps $\mathcal{S}_{k}(\mathcal{G})\oplus\mathcal{S}^{a}_{k}(\mathcal{G})$ onto the parabolic subspace $H^{1}_{\emph{P}}\big{(}\Gamma,\emph{Fun}(\mathcal{G}\backslash\Gamma,V_{k-2,\mathbb{C}})\big{)}$ .

Proof.

The first part of the assertion is a consequence of Theorem 1.1, Lemma 7.2, and Corollary 7.4. The latter half follows from Theorem 5.7 and Proposition 7.1. Note that if $k$ is odd and $-\text{Id}\in\mathcal{G}$ then the induced cohomology vector space is zero (Example 4.5). Thus the second half of the assertion automatically holds in this case. ∎

We conclude the discussion with the explicit expression of the period cocycles at $T$ and $S$ . For $f\in\mathcal{M}_{k}(\mathcal{G})$ define the completed $L$ -function of $f$ by

(7.4)

L^{*}(f,s)=(2\pi)^{-s}\Gamma(s)L(f,s):=\int^{\infty}_{0}\big{(}f(it)-\pi_{\infty}(f)\big{)}t^{s-1}dt.

The integral defining $L^{*}(f,s)$ is absolutely convergent in the region $\text{Re}(s)>k$ . Moreover, we have the following identity involving entire integrals:

		$\displaystyle L^{*}(f,s)=\int_{1}^{\infty}\big{(}f(it)-\pi_{\infty}(f)\big{)}t^{s-1}dt\hskip 56.9055pt(\begin{subarray}{c}\text{Re}(s)>k\end{subarray})$
		$\displaystyle+\int_{0}^{1}\big{(}f(it)-\pi_{\infty}(f\lvert_{S^{-1}})(it)^{-k}\big{)}t^{s-1}dt+i^{-k}\frac{\pi_{\infty}(f\lvert_{S^{-1}})}{s-k}-\frac{\pi_{\infty}(f)}{s}.$

In particular $L^{*}(f,s)$ extends to a meromorphic on the whole complex plane with at most two simple poles supported at $\{0,k\}$ . It also satisfies a functional equation $L^{\ast}(f,s)=i^{k}L^{\ast}(f\lvert_{S},k-s)$ . The next result provides a complete description of the induced cocycle map (7.2). As before we work with the $\Gamma$ -stable set $\mathcal{X}_{\Gamma}$ and consider it as a subset of $\mathcal{M}_{k}(\mathcal{G}_{\text{nc}})$ .

Proposition 7.6.

[14, 8] Let the notation be as above. Then

	$\displaystyle\mathsf{p}_{k}^{*}(T,f\lvert_{\sigma},X)$	$\displaystyle=\pi_{\infty}(f\lvert_{\sigma})\frac{(X+1)^{k-1}-X^{k-1}}{k-1},$
	$\displaystyle\mathsf{p}_{k}^{*}(S,f\lvert_{\sigma},X)$	$\displaystyle=\sum_{r=0}^{k-2}i^{1-r}\binom{k-2}{r}L^{*}(f\lvert_{\sigma},r+1)X^{k-2-r}$

where $f\in\mathcal{M}_{k}(\mathcal{G})$ and $\sigma\in\Gamma$ .

Proof.

The first identity is a direct consequence of Lemma 3.13. We appeal to the formula given in Proposition 3.9 to prove the second identity. In our context $\tau_{0}=i$ and $\gamma=S$ . For brevity write $F=f\lvert_{\sigma}$ . Here

F_{c}(S^{-1})\lvert_{S}(\tau)=F(\tau)-\tau^{-k}\pi_{\infty}(F\lvert_{S^{-1}}).

A simple calculation using the identity below (7.4) yields

		$\displaystyle\int_{i}^{\infty}F_{c}(\xi)(X-\xi)^{k-2}d\xi-\int_{i}^{0}F_{c}(S^{-1})\lvert_{S}(\xi)(X-\xi)^{k-2}d\xi$
		$\displaystyle=\sum_{r=0}^{k-2}i^{1-r}\binom{k-2}{r}\big{(}L^{*}(F,r+1)+i^{-k}\frac{\pi_{\infty}(F\lvert_{S^{-1}})}{k-r-1}+\frac{\pi_{\infty}(F)}{r+1}\big{)}X^{k-2-r}.$

The proof of the identity is now clear. ∎

7.2 Modular forms with nebentypus

The present subsection aims to retrieve the twisted Eichler-Shimura isomorphism theorem for the spaces of modular forms with nebentypus from Theorem 1.1. Let $k$ be a positive integer $\geq 2$ and $\mathcal{G}$ be a finite index subgroup of $\Gamma$ . We say a character $\chi:\mathcal{G}\to\mathbb{C}^{\times}$ is defined over a subfield $\mathbb{K}$ of $\mathbb{C}$ if $\chi(\mathcal{G})\subseteq\mathbb{K}^{\times}$ . Let $\chi$ be a character on $\mathcal{G}$ that is defined over $\mathbb{K}$ . If $V$ be a right $\mathbb{K}[\mathcal{G}]$ -module then $V^{\chi}$ denotes a right $\mathbb{K}[\mathcal{G}]$ -module whose underlying vector space equals $V$ and the $\mathcal{G}$ -action is given by

v\lvert_{\chi,\gamma}:=\chi(\gamma)^{-1}v\lvert_{\gamma}.

Now suppose $\chi$ is an arbitrary character on $\mathcal{G}$ . Set

\text{Hol}(\mathbb{H})^{\mathcal{G}}_{\chi}=\big{\{}f\in\text{Hol}(\mathbb{H})\mid f\lvert_{k,\gamma}=\chi(\gamma)f,\;\forall\gamma\in\mathcal{G}\big{\}}.

Let $\mathcal{X}$ be a $\mathcal{G}$ -stable subset of holomorphic functions so that $\mathcal{X}\subseteq\text{Hol}(\mathbb{H})^{\mathcal{G}}_{\chi}$ . Suppose that $\mathbb{I}_{k,\mathcal{X}}$ is a choice function for $\mathcal{X}$ . With $\mathbb{I}_{k,\mathcal{X}}$ one associates a $\chi$ -twisted Eichler-Shimura cocycle map given by

\text{ES}_{k,\mathcal{X}}^{\chi}:\mathcal{X}\to Z^{1}\big{(}\mathcal{G},V_{k-2,\mathbb{C}}^{\chi}\big{)};\hskip 8.5359ptf\to\big{(}\gamma\mapsto\mathsf{p}_{k,\mathcal{X}}(\gamma,f,X)\big{)}

where $\mathsf{p}_{k,\mathcal{X}}$ is the period function of $\mathbb{I}_{k,\mathcal{X}}$ . The general transformation formula (2.10) ensures that the recipe above indeed defines a cocycle with values in $V_{k-2,\mathbb{C}}^{\chi}$ . As a consequence we obtain the $\chi$ -twisted Eichler-Shimura homomorphism:

[\text{ES}_{k}]_{\mathcal{G},\mathcal{X}}^{\chi}:\mathcal{X}\to H^{1}\big{(}\mathcal{G},V_{k-2,\mathbb{C}}^{\chi}\big{)};\hskip 8.5359ptf\to[\text{ES}_{k,\mathcal{X}}^{\chi}(f)].

Since the period functions attached to two distinct choice functions differ by a coboundary in $\text{Fun}(\mathcal{X},V_{k-2,\mathbb{C}})$ a straightforward calculation verifies that $[\text{ES}_{k}]_{\mathcal{G},\mathcal{X}}^{\chi}$ depends only on $(\mathcal{G},\mathcal{X})$ and is independent of $\mathbb{I}_{k,\mathcal{X}}$ .

