Signed $p$ -adic $L$ -functions of Bianchi modular forms

Mihir V. Deo [email protected] Department of Mathematics and Statistics, University of Ottawa, Ottawa, ON, Canada K1N 6N5

Abstract.

Let $p\geq 3$ be a prime number and $K$ be a quadratic imaginary field in which $p$ splits as $\mathfrak{p}\overline{\mathfrak{p}}$ . Let $\mathcal{F}$ be a cuspidal Bianchi eigenform over $K$ of weight $(k,k)$ , where $k\geq 2$ is an integer, level $\mathfrak{n}$ coprime to $(p)$ , and non-ordinary at both of the primes above $p$ . We assume $\mathcal{F}$ has trivial nebentypus. For $\mathfrak{q}\in\{\mathfrak{p},\overline{\mathfrak{p}}\}$ , let $a_{\mathfrak{q}}$ be the $T_{\mathfrak{q}}$ Hecke eigenvalue of $\mathcal{F}$ and let $\alpha_{\mathfrak{q}},\beta_{\mathfrak{q}}$ be the roots of polynomial $X^{2}-a_{\mathfrak{q}}X+p^{k-1}$ . Then we have four $p$ -stabilizations of $\mathcal{F}$ : $\mathcal{F}^{\alpha_{\mathfrak{p}},\alpha_{\overline{\mathfrak{p}}}},\mathcal{F}^{\alpha_{\mathfrak{p}},\beta_{\overline{\mathfrak{p}}}},\mathcal{F}^{\beta_{\mathfrak{p}},\alpha_{\overline{\mathfrak{p}}}},$ and $\mathcal{F}^{\beta_{\mathfrak{p}},\beta_{\overline{\mathfrak{p}}}}$ which are Bianchi cusp forms of level $p\mathfrak{n}$ . Due to Williams, to each $p$ -stabilization $\mathcal{F}^{*,\dagger}$ , we can attach a locally analytic distribution $L_{p}(\mathcal{F}^{*,\dagger})$ over the ray class group $\text{Cl}(K,p^{\infty})$ . On viewing $L_{p}(\mathcal{F}^{*,\dagger})$ as a two-variable power series with coefficients in some $p$ -adic field having unbounded denominators satisfying certain growth conditions, we decompose this power series into a linear combination of power series with bounded coefficients in the spirit of Pollack, Sprung, and Lei-Loeffler-Zerbes.

Key words and phrases:

Iwasawa theory,

p

-adic

L

-functions,

p

-adic Hodge theory, Bianchi modular forms

1. Introduction

The study and construction of $p$ -adic $L$ -functions of arithmetic objects, like modular forms, is one of the central topics in modern number theory. The analytic $p$ -adic $L$ -functions are distributions on $p$ -adic rings like $\mathbb{Z}_{p}$ . For example, let $f$ be an elliptic modular eigenform of weight $k\geq 2$ , level $N$ and character $\epsilon$ , and let $p$ be a prime such that $p\nmid N$ . Let $\alpha$ be a root of the Hecke polynomial $X^{2}-a_{p}X+\epsilon(p)p^{k-1}$ such that $v_{p}(\alpha)<k-1$ , where $v_{p}$ is the normalized $p$ -adic valuation such that $v_{p}(p)=1$ , and $a_{p}$ is the $T_{p}$ -eigenvalue of $f$ . Then, due to the constructions of Amice-Velu and Vishik (see [1], [33]) we can attach to $f$ a $p$ -adic distribution $L_{p}(f,\alpha)$ of order $v_{p}(\alpha)$ over $\mathbb{Z}_{p}^{\times}$ . This $L_{p}(f,\alpha)$ interpolates the critical values of the complex $L$ -function of $f$ .

We continue with the example of $p$ -adic $L$ -functions of modular forms. When $f$ is $p$ -ordinary, i.e. $v_{p}(a_{p})=0$ , $L_{p}(f,\alpha)$ is a bounded measure and hence an element of Iwasawa algebra $\Lambda_{K}(\Gamma)\cong K\otimes\mathcal{O}_{K}[\Delta][[\Gamma_{1}]]$ , where $K$ is some finite extension of $\mathbb{Q}_{p}$ , $\Gamma=\operatorname{Gal}(\mathbb{Q}_{p}(\mu_{p^{\infty}})/\mathbb{Q}_{p})\cong\Delta\times\Gamma_{1}$ , $\Delta\cong(\mathbb{Z}/p\mathbb{Z})^{\times}$ , and $\Gamma_{1}\cong\mathbb{Z}_{p}$ . In this setting, the arithmetic is well understood and we have an Iwasawa main conjecture which relates this $p$ -adic $L$ -function with the characteristic ideal of the Selmer group of $f$ (proved in many cases by Kato in [15], Skinner-Urban in [31], Wan in [34], etc).

When $f$ is $p$ -non-ordinary, i.e. $a_{p}$ is not a $p$ -adic unit, $L_{p}(f,\alpha)$ is no longer a measure and hence not an element of the Iwasawa algebra. Moreover, it has unbounded denominators and it is an element of a larger algebra known as the distribution algebra $\mathcal{H}_{K}(\Gamma)$ (see section 2 for the definition). When $a_{p}=0$ , Pollack in [30] has given a remedy. If $\alpha_{1},\alpha_{2}$ are the roots of $X^{2}+\epsilon(p)p^{k-1}$ , Pollack showed that there exists a decomposition

L_{p}(f,\alpha_{i})=\log^{+}_{p,k}L_{p}^{+}+\alpha_{i}\log^{-}_{p,k}L_{p}^{-},

where $L_{p}^{\pm}\in\Lambda_{K}(\Gamma)$ , for some finite extension $K$ of $\mathbb{Q}_{p}$ , and $\log^{\pm}_{p,k}$ are some power series in $\mathcal{H}_{\mathbb{Q}_{p}}(\Gamma_{1})$ depending only on $k$ . He also showed that if $k=2$ , then $L_{p}^{\pm}$ have integral coefficients, i.e. they lie in $\mathbb{Z}_{p}[\Delta][[\Gamma]]$ . Later Sprung (for $k=2$ ) in [32] and Lei-Loeffler-Zerbes (for $k\geq 2$ ) in [23] have extended the work of Pollack when $a_{p}\neq 0$ using the method of logarithmic matrix.