Let $N$ be a positive integer and consider $\Gamma_{1}(N)\trianglelefteq\Gamma_{0}(N)$ . Recall that the natural map

\Gamma_{0}(N)\to(\mathbb{Z}/N\mathbb{Z})^{\times},\hskip 8.5359pt\begin{pmatrix}a&b\\ c&d\end{pmatrix}\to d(\text{mod }N)

induces an isomorphism between $\Gamma_{0}(N)/\Gamma_{1}(N)$ and $(\mathbb{Z}/N\mathbb{Z})^{\times}$ . We use this isomorphism to identify each character on $(\mathbb{Z}/N\mathbb{Z})^{\times}$ with a character on $\Gamma_{0}(N)$ . For simplicity write $\widehat{U}_{N}:=\text{Hom}\big{(}(\mathbb{Z}/N\mathbb{Z})^{\times},\mathbb{C}^{\times}\big{)}$ . Our treatment of twisted Eichler-Shimura theory relies on the formalism developed in the paragraph above for the pair $\big{(}\Gamma_{0}(N),\chi\big{)}$ where $\chi\in\widehat{U}_{N}$ . In this setting there is a notion of twisted Hecke action on the functions and the cohomology module [16, 3.5]. Put

\Delta(N)=\big{\{}\gamma\in M_{2}(\mathbb{Z})\mid\begin{subarray}{c}\text{det}(\gamma)>0,\hskip 2.84544pt\gamma\equiv\begin{pmatrix}a&\ast\\ 0&\ast\end{pmatrix}(\text{mod $N)$ with }\gcd(a,N)=1\end{subarray}\big{\}}

where $M_{2}(\cdot)$ refers to the ring of $2\times 2$ matrices. Let $\chi$ be a character on $(\mathbb{Z}/N\mathbb{Z})^{\times}$ defined over $\mathbb{K}$ . With $\chi$ we associate a homomorphism of monoids $\prescript{\iota}{}{\chi}:\Delta(N)\to\mathbb{\mathbb{K}}^{\times}$ by setting $\prescript{\iota}{}{\chi}(\gamma)=\chi(a)$ . If $\gamma\in\Gamma_{0}(N)$ then $\prescript{\iota}{}{\chi}(\gamma)=\chi(\gamma)^{-1}$ . Let $\alpha\in\Delta(N)$ . Write $\Gamma_{0}(N)\alpha\Gamma_{0}(N)=\amalg_{j=1}^{n}\Gamma_{0}(N)\alpha_{j}$ as in Section 4.2. Define

[\Gamma_{0}(N)\alpha\Gamma_{0}(N)]_{k,\chi}:\text{Hol}(\mathbb{H})^{\Gamma_{0}(N)}_{\chi}\to\text{Hol}(\mathbb{H})^{\Gamma_{0}(N)}_{\chi};\hskip 5.69046ptf\to\sum_{j=1}^{n}\prescript{\iota}{}{\chi}(\alpha_{j})f\lvert_{k,\alpha_{j}}.

The double coset operator also acts on the cocycle model of cohomology. Let $V$ be a right $\mathbb{K}[\text{GL}_{2}^{+}(\mathbb{Q})]$ -module. Set

\begin{gathered}{}_{V,\chi}:\text{Fun}\big{(}\Gamma_{0}(N),V^{\chi}\big{)}\to\text{Fun}\big{(}\Gamma_{0}(N),V^{\chi}\big{)},\\ P\to\big{(}\gamma\mapsto\sum_{j=1}^{n}\prescript{\iota}{}{\chi}(\alpha_{j(\gamma)})P(\gamma_{j})\alpha_{j(\gamma)}\big{)}\end{gathered}

where $\gamma_{j}$ -s are as in (4.3) and $\alpha_{j(\gamma)}$ acts on $P(\gamma_{j})$ by the $\text{GL}_{2}^{+}(\mathbb{Q})$ -action on $V$ . This map descends to an endomorphism of $H^{1}\big{(}\Gamma_{0}(N),V^{\chi}\big{)}$ that depends only on the double coset $[\Gamma_{0}(N)\alpha\Gamma_{0}(N)]_{V,\chi}$ . An argument similar to Proposition 4.3 verifies that the twisted Eichler-Shimura map is a homomorphism for the twisted Hecke action. Moreover, a straightforward calculation imitating the untwisted counterpart shows that the twisted Hecke action preserves the parabolic subspace of cohomology. Note that $-\text{Id}\in\Gamma_{0}(N)$ . Therefore $(\Gamma_{0}(N),V^{\chi})$ is a descent module if and only if $-\text{Id}$ acts on $V$ by the scalar $\chi(-1)$ . In particular $(\Gamma_{0}(N),V_{k-2,\mathbb{C}}^{\chi})$ is a descent module if and only if $\chi(-1)=(-1)^{k}$ .

We next relate the $\chi$ -eigenspace in cohomology with the cohomology of the $\chi$ -twisted module using the conjugation action of $\Gamma_{0}(N)$ on the cohomology of $\Gamma_{1}(N)$ . This result allows us to bypass the use of trace maps in Hida’s theory [10, p.177]. Let $\mathbb{K}$ be a subfield of $\mathbb{C}$ and $\chi\in\widehat{U}_{N}$ be a character defined over $\mathbb{K}$ . Suppose that $V$ is a $\mathbb{K}[\Gamma_{0}(N)]$ -module. Since $\Gamma_{1}(N)$ is a normal subgroup of $\Gamma_{0}(N)$ the canonical restriction map induces an isomorphism

(7.5)

H^{1}\big{(}\Gamma_{0}(N),V^{\chi}\big{)}\xrightarrow{\cong}H^{1}\big{(}\Gamma_{1}(N),V^{\chi}\big{)}^{\Gamma_{0}(N)}.

Lemma 7.7.

Let $H^{1}(\Gamma_{1}(N),V)[\chi]$ denote the $\chi$ eigenspace for the $\Gamma_{0}(N)$ -action on $H^{1}(\Gamma_{1}(N),V)$ . Then

(i)

$H^{1}\big{(}\Gamma_{1}(N),V^{\chi}\big{)}^{\Gamma_{0}(N)}=H^{1}\big{(}\Gamma_{1}(N),V\big{)}[\chi]$ .
(ii)

Suppose that $(\Gamma_{0}(N),V^{\chi})$ is a descent module. Then the image of the parabolic subspace $H^{1}_{\emph{P}}(\Gamma_{0}(N),V^{\chi})$ under (7.5) equals $H^{1}_{\emph{P}}\big{(}\Gamma_{1}(N),V\big{)}[\chi]$ .

Proof.

The assertion in part (i) follows from the explicit description of the action provided in (4.5). To prove part (ii) we identify the cohomology modules with their projective counterpart and apply Lemma 5.1 to the restriction map described in (7.5). ∎

The action of diamond operators on the spaces of modular forms on $\Gamma_{1}(N)$ yields decompositions of the form

\mathcal{M}_{k}\big{(}\Gamma_{1}(N)\big{)}=\oplus_{\chi\in\widehat{U}_{N}}\mathcal{M}_{k}(N,\chi),\hskip 8.5359pt\mathcal{S}_{k}\big{(}\Gamma_{1}(N)\big{)}=\oplus_{\chi\in\widehat{U}_{N}}\mathcal{S}_{k}(N,\chi)

where the sum is over all the Dirichlet character modulo $N$ . The space $\mathcal{M}_{k}(N,\chi)$ , resp. $\mathcal{S}_{k}(N,\chi)$ , is zero unless $\chi(-1)=(-1)^{k}$ . Attached to these spaces we have Hecke-equivariant twisted Eichler-Shimura homomorphisms

	$\displaystyle{}_{\text{m},N}^{\chi}$	$\displaystyle:\mathcal{M}_{k}(N,\chi)\to H^{1}\big{(}\Gamma_{0}(N),V_{k-2,\mathbb{C}}^{\chi}\big{)},$
	$\displaystyle[\text{ES}_{k}]_{\text{c},N}^{\chi}$	$\displaystyle:\mathcal{S}_{k}(N,\chi)\to H^{1}\big{(}\Gamma_{0}(N),V_{k-2,\mathbb{C}}^{\chi}\big{)}$

for each $\chi$ . The action of diamond operators on the space of anti-holomorphic cusp forms yields a similar eigenspace decomposition

\mathcal{S}^{a}_{k}\big{(}\Gamma_{1}(N)\big{)}=\oplus_{\chi\in\widehat{U}_{N}}\mathcal{S}^{a}_{k}(N,\chi).