Remark 1.1.

On the algebraic side, we also have notions of signed Selmer groups due to Kobayashi (for $k=2,a_{p}=0$ ) in [16], Sprung (for $k=2,a_{p}\neq 0$ ) in [32], Lei (for $k\geq 2,a_{p}=0$ ) in [20], and Lei-Loeffler-Zerbes (for $k\geq 2,a_{p}\neq 0$ ) in [23]. We also have signed Iwasawa main conjectures relating signed $p$ -adic $L$ -functions of non-ordinary modular forms with signed Selmer groups. See [23] for more details.

In this article, we extend the construction of signed $p$ -adic $L$ -functions (due to Pollack, Sprung, and Lei-Loefller-Zerbes) in the setting of Bianchi modular forms using the two-variable $p$ -adic $L$ -functions constructed by Williams in [35]. Bianchi modular forms are automorphic forms over quadratic imaginary fields. Let $K$ be a quadratic imaginary field. Fix a prime number $p\geq 3$ , which splits in $K$ as $(p)=\mathfrak{p}\overline{\mathfrak{p}}$ , and let $k\geq 2$ be an integer. Let $\mathcal{G}$ be a Bianchi cusp eigenform of weight $(k,k)$ , level $\mathcal{N}$ such that $(p)$ divides $\mathcal{N}$ and $v_{p}(a_{\mathfrak{q}})<(k-1)/e_{\mathfrak{q}}$ , where $a_{\mathfrak{q}}$ is the $U_{\mathfrak{q}}$ Hecke eigenvalue for all primes $\mathfrak{q}$ of $K$ which lie above $p$ and $e_{\mathfrak{q}}$ is the ramification index of the prime $\mathfrak{q}$ . Then Williams has constructed a two-variable $p$ -adic $L$ -function $L_{p}(\mathcal{G})$ (see [35, Theorem 7.4]) of a Bianchi modular cusp form $\mathcal{G}$ using overconvergent modular symbols. More precisely, $L_{p}(\mathcal{G})$ is a locally analytic distribution over the ray class group $\text{Cl}(K,p^{\infty})=G_{p^{\infty}}=\varprojlim_{n}G_{(p)^{n}}$ , where $G_{(p)^{n}}$ is the ray class group of $K$ modulo $(p)^{n}$ . We start with a Bianchi cusp form $\mathcal{F}$ of weight $(k,k)$ , level $\mathfrak{n}$ coprime to $p$ , and $\mathcal{F}$ is non-ordinary at both of the primes above $p$ , that is, $v_{p}(a_{\mathfrak{p}})>0$ and $v_{p}(a_{\overline{\mathfrak{p}}})>0$ , where $a_{\mathfrak{p}}$ and $a_{\overline{\mathfrak{p}}}$ are $T_{\mathfrak{p}}$ and $T_{\overline{\mathfrak{p}}}$ Hecke eigenvalues of $\mathcal{F}$ respectively. For $\mathfrak{q}\in\{\mathfrak{p},\overline{\mathfrak{p}}\}$ , we assume $v_{p}(a_{\mathfrak{q}})>\left\lfloor\dfrac{k-2}{p-1}\right\rfloor$ . Moreover, let $\alpha_{\mathfrak{q}}$ and $\beta_{\mathfrak{q}}$ be the roots of Hecke polynomial $X^{2}-a_{\mathfrak{q}}X+p^{k-1}$ which we assume are distinct. Then we get four $p$ -stabilizations of $\mathcal{F}$ : $\mathcal{F}^{\alpha_{\mathfrak{p}},\alpha_{\overline{\mathfrak{p}}}},\mathcal{F}^{\alpha_{\mathfrak{p}},\beta_{\overline{\mathfrak{p}}}},\mathcal{F}^{\beta_{\mathfrak{p}},\alpha_{\overline{\mathfrak{p}}}}$ and $\mathcal{F}^{\beta_{\mathfrak{p}},\beta_{\overline{\mathfrak{p}}}}$ , which are Bianchi modular forms of the same weight as $\mathcal{F}$ and level $p\mathfrak{n}$ . Thanks to Williams, we can attach a two-variable $p$ -adic $L$ -function to each of the stabilization $L_{*,\dagger}\coloneqq L_{p}(\mathcal{F}^{*,\dagger})$ , for $*\in\{\alpha_{\mathfrak{p}},\beta_{\mathfrak{p}}\}$ and $\dagger\in\{\alpha_{\overline{\mathfrak{p}}},\beta_{\overline{\mathfrak{p}}}\}$ . The main theorem of this article is:

Theorem A (Theorem 8.3).

There exist two variable power series with bounded coefficients, that is, there exist $L_{\sharp,\sharp},L_{\sharp,\flat},L_{\flat,\sharp},L_{\flat,\flat}\in\Lambda_{E}(G_{p^{\infty}})$ such that

\begin{pmatrix}L_{\alpha_{\mathfrak{p}},\alpha_{\overline{\mathfrak{p}}}}&L_{\beta_{\mathfrak{p}},\alpha_{\overline{\mathfrak{p}}}}\\[6.0pt] L_{\alpha_{\mathfrak{p}},\beta_{\overline{\mathfrak{p}}}}&L_{\beta_{\mathfrak{p}},\beta_{\overline{\mathfrak{p}}}}\end{pmatrix}=Q^{-1}_{\overline{\mathfrak{p}}}\underline{M_{\overline{\mathfrak{p}}}}\begin{pmatrix}L_{\sharp,\sharp}&L_{\flat,\sharp}\\[6.0pt] L_{\sharp,\flat}&L_{\flat,\flat}\end{pmatrix}(Q^{-1}_{\mathfrak{p}}\underline{M_{\mathfrak{p}}})^{T},

where $\underline{M_{\mathfrak{p}}}$ and $\underline{M_{\overline{\mathfrak{p}}}}$ are $2\times 2$ logarithmic matrices, $Q_{\mathfrak{q}}=\begin{pmatrix}\alpha_{\mathfrak{q}}&-\beta_{\mathfrak{q}}\\[6.0pt] -p^{k-1}&p^{k-1}\end{pmatrix}$ , and $\Lambda_{E}(G_{p^{\infty}})\cong E[\Delta_{K}]\otimes\mathbb{Z}_{p}[[T_{1},T_{2}]]$ , where $\Delta_{K}$ is a finite abelian group such that $G_{p^{\infty}}\cong\Delta_{K}\times\mathbb{Z}_{p}^{2}$ .

1.1. Plan of the article

We start with the setup and notations in Section 2 which we require throughout the article.

In Section 3, we recall tools from $p$ -adic Hodge theory and Wach modules. In general, we look at crystalline representations, families of Wach modules and the relation between them due to Berger-Li-Zhu [4]. Note that we don’t have much information about $p$ -adic Hodge theoretic properties of the Galois representation attached to Bianchi modular forms, for example conjecturally $p$ -adic Galois representations attached to Bianchi modular forms are crystalline. There are some partial results due to Jorza, see[14] and [13]. Using Berger-Li-Zhu’s construction, we can avoid (rather bypass) the $p$ -adic representation coming from Bianchi modular forms.

In Section 4, we recall the big logarithm map constructed by Perrin-Riou and the $p$ -adic regulator map along with Coleman maps which were introduced by Lei-Loeffler-Zerbes in [23] and [21]. Moreover, we study the relationship between the big logarithm map and the $p$ -adic regulator map. In this section, we also introduce the logarithm matrix $\underline{M}$ , which is an element of $M_{2,2}(\mathcal{H}_{E}(\Gamma_{1}))$ .

Section 5 deals with the factorization of power series in one variable. We first investigate $\underline{M}$ in more depth. Then we prove the following result:

Theorem B (Theorem 5.5).

Let $E$ be a finite extension of $\mathbb{Q}_{p}$ . Let $\alpha,\beta\in\mathcal{O}_{E}$ and $k\geq 2$ be an integer such that $\alpha\beta=p^{k-1}$ . Assume $\alpha\neq\beta$ and $v_{p}(\alpha+\beta)>\left\lfloor\dfrac{k-2}{p-1}\right\rfloor$ . For each $\lambda\in\{\alpha,\beta\}$ , let $F_{\lambda}\in\mathcal{H}_{E,v_{p}(\lambda)}(\Gamma)$ , such that for any integer $0\leq j\leq k-2$ , and for any Dirichlet character $\omega$ of conductor $p^{n}$ , we have $F_{\lambda}(\chi^{j}\omega)=\lambda^{-n}C_{\omega,j}$ , where $\chi$ is the $p$ -adic cyclotomic character and $C_{\omega,j}\in\overline{\mathbb{Q}_{p}}$ that is independent of $\lambda$ . Then there exist $F_{\flat},F_{\sharp}\in\Lambda_{E}(\Gamma)$ such that

\begin{pmatrix}F_{\alpha}\\[6.0pt] F_{\beta}\end{pmatrix}=Q^{-1}\underline{M}\begin{pmatrix}F_{\sharp}\\[6.0pt] F_{\flat}\end{pmatrix}.

Remark 1.2.

In [7, Section 2], the authors proved a similar result as above under the Fontaine-Laiffaille condition ( $p>k$ ). In this article, we are replacing this condition with a weaker condition $v_{p}(\alpha+\beta)>\left\lfloor\dfrac{k-2}{p-1}\right\rfloor$ . We also use different methods than the methods used in [7]. For example, we obtain properties of $\underline{M}$ using the $p$ -adic regulator $\mathcal{L}_{T_{W}}$ and Perrin-Riou’s big logarithm map $\Omega_{T_{W}}$ .

Remark 1.3.

Theorem B can be used in the decomposition of any two power series satisfying certain growth conditions and interpolation properties. In [10], which is an upcoming work regarding $p$ -adic Asai $L$ -distributions of $p$ -non-ordinary Bianchi modular forms, we can use this method to decompose the distributions into bounded measures.

In Section 6, we develop the two-variable setup and recall definitions of ray class groups, Hecke characters, etc.

Bianchi modular forms and their $p$ -adic $L$ -functions are briefly recalled in Section 7.

In the last section, we prove the main theorem (Theorem A) of this article. We generalize and apply results of [19] in our setting.

Acknowledgements

I would like to thank my PhD advisor Antonio Lei for suggesting this problem to me, for answering all my questions, and for his patience, guidance, encouragement, and support. I would like to thank my brother Shaunak Deo for helpful mathematical and non-mathematical conversations. I would also like to thank Katharina Müller for her suggestions on the earlier draft of this article.