It is clear that $\mathcal{S}^{a}_{k}(N,\chi)=\overline{\mathcal{S}_{k}(N,\bar{\chi})}$ . We transport the $\bar{\chi}$ -Hecke action on $\mathcal{S}_{k}(N,\bar{\chi})$ via complex-conjugation to obtain a $\chi$ -Hecke action on $\mathcal{S}^{a}_{k}(N,\chi)$ . This procedure yields a well-defined $\chi$ -Hecke equivariant homomorphism

[\text{ES}_{k}]_{\text{c},N}^{a,\chi}:\mathcal{S}^{a}_{k}(N,\chi)\to H^{1}\big{(}\Gamma_{0}(N),V_{k-2,\mathbb{C}}^{\chi}\big{)};\hskip 5.69046ptf\to\#[\text{ES}_{k}]_{\text{c},N}^{\bar{\chi}}(\bar{f})

where $\#$ is the complex conjugation defined in Section 5.2.

Theorem 7.8.

Let $N$ be an integer $\geq 1$ and $\chi$ be a character on $(\mathbb{Z}/N\mathbb{Z})^{\times}$ . The $\chi$ -Hecke-equivariant homomorphism

(7.6)

[\emph{ES}_{k}]_{\emph{m},N}^{\chi}\oplus[\emph{ES}_{k}]_{\emph{c},N}^{a,\chi}:\mathcal{M}_{k}(N,\chi)\oplus\mathcal{S}^{a}_{k}(N,\chi)\to H^{1}\big{(}\Gamma_{0}(N),V_{k-2,\mathbb{C}}^{\chi}\big{)}

is an isomorphism. Moreover the image of $\mathcal{S}_{k}(N,\chi)\oplus\mathcal{S}^{a}_{k}(N,\chi)$ under this map equals $H^{1}_{\emph{P}}\big{(}\Gamma_{0}(N),V_{k-2,\mathbb{C}}^{\chi}\big{)}$ .

Proof.

We start with the untwisted Eichler-Shimura isomorphism

(7.7)

\mathcal{M}_{k}\big{(}\Gamma_{1}(N)\big{)}\oplus\mathcal{S}^{a}_{k}\big{(}\Gamma_{1}(N)\big{)}\xrightarrow{\cong}H^{1}\big{(}\Gamma_{1}(N),V_{k-2,\mathbb{C}}\big{)}.

One considers the $\chi$ -eigenspaces for the action of diamond operators to discover that the isomorphism (7.7) maps $\mathcal{M}_{k}(N,\chi)\oplus\mathcal{S}^{a}_{k}(N,\chi)$ isomorphically onto $H^{1}\big{(}\Gamma_{1}(N),V_{k-2,\mathbb{C}}\big{)}[\chi]$ . Now the Eichler-Shimura homomorphism and its twisted avatar together yield a commutative diagram

(7.8)

where the vertical arrow is the restriction map. Note that the vertical arrow is injective and its image equals $H^{1}\big{(}\Gamma_{1}(N),V_{k-2,\mathbb{C}}\big{)}[\chi]$ (Lemma 7.7). Therefore the twisted homomorphism (7.6) is an isomorphism. If $\chi(-1)=(-1)^{k+1}$ then the domain and codomain of (7.6) are zero (Example 4.5). Now suppose $\chi(-1)=(-1)^{k}$ . Observe that the image of $\mathcal{S}_{k}(N,\chi)\oplus\mathcal{S}^{a}_{k}(N,\chi)$ under (7.7) is $H^{1}_{\text{P}}\big{(}\Gamma_{1}(N),V_{k-2,\mathbb{C}}\big{)}[\chi]$ . Hence, by Lemma 7.7(ii), the isomorphism (7.6) must map $\mathcal{S}_{k}(N,\chi)\oplus\mathcal{S}^{a}_{k}(N,\chi)$ onto $H^{1}_{\text{P}}\big{(}\Gamma_{0}(N),V_{k-2,\mathbb{C}}^{\chi}\big{)}$ . ∎

The space of Eisenstein series with nebentypus $\chi$ is given by

(7.9)

\mathcal{E}_{k}(N,\chi):=\mathcal{E}_{k}\big{(}\Gamma_{1}(N)\big{)}\cap\mathcal{M}_{k}(N,\chi).

If $\chi$ is trivial then $\mathcal{E}_{k}(N,\chi)=\mathcal{E}_{k}\big{(}\Gamma_{0}(N)\big{)}$ . Without loss of generality assume that $\chi$ is a nontrivial character with $\chi(-1)=(-1)^{k}$ . In this setting, the author has recently constructed an explicit basis $\mathcal{B}_{k,\chi}$ that is parametrized by the cusps of $\Gamma_{0}(N)$ and established a weaker version of Theorem 7.8 [15, 7.3]. By construction the basis functions in $\mathcal{B}_{k,\chi}$ lie inside $\mathcal{E}_{k}\big{(}\Gamma_{1}(N),\mathbb{K}\big{)}$ if $\mathbb{Q}_{\varphi(N)}\subseteq\mathbb{K}$ . Here $\varphi$ is Euler’s totient function and $\mathbb{Q}_{\varphi(N)}$ is the $\varphi(N)$ -th cyclotomic extension of $\mathbb{Q}$ in $\mathbb{C}$ . Recall the spectral basis $\mathcal{B}_{\mathcal{G},k}$ from Section 5.3. For an arbitrary character $\chi$ set

(7.10)

\mathcal{B}_{k,\chi}=\begin{cases}\emptyset,&\text{if $\chi(-1)=(-1)^{k+1}$};\\ \mathcal{B}_{\Gamma_{0}(N),k},&\text{if $k$ is even and $\chi$ is trivial};\\ \text{$\mathcal{B}_{k,\chi}$ above},&\text{if $\chi$ is nontrivial and $\chi(-1)=(-1)^{k}$.}\end{cases}

The compatibility with restriction property described in (5.7) shows that $\mathcal{E}_{k}(\Gamma_{0}(N),\mathbb{K})\subseteq\mathcal{E}_{k}(\Gamma_{1}(N),\mathbb{K})$ for each $\mathbb{K}$ . Therefore $\mathcal{B}_{k,\chi}\subseteq\mathcal{E}_{k}\big{(}\Gamma_{1}(N),\mathbb{K}\big{)}$ for all $\chi$ whenever $\mathbb{Q}_{\varphi(N)}\subseteq\mathbb{K}$ . We study rationality of the image of this basis under $[\text{ES}_{k}]_{\text{m},N}^{\chi}$ in Section 8.2.

8 Computation of Eichler-Shimura map

Next, we use the induced picture, in particular, Proposition 7.6 to compute the Eichler-Shimura map for the space of Eisenstein series on principal congruence subgroups. This calculation enables us to prove algebraicity results regarding the image of the Eichler-Shimura map attached to the space of Eisenstein series on an arbitrary congruence subgroup.

8.1 Eisenstein series on principal congruence subgroups

8.1.1 Eisenstein series on the universal elliptic curve

This subsection aims to study a special family of Eisenstein series arising from the $N$ -torsion points of the universal elliptic curve that exhibits remarkable rationality properties regarding period polynomials. We begin with a brief review of Hecke’s theory of Eisenstein series on $\Gamma(N)$ .