2. Setup and notations

Fix an odd prime $p$ . Let $E$ be a finite extension of $\mathbb{Q}_{p}$ with the ring of integers $\mathcal{O}_{E}$ . Let $\alpha,\beta\in\mathcal{O}_{E}$ such that $v_{p}(a)>0$ , where $a=\alpha+\beta$ . Let $v\in\mathcal{O}_{E}$ and $k\geq 2$ be an integer such that $\alpha\beta=vp^{k-1}$ for some $v$ such that $v^{1/2}\in\mathcal{O}_{E}^{\times}$ . We denote $\operatorname{Gal}(\overline{\mathbb{Q}_{p}}/\mathbb{Q}_{p})$ by $G_{\mathbb{Q}_{p}}$ .

Assumption 2.1.

$v_{p}(a)>m=\left\lfloor\dfrac{k-2}{p-1}\right\rfloor$ and $\alpha\neq\beta$ .

We fix $a,\alpha,\beta,$ and $v$ for the rest of the article.

Iwasawa algebras

Let $\mathbb{Q}_{p,n}=\mathbb{Q}_{p}(\mu_{p^{n}})$ , where $\mu_{p^{n}}$ is the set of all $p^{n}$ -th roots of unity. Let $\mathbb{Q}_{p,\infty}=\bigcup_{n\geq 1}\mathbb{Q}_{p,n}$ . Then $\Gamma=\operatorname{Gal}(\mathbb{Q}_{p,\infty}/\mathbb{Q}_{p})\cong\Delta\times\mathbb{Z}_{p}$ , where $\Delta$ is the torsion group of $\Gamma$ of order $p-1$ . Let $\Gamma_{1}$ be a subgroup of $\Gamma$ such that $\Gamma_{1}\cong\Gamma/\Delta\cong\mathbb{Z}_{p}$ . In other words, $\Gamma_{1}$ is the Galois group of $\mathbb{Q}_{p,\infty}$ over $\mathbb{Q}_{p,1}$ . We denote the Iwasawa algebra $\mathcal{O}_{E}\otimes\mathbb{Z}_{p}[[\Gamma]]\cong\mathcal{O}_{E}[[\Gamma]]$ over $\mathcal{O}_{E}$ by $\Lambda_{\mathcal{O}_{E}}(\Gamma)$ . Fix a topological generator $\gamma_{0}$ of $\Gamma_{1}$ . Then we can identify $\mathcal{O}_{E}[[\Gamma_{1}]]$ with $\mathcal{O}_{E}[[X]]$ via identification $\gamma_{0}\mapsto 1+X$ . This can be extended to $\Lambda_{\mathcal{O}_{E}}(\Gamma)\cong\mathcal{O}_{E}[\Delta][[X]]$ . We further write $\Lambda_{E}(\Gamma_{1})=E\otimes\Lambda_{\mathcal{O}_{E}}(\Gamma_{1})$ and $\Lambda_{E}(\Gamma)=E\otimes\Lambda_{\mathcal{O}_{E}}(\Gamma)$ . Fix a topological generator $u$ of $1+p\mathbb{Z}_{p}$ and let $\chi$ be the $p$ -adic cyclotomic character on $\Gamma$ such that $\chi(\gamma_{0})=u$ .

Power series rings

Given any power series $F\in E[[X]]$ and $0<\rho<1$ , we define the sup norm $\|F\|_{\rho}=\text{sup}_{|z|_{p}\leq\rho}|F(z)|_{p}$ . For any real number $r\geq 0$ , we define

\mathcal{H}_{r}=\{F\in E[[X]]:\operatorname{sup_{t}}(p^{-tr}\|F\|_{\rho_{t}})<\infty\},

where $\rho_{t}=p^{-1/p^{t-1}(p-1)}$ and $t\geq 1$ is an integer. We write $\mathcal{H}_{E}=\bigcup_{r\geq 0}\mathcal{H}_{r}$ . We define $\mathcal{H}_{E,r}(\Gamma)$ to be the set of power series $\sum_{n\geq 0,\sigma\in\Delta}c_{n,\sigma}\cdot\sigma\cdot(\gamma_{0}-1)^{n},$ such that $\sum_{n\geq 0}c_{n,\sigma}X^{n}\in\mathcal{H}_{r}$ for all $\sigma\in\Delta$ . In other words, the elements of $\mathcal{H}_{E,r}(\Gamma)$ are the power series in $\gamma_{0}-1$ over $E[\Delta]$ with growth rate $O(\log_{p}^{r})$ . Write $\mathcal{H}_{E}(\Gamma)=\bigcup_{r\geq 0}\mathcal{H}_{E,r}(\Gamma)$ . We call $\mathcal{H}_{E}(\Gamma)$ the space of distributions on $\Gamma$ . We can identify $\mathcal{H}_{E}(\Gamma)$ with

\{F\in E[\Delta][[X]]:F\text{ converges everywhere on the open unit }p\text{-adic disk}\}

where $X$ corresponds to $\gamma_{0}-1$ .

Fontaine’s rings

Let $\pi$ be a variable, $\mathbb{A}_{\mathbb{Q}_{p}}^{+}=\mathbb{Z}_{p}[[\pi]]$ and $\mathbb{B}^{+}_{\mathbb{Q}_{p}}=\mathbb{A}_{\mathbb{Q}_{p}}^{+}[1/p]$ . Let $\mathbb{A}_{\mathbb{Q}_{p}}$ be the ring of power series $\sum_{i=-\infty}^{+\infty}a_{i}\pi^{i}$ such that $a_{i}\in\mathbb{Z}_{p}$ and $a_{i}\to 0$ as $i\to-\infty$ . Write $\mathbb{B}^{+}_{\mathrm{rig},\mathbb{Q}_{p}}$ for the ring of power series $f(\pi)\in\mathbb{Q}_{p}[[\pi]]$ such that $f(X)$ converges everywhere in the open unit $p$ -adic disk. We equip $\mathbb{B}^{+}_{\mathrm{rig},\mathbb{Q}_{p}}$ with actions of a Frobenius operator $\varphi$ and $\Gamma$ by $\varphi:\pi\mapsto(1+\pi)^{p}-1$ and $\sigma:\pi\mapsto(1+\pi)^{\chi(\sigma)}-1$ for all $\sigma\in\Gamma$ . We then write $\mathbb{B}^{+}_{\mathrm{rig},E}$ for the power series ring $E\otimes\mathbb{B}^{+}_{\mathrm{rig},\mathbb{Q}_{p}}$ . We can define a left inverse $\psi$ of $\varphi$ such that

\varphi\circ\psi(f(\pi))=\dfrac{1}{p}\sum_{\zeta^{p}=1}f(\zeta(1+\pi)-1).

Inside $\mathbb{B}^{+}_{\mathrm{rig},E}$ , we have subrings $\mathbb{A}^{+}_{E}=\mathcal{O}_{E}[[\pi]]$ and $\mathbb{B}^{+}_{E}=E\otimes\mathbb{A}^{+}_{E}$ . The actions of $\varphi,\psi$ , and $\Gamma$ preserve these subrings. Write $t=\log(1+\pi)\in\mathbb{B}^{+}_{\mathrm{rig},E}$ and $q=\varphi(\pi)/\pi\in\mathbb{A}^{+}_{E}$ . Note that $\varphi(t)=pt$ and $\sigma(t)=\chi(\sigma)t$ for all $\sigma\in\Gamma$ .

Mellin transform

We have a $\Lambda_{E}(\Gamma)$ -module isomorphism between $\mathcal{H}_{E}(\Gamma)$ and $(\mathbb{B}^{+}_{\mathrm{rig},E})^{\psi=0}$ due to the action of $\Gamma$ on $\mathbb{B}^{+}_{\mathrm{rig},E}$ , called the Mellin transform. The isomorphism is given by

	$\displaystyle\mathfrak{M}:\mathcal{H}_{E}(\Gamma)$	$\displaystyle\to(\mathbb{B}^{+}_{\mathrm{rig},E})^{\psi=0}$
	$\displaystyle f(\gamma_{0}-1)$	$\displaystyle\mapsto f(\gamma_{0}-1)\cdot(1+\pi).$

Moreover, $\Lambda_{\mathcal{O}_{E}}(\Gamma)$ corresponds to $(\mathbb{A}^{+}_{E})^{\psi=0}$ and $\Lambda_{\mathcal{O}_{E}}(\Gamma_{1})$ corresponds to $(1+\pi)\varphi(\mathbb{A}^{+}_{E})$ under $\mathfrak{M}$ . Let $\mathcal{H}_{E}(\Gamma_{1})=\{f(\gamma_{0}-1):f\in\mathcal{H}_{E}\}$ , then $\mathcal{H}_{E}(\Gamma_{1})$ corresponds to $(1+\pi)\varphi(\mathbb{B}^{+}_{\mathrm{rig},E})$ . See [28, Section B.2.8] for more details.

3. Crystalline representations and Wach modules

In this section, we recall definitions of crystalline representations and Wach modules. Furthermore, we recall the construction of families of Wach modules from [4]. The primary reference for this section is [4, Sections 1, 2, and 3].

3.1. Crystalline representations

Let $\mathbb{B}_{\mathrm{crys}}$ be the Fontaine’s period ring. Recall that we call a $\mathbb{Q}_{p}$ -linear $G_{\mathbb{Q}_{p}}$ -representation $V$ a crystalline representation if $V$ is $\mathbb{B}_{\mathrm{crys}}$ -admissible. In other words, $V$ is a crystalline representation if the dimension of the filtered $\varphi$ -module $\mathbb{D}_{\mathrm{crys}}(V)=(\mathbb{B}_{\mathrm{crys}}\otimes V)^{G_{\mathbb{Q}_{p}}}$ is $\text{dim}_{\mathbb{Q}_{p}}V$ . For any integer $j$ , we take $\mathbb{Q}_{p}(j)=\mathbb{Q}_{p}\cdot e_{j}$ , where $G_{\mathbb{Q}_{p}}$ acts on $e_{j}$ via $\chi^{j}$ . We know that $\mathbb{Q}_{p}(j)$ is a crystalline representation. Then for any crystalline representation $V$ , the representation $V(j)=V(\chi^{j})=V\otimes\mathbb{Q}_{p}(j)$ is again a crystalline representation. Moreover,we have $\mathbb{D}_{\mathrm{crys}}(V(j))=t^{-j}\mathbb{D}_{\mathrm{crys}}(V)\otimes e_{j}$ . We say a crystalline (or more generally a Hodge-Tate) representation $V$ is positive if its Hodge-Tate weights are negative.

Let $E$ be a finite extension of $\mathbb{Q}_{p}$ . We say that an $E$ -linear $G_{\mathbb{Q}_{p}}$ -representation $V$ is crystalline if and only if the underlying $\mathbb{Q}_{p}$ -linear representation is crystalline. In this case, $\mathbb{D}_{\mathrm{crys}}(V)$ is an $E$ -vector space with $E$ -linear Frobenius and a filtration of $E$ -vector spaces. More precisely, $\mathbb{D}_{\mathrm{crys}}(V)$ is an admissible $E$ -linear filtered $\varphi$ -module and the functor $V\mapsto\mathbb{D}_{\mathrm{crys}}(V)$ is an equivalence of categories from the category of crystalline $E$ -linear representations to the category of admissible $E$ -linear filtered $\varphi$ -module (see [9] for more details).