Let $k$ be an integer $\geq 2$ and $\Lambda=\mathbb{Z}^{2}$ . With each $\bar{\lambda}\in\Lambda/N\Lambda$ one associates an Eisenstein series described as

G_{k}(\tau,\bar{\lambda},N;s):=\sum_{\begin{subarray}{c}(c,d)\equiv\lambda(N),\\ (c,d)\neq(0,0)\end{subarray}}\frac{\text{Im}(\tau)^{s}}{(c\tau+d)^{k}\lvert c\tau+d\rvert^{2s}}\hskip 8.5359pt(\begin{subarray}{c}\tau\in\mathbb{H}\end{subarray})

where $\lambda$ is lift of $\bar{\lambda}$ and $\text{Im}(\tau)$ is the imaginary part of $\tau$ . This series converges uniformly on the compact subsets of $\text{Re}(k+2s)>2$ to define an analytic function of $s$ in this region that admits an analytic continuation to the whole $s$ plane. Set $G_{k}(\tau,\bar{\lambda},N):=G_{k}(\tau,\bar{\lambda},N;0)$ . As a function on $\mathbb{H}$ the Fourier expansion of the $G$ -series is $G_{k}(\tau,\bar{\lambda},N)=\sum_{j\geq 0}A_{k,j}(\bar{\lambda})\mathbf{e}(\frac{j\tau}{N})+P_{k}(\tau)$ where

(8.1)

\begin{gathered}A_{k,0}(\bar{\lambda}):=\mathbbm{1}_{\mathbb{Z}/N\mathbb{Z}}(\bar{\lambda}_{1},0)\sum_{\begin{subarray}{c}n\in\mathbb{Z}-\{0\},\\ n\equiv\lambda_{2}(N)\end{subarray}}\frac{1}{n^{k}},\;P_{k}(\tau):=-\mathbbm{1}_{\mathbb{Z}}(k,2)\frac{\pi}{N^{2}\text{Im}(\tau)}\\ A_{k,j}(\bar{\lambda}):=\frac{(-2\pi i)^{k}}{(k-1)!N^{k}}\sum_{\begin{subarray}{c}r\in\mathbb{Z}-\{0\},\\ r\mid j,\frac{j}{r}\equiv\lambda_{1}(N)\end{subarray}}\text{sgn}(r)r^{k-1}\mathbf{e}(\frac{r\lambda_{2}}{N})\hskip 11.38092pt(\begin{subarray}{c}j\geq 1\end{subarray})\end{gathered}

and $(\lambda_{1},\lambda_{2})$ is a lift of $\bar{\lambda}$ to $\Lambda$ [17, 9.1]. Recall the rational structure on the space of Eisenstein series from Section 5.3 arising from the spectral basis. Let $\mathbb{Q}_{N}$ denote the $N$ -th cyclotomic field $\mathbb{Q}\big{(}\mathbf{e}(\frac{1}{N})\big{)}$ . Suppose that $\mathbb{K}$ is a subfield of $\mathbb{C}$ containing $\mathbb{Q}_{N}$ . Then the $\mathbb{K}$ -span of $\{G_{k}(\bar{\lambda},N)\mid\bar{\lambda}\in\Lambda/N\Lambda\}$ equals $(2\pi i)^{k}\mathcal{E}_{k}^{\ast}\big{(}\Gamma(N),\mathbb{K}\big{)}$ and its intersection with $\text{Hol}(\mathbb{H})$ is $(2\pi i)^{k}\mathcal{E}_{k}\big{(}\Gamma(N),\mathbb{K}\big{)}$ [15, 6.2]. The explicit Fourier expansion makes $G$ -series accessible for the computation of $L$ -function and $L$ -values. A standard approach towards this problem uses a cyclotomic linear combination of $G$ -series to write down a Hecke eigenfunction on $\Gamma(N)$ whose $L$ -function is given by a product of two Dirichlet $L$ -functions [13, IV-39]. However, it is difficult to study the rationality properties of special values of such product $L$ -functions.

One defines the Eisenstein series on the universal elliptic curve as

e_{k}:\mathbb{C}\times\mathbb{H}\to\mathbb{C},\hskip 8.5359pte_{k}(\eta,\tau)=\sum_{w\in\Lambda_{\tau}-\{0\}}\frac{\chi_{\eta}(w)}{w^{k}}\hskip 14.22636pt(\begin{subarray}{c}k\geq 3\end{subarray})

where $\Lambda_{\tau}=\{(c\tau+d)\mid(c,d)\in\Lambda\}$ and $\chi_{\eta}(w)=\mathbf{e}(\frac{w\bar{\eta}-\bar{w}\eta}{\tau-\bar{\tau}})$ [11, 4]. This series converges uniformly on compact subsets to define a real analytic function that transforms like a Jacobi form of weight $k$ and index $0$ . In particular $e_{k}(\eta+c\tau+d,\tau)=e_{k}(\eta,\tau)$ for each $(c,d)\in\Lambda$ . With $\bar{\lambda}\in\Lambda/N\Lambda$ one associates the division value

e_{k}(\tau,\bar{\lambda},N):=e_{k}(\frac{\lambda_{1}\tau}{N}+\frac{\lambda_{2}}{N},\tau)=\sum_{\begin{subarray}{c}(c,d)\in\Lambda,\\ (c,d)\neq(0,0)\end{subarray}}\frac{\mathbf{e}(\frac{c\lambda_{2}-d\lambda_{1}}{N})}{(c\tau+d)^{k}};\hskip 8.5359pt\bar{\lambda}=\overline{(\lambda_{1},\lambda_{2})}.

Let $\mathbf{b}:\Lambda/N\Lambda\times\Lambda/N\Lambda\to\mathbb{C}$ be the nondegenerate bilinear form defined by

\mathbf{b}(\bar{\lambda},\bar{\theta})=\mathbf{e}(\frac{\theta_{1}\lambda_{2}-\lambda_{1}\theta_{2}}{N});\hskip 8.5359pt\text{$\bar{\lambda}=\overline{(\lambda_{1},\lambda_{2})}$ and $\bar{\theta}=\overline{(\theta_{1},\theta_{2})}$}.

Then the relation between $e$ -series and $G$ -series is as follows:

(8.2)

e_{k}(\bar{\lambda},N)=\sum_{\bar{\theta}\in\Lambda/N\Lambda}\mathbf{b}(\bar{\lambda},\bar{\theta})G_{k}(\bar{\theta},N).

Let $k$ be an integer $\geq 2$ and $\bar{\lambda}\in\Lambda/N\Lambda$ . We define the elliptic Eisenstein series of weight $k$ and parameter $\bar{\lambda}$ , denoted $e_{k}(\bar{\lambda},N)$ , using the identity (8.2). Note that $\mathbf{b}$ is compatible with the natural right action of $\Gamma$ on $\Lambda$ , i.e., $\mathbf{b}(\bar{\lambda}\gamma,\bar{\theta}\gamma)=\mathbf{b}(\bar{\lambda},\bar{\theta})$ for each $\gamma\in\Gamma$ . Thus $e_{k}(\bar{\lambda},N)\lvert_{\gamma}=e_{k}(\bar{\lambda}\gamma,N)$ . In particular $e_{k}(\bar{\lambda},N)$ is invariant under $\Gamma(N)$ . One uses the orthogonality of characters for $\Lambda/N\Lambda$ to invert the linear relation (8.2) and discovers that

G_{k}(\bar{\theta},N)=\frac{1}{N^{2}}\sum_{\bar{\lambda}\in\Lambda/N\Lambda}\mathbf{b}(\bar{\theta},\bar{\lambda})e_{k}(\bar{\lambda},N).

Therefore the $\mathbb{K}$ -span of $\{e_{k}(\bar{\lambda},N)\mid\bar{\lambda}\in\Lambda/N\Lambda\}$ equals the $\mathbb{K}$ -span of $\{G_{k}(\bar{\lambda},N)\mid\bar{\lambda}\in\Lambda/N\Lambda\}$ whenever $\mathbb{Q}_{N}\subseteq\mathbb{K}$ .

We next record the Fourier expansion formula for the elliptic Eisenstein series. Let $B_{\bullet}(-)$ be the Bernoulli polynomial defined by the generating function identity

\frac{X\exp(tX)}{\exp(X)-1}=\sum_{n\geq 0}\frac{B_{n}(t)}{n!}X^{n}.