3.1.1. Crystalline representations as filtered $\varphi$ -modules

Let $D_{k,v^{1/2}a}$ be a filtered $\varphi$ -module given by $D_{k,v^{1/2}a}=Ee_{1}\oplus Ee_{2}$ where:

\displaystyle\begin{matrix}\begin{cases}\varphi(e_{1})=p^{k-1}e_{2}\\ \varphi(e_{2})=-e_{1}+(v^{1/2}a)e_{2}\end{cases}&\text{and }&\mathrm{Fil}^{i}D_{k,v^{1/2}a}=\begin{cases}D_{k,v^{1/2}a}\text{ if }i\leq 0\\ Ee_{1}\text{ if }1\leq i\leq k-1\\ 0\text{ if }i\geq k.\end{cases}\end{matrix}

Take $e^{\prime}_{1}=v^{1/2}e_{1}$ and $e^{\prime}_{2}=e_{2}$ . Thus, $e^{\prime}_{1},e^{\prime}_{2}$ is another $E$ -basis of $D_{k,v^{1/2}a}$ . The matrix of $\varphi$ with respect to basis $e^{\prime}_{1},e^{\prime}_{2}$ is

\tilde{A}_{\varphi}=\begin{pmatrix}0&-v^{-1/2}\\[6.0pt] v^{1/2}p^{k-1}&v^{-1/2}a\end{pmatrix}.

Theorem 3.1 (Colmez-Fontaine[9], Berger-Li-Zhu[4]).

There exists a crystalline $E$ -linear representation $V_{k,v^{1/2}a}$ , such that $\mathbb{D}_{\mathrm{crys}}(V^{*}_{k,v^{1/2}a})=D_{k,v^{1/2}a}$ , where $V^{*}_{k,v^{1/2}a}=\mathrm{Hom}(V_{k,v^{1/2}a},E)$ .

Proof.

See [4, Section I and Proposition 3.2.4]∎

From the above theorem, we get

\mathbb{D}_{\mathrm{crys}}(V^{*}_{k,v^{1/2}a})=Ee_{1}\oplus Ee_{2}=Ee^{\prime}_{1}\oplus Ee^{\prime}_{2}.

The Hodge-Tate weights of $V_{k,v^{1/2}a}$ are $0$ and $k-1$ , and thus the Hodge-Tate weights of $V^{*}_{k,v^{1/2}a}$ are $0$ and $1-k$ . Let $W=V^{*}_{k,v^{1/2}a}(\chi^{k-1}\otimes\eta)$ , where $\eta:G_{\mathbb{Q}_{p}}\to\overline{\mathbb{Q}_{p}}^{\times}$ is an unramified character such that $\eta(\text{Frob}_{p})=v^{1/2}$ . In other words, $W$ is the representation we get after twisting $V^{*}_{k,v^{1/2}a}$ by the character $\chi^{k-1}\otimes\eta$ . Therefore $W$ is a crystalline representation with Hodge-Tate weights $0$ and $k-1$ .

Let $w_{i}=e^{\prime}_{i}\otimes t^{-(k-1)}e_{k-1}\otimes e_{\eta}$ , for $i=1,2$ , where $e_{\eta}$ is a basis of $\mathbb{Q}_{p}(\eta)$ and the action of $\varphi$ on $e_{\eta}$ is given by $\varphi(e_{\eta})=\eta(\text{Frob}^{-1}_{p})e_{\eta}$ . Then $w_{1},w_{2}$ is a basis of $\mathbb{D}_{\mathrm{crys}}(W)$ . The action of $\varphi$ on $w_{i}$ can be calculated as

\displaystyle\begin{cases}\varphi(w_{1})=w_{2}\\ \varphi(w_{2})=(-1/vp^{k-1})w_{1}+(a/vp^{k-1})w_{2}.\end{cases}

Thus, the matrix of $\varphi$ with respect to basis $w_{1},w_{2}$ is

A_{\varphi}=\begin{pmatrix}0&\dfrac{-1}{vp^{k-1}}\\[12.0pt] 1&\dfrac{a}{vp^{k-1}}\end{pmatrix}.

3.2. Wach modules

An étale $(\varphi,\Gamma)$ -module over $\mathbb{A}_{\mathbb{Q}_{p}}$ is a finitely generated $\mathbb{A}_{\mathbb{Q}_{p}}$ - module $M$ , with semilinear action of $\varphi$ and a continuous action of $\Gamma$ commuting with each other, such that $\varphi(M)$ generates $M$ as an $\mathbb{A}_{\mathbb{Q}_{p}}$ -module. In [11, A.3.4], Fontaine has constructed a functor $T\mapsto\mathbb{D}(T)$ which associates to every $\mathbb{Z}_{p}$ -linear representation an étale $(\varphi,\Gamma)$ -module over $\mathbb{A}_{\mathbb{Q}_{p}}$ . The $(\varphi,\Gamma)$ -module $\mathbb{D}(T)$ is defined as $(\mathbb{A}\otimes T)^{H_{\mathbb{Q}_{p}}}$ , where $\mathbb{A}$ is the ring defined in [11] and $H_{\mathbb{Q}_{p}}=\operatorname{Gal}(\overline{\mathbb{Q}_{p}}/\mathbb{Q}_{p,\infty})$ . He also shows that the functor $T\mapsto\mathbb{D}(T)$ is an equivalence of categories. By inverting $p$ , we also get an equivalence of categories between the category of $\mathbb{Q}_{p}$ -linear $G_{\mathbb{Q}_{p}}$ -representations and the category of étale $(\varphi,\Gamma)$ -module over $\mathbb{B}_{\mathbb{Q}_{p}}=\mathbb{A}_{\mathbb{Q}_{p}}[p^{-1}]$ .

If $E$ is a finite extension of $\mathbb{Q}_{p}$ , we extend the Frobenius and the action of $\Gamma$ to $E\otimes\mathbb{B}_{\mathbb{Q}_{p}}$ by $E$ -linearity. We then get an equivalence of categories between $\mathcal{O}_{E}$ -modules (or $E$ -linear $G_{\mathbb{Q}_{p}}$ -representations) and the category of $(\varphi,\Gamma)$ -modules over $\mathcal{O}_{E}\otimes\mathbb{A}_{\mathbb{Q}_{p}}$ (or over $E\otimes\mathbb{B}_{\mathbb{Q}_{p}}$ ), given by $T\mapsto\mathbb{D}(T)$ .

In [3], Berger shows that if $V$ is an $E$ -linear $G_{\mathbb{Q}_{p}}$ -representation, then $V$ is crystalline with Hodge-Tate weights in $[a,b]$ if and only if there exists a unique $E\otimes\mathbb{B}^{+}_{\mathbb{Q}_{p}}$ -module $\mathbb{N}(V)\subset\mathbb{D}(V)$ such that:

(1)

$\mathbb{N}(V)$ is free of rank $d=\dim_{E}(V)$ over $E\otimes\mathbb{B}^{+}_{\mathbb{Q}_{p}}$ ;
(2)

The action of $\Gamma$ preserves $\mathbb{N}(V)$ and is trivial on $\mathbb{N}(V)/\pi\mathbb{N}(V)$ ;
(3)

$\varphi(\pi^{b}\mathbb{N}(V))\subset\pi^{b}\mathbb{N}(V)$ and $\pi^{b}\mathbb{N}(V)/\varphi^{*}(\pi^{b}\mathbb{N}(V))$ is killed by $q^{b-a},$ where $q=\frac{\varphi(\pi)}{\pi}$ .

Moreover, if $V$ is crystalline and positive, then we can take $b=0$ . In this case, $\mathbb{N}(V)/\pi\mathbb{N}(V)$ is a filtered $E$ -module and there exists an isomorphism $\mathbb{N}(V)/\pi\mathbb{N}(V)\cong\mathbb{D}_{\mathrm{crys}}(V)$ . See [3, Section III.4] for more details.

Let $T$ be a $G_{\mathbb{Q}_{p}}$ -stable lattice in $V$ . Then $\mathbb{N}(T)=\mathbb{D}(T)\cap\mathbb{N}(V)$ is an $\mathcal{O}_{E}\otimes\mathbb{A}^{+}_{\mathbb{Q}_{p}}$ -lattice in $\mathbb{N}(V)$ . By [3], the functor $T\mapsto\mathbb{N}(T)$ gives a bijection between the $G_{\mathbb{Q}_{p}}$ -stable lattices $T$ in $V$ and the $\mathcal{O}_{E}\otimes\mathbb{A}^{+}_{\mathbb{Q}_{p}}$ -lattices in $\mathbb{N}(V)$ which satisfy

(1)

$\mathbb{N}(T)$ is free of rank $d=\dim_{E}(V)$ over $\mathcal{O}_{E}\otimes\mathbb{A}^{+}_{\mathbb{Q}_{p}}$ ;
(2)

The action of $\Gamma$ preserves $\mathbb{N}(T)$ ;
(3)

$\varphi(\pi^{b}\mathbb{N}(T))\subset\pi^{b}\mathbb{N}(T)$ and $\pi^{b}\mathbb{N}(T)/\varphi^{*}(\pi^{b}\mathbb{N}(T))$ is killed by $q^{b-a}$ .

The $E\otimes\mathbb{B}^{+}_{\mathbb{Q}_{p}}$ -module $\mathbb{N}(V)\subset\mathbb{D}(V)$ as well as $\mathcal{O}_{E}\otimes\mathbb{A}^{+}_{\mathbb{Q}_{p}}$ -module $\mathbb{N}(T)\subset\mathbb{D}(T)$ are called Wach modules.

3.2.1. Famillies of Wach modules

In this section, we recall some results from [4].

Recall $q=\varphi(\pi)/\pi$ . We define $\lambda_{+}$ and $\lambda_{-}$ as

\lambda_{+}=\prod_{n\geq 0}\dfrac{\varphi^{2n+1}(q)}{p}

and

\lambda_{-}=\prod_{n\geq 0}\dfrac{\varphi^{2n}(q)}{p}.

Lemma 3.2.

Write $p^{m}(\lambda_{-}/\lambda_{+})^{k-1}=\sum_{i\geq 0}z_{i}\pi^{i}$ , where $m=\left\lfloor\dfrac{k-2}{p-1}\right\rfloor$ and define $z=z_{0}+z_{1}\pi+\cdots z_{k-2}\pi^{k-2}$ . Then $z\in\mathbb{Z}_{p}[[\pi]]$ .

Proof.

See [4, Proposition 3.1.1]. ∎

Let $Y$ be a variable. Define a matrix

P(Y)=\begin{pmatrix}0&-v^{-1/2}\\[6.0pt] v^{1/2}q^{k-1}&Yv^{-1/2}z\end{pmatrix}.

Then by [4, Proposition 3.1.3], for $\gamma\in\Gamma$ , there exists a matrix $G_{\gamma}(Y)\in I_{2}+\pi M_{2,2}(\mathbb{Z}_{p}[[\pi,Y]])$ such that

P(Y)\varphi(G_{\gamma}(Y))=G_{\gamma}(Y)\gamma(P(Y)).