Our calculation requires the Fourier expansion formula for the Bernoulli polynomials [17, 4.5]:

(8.3)

B_{k}(t)=-\frac{k!}{(2\pi i)^{k}}\sum_{n\in\mathbb{Z}-\{0\}}\frac{\mathbf{e}(nt)}{n^{k}}.\hskip 14.22636pt(\begin{subarray}{c}k\geq 2,\;0\leq t\leq 1\end{subarray})

The formula is valid even for $k=1$ if $0<t<1$ and one interprets the sum as $\lim_{m\to\infty}\sum_{0<\lvert n\rvert\leq m}$ .

Lemma 8.1.

Let $k$ be an integer $\geq 2$ and $\bar{\lambda}\in\Lambda/N\Lambda$ . Suppose that $(\lambda_{1},\lambda_{2})$ is a lift of $\bar{\lambda}$ that satisfies $0\leq\lambda_{1}<N$ . The Fourier series for $e_{k}(\bar{\lambda},N)$ is

	$\displaystyle e_{k}(\bar{\lambda},N)$	$\displaystyle=-\frac{(-2\pi i)^{k}}{k!}B_{k}(\frac{\lambda_{1}}{N})-\mathbbm{1}_{\mathbb{Z}}(k,2)\mathbbm{1}_{\Lambda/N\Lambda}(\bar{\lambda},0)\frac{\pi}{\emph{Im}(\tau)}$
		$\displaystyle+\frac{(-2\pi i)^{k}}{(k-1)!N^{k-1}}\sum_{\begin{subarray}{c}m,n\geq 1,\\ n\equiv\lambda_{1}(N)\end{subarray}}n^{k-1}\mathbf{e}(\frac{mn\tau}{N}+\frac{m\lambda_{2}}{N})$
		$\displaystyle+\frac{(2\pi i)^{k}}{(k-1)!N^{k-1}}\sum_{\begin{subarray}{c}m,n\geq 1,\\ n\equiv-\lambda_{1}(N)\end{subarray}}n^{k-1}\mathbf{e}(\frac{mn\tau}{N}-\frac{m\lambda_{2}}{N}).$

Proof.

We substitute the Fourier expansion formula for the $G$ -series (8.1) into the defining relation (8.2) and simplify the expression to obtain the identity above. The constant term is a consequence of (8.3) while the real analytic term follows from a straightforward calculation. ∎

Corollary 8.2.

Let $\bar{\lambda}\in\Lambda/N\Lambda$ . Then $e_{2}(\bar{\lambda},N)$ is holomorphic if and only if $\bar{\lambda}\neq 0$ . Moreover $\{e_{2}(\bar{\lambda},N)\mid\bar{\lambda}\neq 0\}$ is a spanning set for the holomorphic subspace $(2\pi i)^{2}\mathcal{E}_{2}\big{(}\Gamma(N),\mathbb{K}\big{)}$ whenever $\mathbb{Q}_{N}\subseteq\mathbb{K}$ .

Proof.

The first part of the assertion is obvious. The extended space of $G$ -series $(2\pi i)^{2}\mathcal{E}^{\ast}_{2}\big{(}\Gamma(N),\mathbb{K}\big{)}$ contains $(2\pi i)^{2}\mathcal{E}_{2}\big{(}\Gamma(N),\mathbb{K}\big{)}$ as a subspace of codimension $1$ . But $\{e_{2}(\bar{\lambda},N)\mid\bar{\lambda}\neq 0\}$ is already a subset of holomorphic subspace. Therefore this collection spans $(2\pi i)^{2}\mathcal{E}_{2}\big{(}\Gamma(N),\mathbb{K}\big{)}$ . ∎

For convenience, we introduce the notation

\mathcal{I}_{N,k}:=\begin{cases}\Lambda/N\Lambda,&\text{$k$ even and $\geq 4$;}\\ \Lambda/N\Lambda,&\text{$k$ odd and $N\geq 3$;}\\ \Lambda/N\Lambda-\{0\},&k=2;\\ \emptyset,&\text{otherwise.}\end{cases}

8.1.2 $L$ -function and $L$ -values of the elliptic Eisenstein series

Our next job is explicitly writing down the completed $L$ -function (7.4) attached to the elliptic Eisenstein series and determining its values at $\{1,\ldots,k-1\}$ . Let $x\in\mathbb{R}$ and $0<a\leq 1$ . One defines the Lerch zeta function [1] by

(8.4)

\phi(x,a,s)=\sum_{n\geq 0}\frac{\mathbf{e}(nx)}{(n+a)^{s}}.

The series above converges uniformly on the compact subsets of $\{s\mid\text{Re}(s)>1\}$ to define an analytic function in this region. The Lerch series admits a meromorphic continuation to the whole $s$ -plane with a simple pole at $s=1$ whenever $x$ is an integer. If $x$ is not an integer then $\phi(x,a,\cdot)$ extends to an entire function on the $s$ -plane. Our treatment requires two special avatars of the Lerch function. The Lerch series reduces to the familiar Hurwitz zeta function $\zeta(a,s)$ for $x=0$ , i.e., $\zeta(a,s)=\phi(0,a,s)$ . The value of the Hurwitz zeta function at nonnegative integers [2, 12.11] is given by

(8.5)

\zeta(a,-n)=-\frac{B_{n+1}(a)}{n+1}.\hskip 8.5359pt(\begin{subarray}{c}0<a\leq 1,\;n\geq 0\end{subarray})

On the other hand setting $a=1$ in (8.4) yields the polylogarithm function given by the convergent series

\text{Li}_{s}\big{(}\mathbf{e}(x)\big{)}:=\sum_{n\geq 1}\frac{\mathbf{e}(nx)}{n^{s}}=\mathbf{e}(x)\phi(x,1,s)

in the region $\text{Re}(s)>1$ . The meromorphic continuation of the Lerch function provides a meromorphic continuation for Li. If $x$ is not an integer then $\text{Li}_{1}\big{(}\mathbf{e}(x)\big{)}$ equals the conditionally convergent logarithmic series $\sum_{n\geq 1}\frac{\mathbf{e}(nx)}{n}$ . The Fourier expansion formula for the Bernoulli polynomials (8.3) shows that

(8.6)

\text{Li}_{k}\big{(}\mathbf{e}(x)\big{)}+(-1)^{k}\text{Li}_{k}\big{(}\mathbf{e}(-x)\big{)}=-\frac{(2\pi i)^{k}}{k!}B_{k}(x)\hskip 8.5359pt(\begin{subarray}{c}0\leq x\leq 1\end{subarray})

for each $k\geq 2$ . The identity above is valid even for $k=1$ if $0<x<1$ .

Lemma 8.3.

Let $k\geq 2$ and $\bar{\lambda}\in\mathcal{I}_{N,k}$ . Suppose that $(\lambda_{1},\lambda_{2})$ is a lift of $\bar{\lambda}$ that satisfies $0\leq\lambda_{1}<N$ . Then

(8.7)			$\displaystyle\frac{(2\pi)^{s}}{\Gamma(s)}L^{\ast}\big{(}e_{k}(\bar{\lambda},N),s\big{)}$
			$\displaystyle=\frac{(-2\pi i)^{k}}{(k-1)!}\big{\{}\emph{Li}_{s}\big{(}\mathbf{e}(\frac{\lambda_{2}}{N})\big{)}\zeta^{\ast}(\frac{\lambda_{1}}{N},s-k+1)+$
			$\displaystyle\hskip 85.35826pt(-1)^{k}\emph{Li}_{s}\big{(}\mathbf{e}(-\frac{\lambda_{2}}{N})\big{)}\zeta(1-\frac{\lambda_{1}}{N},s-k+1)\big{\}}$

where

\zeta^{\ast}(\frac{\lambda_{1}}{N},s)=\begin{cases}\zeta(\frac{\lambda_{1}}{N},s),&\text{if $0<\lambda_{1}<N$;}\\ \zeta(1,s),&\text{if $\lambda_{1}=0$.}\end{cases}

Proof.

We first assume that $\text{Re}(s)>k$ . Then the completed $L$ -function admits convergent integral representation and the identity (8.7) is a straightforward consequence of Lemma 8.1. Observe that both the sides of (8.7) meromorphically continue to the whole plane. Thus the identity holds due to the principle of analytic continuation. ∎

Proposition 8.4.

Let the notation be as in Lemma 8.3. We further assume that $0\leq\lambda_{2}<N$ . Suppose that $1\leq r\leq k-1$ . Then

	$\displaystyle L^{\ast}\big{(}e_{k}(\bar{\lambda},N),r\big{)}$	$\displaystyle=i^{r}\frac{(-2\pi i)^{k}}{(k-1)!}\frac{B_{k-r}(\frac{\lambda_{1}}{N})B_{r}(\frac{\lambda_{2}}{N})}{r(k-r)}$
		$\displaystyle+(-1)^{k-1}i^{k}\mathbbm{1}_{\mathbb{Z}/N\mathbb{Z}}(\lambda_{1},0)\mathbbm{1}_{\mathbb{Z}}(r,k-1)\frac{2\pi}{k-1}\emph{Li}_{k-1}\big{(}\mathbf{e}(\frac{\lambda_{2}}{N})\big{)}$
		$\displaystyle-\mathbbm{1}_{\mathbb{Z}/N\mathbb{Z}}(\lambda_{2},0)\mathbbm{1}_{\mathbb{Z}}(r,1)\frac{2\pi}{k-1}\emph{Li}_{k-1}\big{(}\mathbf{e}(-\frac{\lambda_{1}}{N})\big{)}.$

The proof of the identity involves the following standard properties of the Bernoulli polynomials [17, 4.5]:

(8.8)

\begin{gathered}B_{n}(1-t)=(-1)^{n}B_{n}(t),\hskip 8.5359pt(\begin{subarray}{c}0\leq t\leq 1,\;n\geq 0\end{subarray}),\\ B_{n}(1+t)=B_{n}(t)+nt^{n-1}.\hskip 8.5359pt(\begin{subarray}{c}0\leq t\leq 1,\;n\geq 1\end{subarray})\end{gathered}

In particular

B_{n}(1)=\begin{cases}B_{n}(0),&\text{if $n\geq 2$;}\\ B_{n}(0)+1,&\text{if $n=1$.}\end{cases}

Proof.

If $0<\lambda_{1},\lambda_{2}<N$ then the identity is an easy consequence of the formulas given in (8.5)-(8.8):

	$\displaystyle L^{\ast}\big{(}e_{k}(\bar{\lambda},N),r\big{)}$
	$\displaystyle=-\frac{(r-1)!}{(2\pi)^{r}}\frac{(-2\pi i)^{k}}{(k-1)!(k-r)}\Big{\{}\text{Li}_{r}\big{(}\mathbf{e}(\frac{\lambda_{2}}{N})\big{)}B_{k-r}(\frac{\lambda_{1}}{N})+$
	$\displaystyle\hskip 170.71652pt(-1)^{k}\text{Li}_{r}\big{(}\mathbf{e}(-\frac{\lambda_{2}}{N})\big{)}B_{k-r}(1-\frac{\lambda_{1}}{N})\Big{\}}$
	$\displaystyle=i^{r}\frac{(-2\pi i)^{k}}{(k-1)!}\frac{B_{k-r}(\frac{\lambda_{1}}{N})B_{r}(\frac{\lambda_{2}}{N})}{r(k-r)}.$

The extremal case of (8.8) together with (8.6) guarantee that the calculation for $0<\lambda_{1},\lambda_{2}<N$ extends to the following situations without any difficulty:

•

all $(\lambda_{1},\lambda_{2})$ if $2\leq r\leq k-2$ ,
•

$\{(0,\lambda_{2})\mid\lambda_{2}\neq 0\}$ if $r\neq k-1$ , $\{(\lambda_{1},0)\mid\lambda_{1}\neq 0\}$ if $r\neq 1$ .

Suppose that $r=k-1$ and $\lambda_{1}=0$ . The hypothesis on $\bar{\lambda}$ implies that if $k=2$ then $\lambda_{2}$ is automatically nonzero. In this situation $\zeta^{\ast}$ contributes $B_{1}(1)$ instead of $B_{1}(0)$ and this discrepancy gives rise to the desired term involving $\text{Li}_{k-1}\big{(}\mathbf{e}(\frac{\lambda_{2}}{N})\big{)}$ . It remains to verify the identity for $r=1$ and $\lambda_{2}=0$ . As before, if $k=2$ then $\lambda_{1}\neq 0$ . The functional equation for the completed $L$ -function implies that

	$\displaystyle L^{\ast}\big{(}e_{k}\big{(}\overline{(\lambda_{1},0)},N\big{)},1\big{)}$
	$\displaystyle=(-i)^{k}L^{\ast}\big{(}e_{k}\big{(}\overline{(0,\lambda_{1})},N\big{)},k-1\big{)}$
	$\displaystyle=i\frac{(-2\pi i)^{k}}{(k-1)!}\frac{B_{k-1}(\frac{\lambda_{1}}{N})B_{1}(0)}{k-1}-\frac{2\pi}{k-1}\text{Li}_{k-1}\big{(}\mathbf{e}(-\frac{\lambda_{1}}{N})\big{)}.$

Here, in the last step, we are using (8.6) to simplify the expression. ∎

8.2 Algebraicity of cohomology classes

Let the notation be as in Section 8.1. The explicit expressions for the special values of the completed $L$ function in Proposition 8.4 motivates us to normalize the elliptic Eisenstein series as

\tilde{e}_{k}(\bar{\lambda},N):=\frac{1}{(-2\pi i)^{k}}e_{k}(\bar{\lambda},N).\hskip 8.5359pt(\begin{subarray}{c}\bar{\lambda}\in\Lambda/N\Lambda\end{subarray})

The following lemma computes the value of the period function attached to the $\Gamma$ -stable subset $\mathcal{M}_{k}\big{(}\Gamma(N)\big{)}$ at the normalized elliptic Eisenstein series.

Lemma 8.5.

Let $k\geq 2$ and $\bar{\lambda}\in\mathcal{I}_{N,k}$ . Suppose that $(\lambda_{1},\lambda_{2})$ is a lift of $\bar{\lambda}$ satisfying $0\leq\lambda_{1},\lambda_{2}<N$ . Then

	$\displaystyle\mathsf{p}_{k}^{\ast}\big{(}T,\tilde{e}_{k}(\bar{\lambda},N),X\big{)}=-\frac{B_{k}(\frac{\lambda_{1}}{N})}{k!(k-1)}\big{(}(X+1)^{k-1}-X^{k-1}\big{)},$
	$\displaystyle\mathsf{p}_{k}^{\ast}\big{(}S,\tilde{e}_{k}(\bar{\lambda},N),X\big{)}=-\sum_{r=0}^{k-2}\binom{k}{r+1}\frac{B_{k-r-1}(\frac{\lambda_{1}}{N})B_{r+1}(\frac{\lambda_{2}}{N})}{k!(k-1)}X^{k-2-r}+(-1)^{k-1}$
	$\displaystyle\mathbbm{1}_{\mathbb{Z}/N\mathbb{Z}}(\lambda_{1},0)\frac{\emph{Li}_{k-1}\big{(}\mathbf{e}(\frac{\lambda_{2}}{N})\big{)}}{(k-1)(-2\pi i)^{k-1}}+\mathbbm{1}_{\mathbb{Z}/N\mathbb{Z}}(\lambda_{2},0)\frac{\emph{Li}_{k-1}\big{(}\mathbf{e}(-\frac{\lambda_{1}}{N})\big{)}}{(k-1)(-2\pi i)^{k-1}}X^{k-2}.$

Proof.

The demonstration is a straightforward exercise using the formulas in Proposition 7.6, Lemma 8.1, and Proposition 8.4. ∎

Note that the value of the period polynomial at $S$ involves polylogarithm functions. Thus it is difficult to examine the rationality properties of the period polynomials described above. We next explain how to change the Eichler-Shimura cocycle attached to a function by a coboundary so that the resulting cocycle has coefficients in $\mathbb{Q}$ ; cf. [20, 7]. Let

(8.9)

[\text{ES}_{k}]_{\text{m},\Gamma(N)}:\mathcal{M}_{k}\big{(}\Gamma(N)\big{)}\to H^{1}\big{(}\Gamma(N),V_{k-2,\mathbb{C}}\big{)}

be the Eichler-Shimura map attached to the space of modular forms.

Proposition 8.6.

Let $k\geq 2$ and $\bar{\lambda}\in\mathcal{I}_{N,k}$ . Then the image of $\tilde{e}_{k}(\bar{\lambda},N)$ under the Eichler-Shimura map (8.9) is definable over $\mathbb{Q}$ .

Proof.

We use the induced picture of Section 7.1 to compute the Eichler-Shimura map. In the light of the explicit Shapiro map (7.1) it suffices to check that the image of $\tilde{e}_{k}(\bar{\lambda},N)$ under $\text{Ind-}[\text{ES}_{k}]$ lies inside the rational subspace $H^{1}\big{(}\Gamma,\text{Fun}(\Gamma(N)\backslash\Gamma,V_{k-2,\mathbb{Q}})\big{)}$ . For $\sigma\in\Gamma$ let $(\lambda_{1}^{\sigma},\lambda_{2}^{\sigma})$ denote the unique lift of $\bar{\lambda}\sigma$ in $[0,N)\times[0,N)$ . Note that $(\lambda_{1}^{\sigma},\lambda_{2}^{\sigma})$ depends only on the coset of $\sigma$ in $\Gamma(N)\backslash\Gamma$ .

Case I. $k\geq 3$ .

Define $F\in\text{Fun}\big{(}\Gamma(N)\backslash\Gamma,V_{k-2,\mathbb{C}}\big{)}$ by setting

F\big{(}\Gamma(N)\sigma\big{)}=i^{3-k}L^{\ast}\big{(}\tilde{e}_{k}(\bar{\lambda}\sigma,N),k-1\big{)}.

A straight-forward computation using the formulas in Proposition 8.4 and Lemma 8.5 shows that the modified cocycle

\Gamma\to\text{Fun}\big{(}\Gamma(N)\backslash\Gamma,V_{k-2,\mathbb{C}}\big{)},\;\gamma\mapsto\text{Ind-}\text{ES}^{\ast}_{k}(\tilde{e}_{k}(\bar{\lambda},N))(\gamma)+F\lvert_{\gamma}-F,

takes values in $\text{Fun}\big{(}\Gamma(N)\backslash\Gamma,V_{k-2,\mathbb{Q}}\big{)}$ at $T$ and $S$ .

Case II. $k=2$ .

The problem, in this case, is that both the transcendental terms in the special $L$ -value may simultaneously contribute a term to the calculation. We construct another function $F\in\text{Fun}\big{(}\Gamma(N)\backslash\Gamma,\mathbb{C}\big{)}$ by putting

F\big{(}\Gamma(N)\sigma\big{)}=\begin{cases}0,&\text{if $\lambda_{1}^{\sigma}\neq 0$;}\\ iL^{\ast}\big{(}\tilde{e}_{2}(\bar{\lambda}\sigma,N),1\big{)},&\text{if $\lambda_{1}^{\sigma}=0$.}\end{cases}

Then the modified cocycle

\Gamma\to\text{Fun}\big{(}\Gamma(N)\backslash\Gamma,\mathbb{C}\big{)},\;\gamma\mapsto\text{Ind-}\text{ES}^{\ast}_{2}(\tilde{e}_{2}(\bar{\lambda},N))(\gamma)+F\lvert_{\gamma}-F,

again takes values in $\text{Fun}\big{(}\Gamma(N)\backslash\Gamma,\mathbb{Q}\big{)}$ at $T$ and $S$ . ∎

Let $\mathbb{K}$ be a subfield of $\mathbb{C}$ containing $\mathbb{Q}_{N}$ . Recall that the $\mathbb{K}$ -span of $\{e_{k}(\bar{\lambda},N)\mid\bar{\lambda}\in\mathcal{I}_{N,k}\}$ equals $(2\pi i)^{k}\mathcal{E}_{k}\big{(}\Gamma(N),\mathbb{K}\big{)}$ . Therefore the collection

\{\tilde{e}_{k}(\bar{\lambda},N)\mid\bar{\lambda}\in\mathcal{I}_{N,k}\}

is a $\mathbb{K}$ -spanning set of $\mathcal{E}_{k}\big{(}\Gamma(N),\mathbb{K}\big{)}$ .

Corollary 8.7.

The homomorphism $[\emph{ES}_{k}]_{\emph{m},\Gamma(N)}$ maps $\mathcal{E}_{k}\big{(}\Gamma(N),\mathbb{K}\big{)}$ inside the canonical $\mathbb{K}$ -form $H^{1}\big{(}\Gamma(N),V_{k-2,\mathbb{K}}\big{)}$ .

Proof.

Proposition 8.6 demonstrates that

[\text{ES}_{k}]_{\text{m},\Gamma(N)}\big{(}\tilde{e}_{k}(\bar{\lambda},N)\big{)}\in H^{1}\big{(}\Gamma(N),V_{k-2,\mathbb{Q}}\big{)}

for each $\bar{\lambda}\in\mathcal{I}_{N,k}$ . Thus the assertion follows from the discussion above. ∎

Suppose that $\mathcal{G}$ is a congruence subgroup containing the principal level $\Gamma(N)$ . One considers the $\mathcal{G}$ -invariants of the domain and codomain of (8.9) to retrieve the Eichler-Shimura homomorphism for $\mathcal{G}$ :

(8.10)

[\text{ES}_{k}]_{\text{m},\mathcal{G}}:\mathcal{M}_{k}(\mathcal{G})\to H^{1}\big{(}\mathcal{G},V_{k-2,\mathbb{C}}\big{)}.

Theorem 8.8.

Let $\mathbb{K}$ be a subfield of $\mathbb{C}$ containing $\mathbb{Q}_{N}$ . Then the image of $\mathcal{E}_{k}(\mathcal{G},\mathbb{K})$ under the Eichler-Shimura map (8.10) lies inside $H^{1}\big{(}\mathcal{G},V_{k-2,\mathbb{K}}\big{)}$ .

Proof.

From Corollary 8.7 we know that the Eichler-Shimura homomorphism (8.9) maps $\mathcal{E}_{k}\big{(}\Gamma(N),\mathbb{K}\big{)}$ inside $H^{1}\big{(}\Gamma(N),V_{k-2,\mathbb{K}}\big{)}$ . But the $\mathcal{G}$ -invariant subspace of $\mathcal{E}_{k}\big{(}\Gamma(N),\mathbb{K}\big{)}$ , resp. $H^{1}\big{(}\Gamma(N),V_{k-2,\mathbb{K}}\big{)}$ , equals $\mathcal{E}_{k}(\mathcal{G},\mathbb{K})$ , resp. $H^{1}\big{(}\mathcal{G},V_{k-2,\mathbb{K}}\big{)}$ . Therefore the image of $\mathcal{E}_{k}(\mathcal{G},\mathbb{K})$ under $[\text{ES}_{k}]_{\text{m},\mathcal{G}}$ must lie in $H^{1}\big{(}\mathcal{G},V_{k-2,\mathbb{K}}\big{)}$ . ∎

Proof of Theorem 1.5:

Let the notation be as in Theorem 1.5. Suppose that $f\in\mathcal{B}_{\mathcal{G},k}$ . From the definition of rational structure it is clear that $f\in\mathcal{E}_{k}(\mathcal{G},\mathbb{Q}_{N})$ . Hence the assertion is a consequence of Theorem 8.8. $\square$

Remark 8.9.

The space of Eisenstein series $\mathcal{E}_{k}(\mathcal{G})$ possesses a computable basis (different from $\mathcal{B}_{\mathcal{G},k}$ ) so that each basis element lies in $\mathcal{E}_{k}(\mathcal{G},\mathbb{Q}_{N})$ and admits Fourier expansion with coefficients in $\mathbb{Q}_{N}$ [15]. Theorem 8.8 shows that the image of this basis under the Eichler-Shimura map is also definable over $\mathbb{Q}_{N}$ .

Observe that the $\mathbb{K}$ -subspace $H^{1}(\mathcal{G},V_{k-2,\mathbb{K}})$ of $H^{1}(\mathcal{G},V_{k-2,\mathbb{C}})$ is stable under the action of a double coset operator (Section 4.2). Thus one can use Theorem 8.8 to conclude that the action of a double coset operator is definable on our rational spaces of Eisenstein series; cf. Theorem 7.3 in [15].

Corollary 8.10.

Let the notation be as in Theorem 8.8. Then $\mathcal{E}_{k}(\mathcal{G},\mathbb{K})$ is stable under the action of double coset operators.

Proof.

Follows from the Hecke equivariance of the Eichler-Shimura isomorphism and Theorem 8.8. ∎

Next, we discuss the rationality of the twisted Eichler-Shimura homomorphism on the space of Eisenstein series with nebentypus $\mathcal{E}_{k}(N,\chi)$ . Let $N$ be an integer $\geq 1$ and $\chi\in\widehat{U}_{N}$ . Suppose that $\chi$ is defined over a subfield $\mathbb{K}$ of $\mathbb{C}$ . Then $V_{k-2,\mathbb{K}}^{\chi}$ is a $\mathbb{K}$ -form of $V_{k-2,\mathbb{C}}^{\chi}$ and the inclusion $V_{k-2,\mathbb{K}}^{\chi}\hookrightarrow V_{k-2,\mathbb{C}}^{\chi}$ induces a change of scalar isomorphism:

H^{1}\big{(}\Gamma_{0}(N),V_{k-2,\mathbb{K}}^{\chi}\big{)}\otimes_{\mathbb{K}}\mathbb{C}\cong H^{1}\big{(}\Gamma_{0}(N),V_{k-2,\mathbb{C}}^{\chi}\big{)}.

Note that $H^{1}\big{(}\Gamma_{1}(N),V_{k-2,\mathbb{K}}\big{)}$ is a $\mathbb{K}$ -form of $H^{1}\big{(}\Gamma_{1}(N),V_{k-2,\mathbb{C}}\big{)}$ . Since $\chi$ is defined over $\mathbb{K}$ the $\chi$ -eigenspace for the $\Gamma_{0}(N)$ on $H^{1}\big{(}\Gamma_{1}(N),V_{k-2,\mathbb{C}}\big{)}$ is also defined over $\mathbb{K}$ , i.e., $H^{1}\big{(}\Gamma_{1}(N),V_{k-2,\mathbb{K}}\big{)}[\chi]\otimes_{\mathbb{K}}\mathbb{C}\cong H^{1}\big{(}\Gamma_{1}(N),V_{k-2,\mathbb{C}}\big{)}[\chi]$ . Lemma 7.7 shows that the natural restriction map induces an isomorphism

H^{1}\big{(}\Gamma_{0}(N),V_{k-2,\mathbb{C}}^{\chi}\big{)}\cong H^{1}\big{(}\Gamma_{1}(N),V_{k-2,\mathbb{C}}\big{)}[\chi]

that maps $H^{1}\big{(}\Gamma_{0}(N),V_{k-2,\mathbb{K}}^{\chi}\big{)}$ onto $H^{1}\big{(}\Gamma_{1}(N),V_{k-2,\mathbb{K}}\big{)}[\chi]$ . Recall the basis $\mathcal{B}_{k,\chi}$ of $\mathcal{E}_{k}(N,\chi)$ defined in (7.10).

Theorem 8.11.

Let the notation be as above. The image of $\mathcal{B}_{k,\chi}$ under $[\emph{ES}_{k}]_{\emph{m},N}^{\chi}$ lies inside $H^{1}\big{(}\Gamma_{0}(N),V_{k-2,\mathbb{K}}^{\chi}\big{)}$ where $\mathbb{K}=\mathbb{Q}_{\varphi(N)}\mathbb{Q}_{N}=\mathbb{Q}_{\emph{lcm}(N,\varphi(N))}$ .

Proof.

Let $\mathbb{K}$ be as in the statement of the theorem. We know that $\mathcal{B}_{k,\chi}\subseteq\mathcal{E}_{k}\big{(}\Gamma_{1}(N),\mathbb{K}\big{)}$ . Therefore, by Theorem 8.8, the image of $\mathcal{B}_{k,\chi}$ under the Eichler-Shimura isomorphism for $\Gamma_{1}(N)$ (7.7) lies in $H^{1}\big{(}\Gamma_{1}(N),V_{k-2,\mathbb{K}}\big{)}[\chi]$ . Thus the assertion is a consequence of the diagram in (7.8) and the discussion before the theorem. ∎

One can extract a Hecke stability statement from the theorem above.

Corollary 8.12.

Let $\mathbb{K}$ be a subfield of $\mathbb{C}$ containing $\mathbb{Q}_{\varphi(N)}\mathbb{Q}_{N}$ . Then the $\mathbb{K}$ -span of $\mathcal{B}_{k,\chi}$ is stable under the action of twisted double coset operators defined in Section 7.2.

Proof.

Follows from the $\chi$ -Hecke equivariance of the twisted Eichler-Shimura map $[\text{ES}_{k}]_{\text{m},N}^{\chi}$ and Theorem 8.11. ∎

Acknowledgments.

The author wishes to thank Don Zagier for introducing him to the Eichler-Shimura theory and for lots of valuable advice. He is grateful to the Max Planck Institute for Mathematics, Bonn for hospitality and financial support during the visit (2018-2019). The author is also indebted to the School of Mathematics at Tata Institute of Fundamental Research for providing an exciting environment to study and work.

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Eichler-Shimura theory for general congruence subgroups

Abstract

1 Introduction

Theorem 1.1.

Proposition 1.2.

Theorem 1.3.

Theorem 1.4.

Theorem 1.5.

1.1 Notation and conventions

2 Construction of period cocycle

2.1 Preliminaries

Group action on ℍ\mathbb{H}.

Group action on function spaces.

Representations of GL2​(ℤ)\text{GL}_{2}(\mathbb{Z}).

Differential operators.

2.2 Period cocycle and Eichler-Shimura class

Definition 2.1.

Example 2.2.

Lemma 2.3.

Proof.

Definition 2.4.

Example 2.5.

Lemma 2.6.

Proof.

Proposition 2.7.

Proof.

Definition 2.8.

Global Eichler-Shimura class.

2.3 Variations of formalism

2.3.1 General subgroups of GL2+​(ℝ)\text{GL}_{2}^{+}(\mathbb{R})

2.3.2 Linear cocycles and projectivization

Example 2.9.

Lemma 2.10.

Proof.

3 Base point at the boundary

3.1 Growth conditions at the boundary

Definition 3.1.

Example 3.2.

Growth rate at infinity.

Lemma 3.3.

Proof.

Growth near the real line.

Exponential polynomials.

Lemma 3.4.

Proof.

Lemma 3.5.

Proof.

3.2 Eichler-Shimura integrals with base point at ∞\infty

Lemma 3.6.

Proof.

Corollary 3.7.

Proof.

Lemma 3.8.

Proof.

Proposition 3.9.

Proof.

Corollary 3.10.

Proof.

3.3 Connections with the theory of modular forms

Lemma 3.11.

Proof.

Corollary 3.12.

Lemma 3.13.

Proof.

4 Eichler-Shimura homomorphism

4.1 Definition and basic properties

Definition 4.1.

Proposition 4.2.

Proof.

4.2 The action of double coset operators

Proposition 4.3.

Proof.

4.3 Equivariance under group action

Lemma 4.4.

Proof.

Example 4.5.

Remark 4.6.

Proposition 4.7.

Proof.

Remark 4.8.

Group action on $\mathbb{H}$ .

Representations of $\text{GL}_{2}(\mathbb{Z})$ .

2.3.1 General subgroups of $\text{GL}_{2}^{+}(\mathbb{R})$

3.2 Eichler-Shimura integrals with base point at $\infty$

8.1.2 $L$ -function and $L$ -values of the elliptic Eisenstein series