Note that $\varphi$ and $\gamma\in\Gamma$ acts trivially on $Y$ .

Lemma 3.3.

For $\delta=\dfrac{a}{p^{m}}$ and $\gamma,\gamma^{\prime}\in\Gamma$ , we have $G_{\gamma\gamma^{\prime}}(\delta)=G_{\gamma}(\delta)\gamma^{\prime}(G_{\gamma^{\prime}}(\delta))$ and $P(\delta)\varphi(G_{\gamma}(\delta))=G_{\gamma}(\delta)\gamma(P(\delta))$ . Therefore, one can use the matrices $P(\delta)$ and $G_{\gamma}(\delta)$ to define a Wach module $\mathbb{N}_{k}(\delta)$ over $\mathcal{O}_{E}[[\pi]]$ .

Proof.

Define the free $\mathcal{O}_{E}[[\pi]]$ -module of rank $2$ with basis $n_{1},n_{2}$ as: $\mathbb{N}_{k}(\delta)=\mathcal{O}_{E}[[\pi]]n_{1}\oplus\mathcal{O}_{E}[[\pi]]n_{2}$ . Endow it with Frobenius $\varphi$ and an action of $\gamma\in\Gamma$ such that the matrix of $\varphi$ with respect to the basis $n_{1},n_{2}$ is $P(\delta)=\begin{pmatrix}0&-v^{-1/2}\\[6.0pt] v^{1/2}q^{k-1}&v^{-1/2}\delta\cdot z\end{pmatrix}$ and the matrix of $\gamma$ is $G_{\gamma}(\delta)$ . See [4, Proposition 3.2.1] for details. ∎

The above lemma implies $E\otimes\mathbb{N}_{k}(\delta)=\mathbb{N}(V^{*}_{k,v^{1/2}a})$ , where $\delta=a/p^{m}$ , and $V_{k,v^{1/2}a}$ is crystalline $E$ -linear representation which is described in subsection 3.1.1 above. Here $\mathbb{N}(V^{*}_{k,v^{1/2}a})$ is the Wach module associated to the crystalline representation $V^{*}_{k,v^{1/2}a}$ . More precisely:

Theorem 3.4.

The filtered $\varphi$ -module $E\otimes_{\mathcal{O}_{E}}(\mathbb{N}_{k}(\delta)/\pi\mathbb{N}_{k}(\delta))$ is isomorphic to the $\varphi$ -module $D_{k,v^{1/2}a}$ which is described in the subsection 3.1.

Proof.

This can be proved using [4, Proposition 3.2.4]. ∎

We adapt the above machinery in our setting. Recall that $W=V^{*}_{k,v^{1/2}a}(\chi^{k-1}\otimes\eta)$ . Let $T_{W}$ be an $\mathcal{O}_{E}$ -lattice in $W$ . Then $T_{W}=T(\chi^{k-1}\otimes\eta)$ , where $T$ is an $\mathcal{O}_{E}$ -lattice in $V_{k,v^{1/2}a}$ such that $\mathbb{N}_{k}(\delta)=\mathbb{N}(T^{*})$ . By an abuse of notation, we write $\mathbb{N}_{k}(\delta)=\mathbb{N}(T_{W})=\mathcal{O}_{E}[[\pi]]n^{\prime}_{1}\oplus\mathcal{O}_{E}[[\pi]]n^{\prime}_{2}$ , where $n^{\prime}_{1},n^{\prime}_{2}$ is a basis after twisting the basis $n_{1},n_{2}$ of $\mathbb{N}(T)$ with $\chi^{k-1}\otimes\eta$ . Then the matrix of $\varphi$ with respect to $\{n^{\prime}_{1},n^{\prime}_{2}\}$ is

P=\begin{pmatrix}0&-1/vq^{k-1}\\[6.0pt] 1&\delta\cdot z/vq^{k-1}\end{pmatrix}.

Note that $P\equiv A_{\varphi}\operatorname{mod}\pi,$ since $q\equiv p\operatorname{mod}\pi$ and $\delta\cdot z\equiv a\operatorname{mod}\pi$ . We fix this $\mathcal{O}_{E}[[\pi]]$ -basis $n^{\prime}_{1},n^{\prime}_{2}$ for $\mathbb{N}(T_{W})$ for the rest of the article.

4. Perrin-Riou’s big logarithm map, Coleman maps, and the $p$ -adic regulator

We recall definitions of Perrin-Rious’s big logarithm map, $p$ -adic regulator, and Coleman maps. We also explicitly study these maps and the relationship between them after fixing some basis. The preliminary reference for this section is [21].

4.1. Iwasawa cohomology and Wach modules

Let $V$ be any crystalline $E$ -linear representation of $G_{\mathbb{Q}_{p}}$ and let $T$ be an $\mathcal{O}_{E}$ -lattice inside $V$ . The Iwasawa cohomology group $H^{1}_{\mathrm{Iw}}(\mathbb{Q}_{p},T_{W})$ is defined by

H^{1}_{\mathrm{Iw}}(\mathbb{Q}_{p},T)=\varprojlim H^{1}(\mathbb{Q}_{p,n},T),

where the inverse limit is taken with respect to the corestriction maps. Then, due to Fontaine (see [8, Section II.1]), there exists a canonical $\Lambda_{\mathcal{O}_{E}}(\Gamma)$ -module isomorphism

h^{1}_{\mathrm{Iw}}\colon\mathbb{D}(T)^{\psi=1}\to H^{1}_{\mathrm{Iw}}(\mathbb{Q}_{p},T),

where $\mathbb{D}(T)$ is a $(\varphi,\Gamma)$ -module associated to $T$ .

From now on, we fix $p$ -adic representation $W$ from the previous section unless mentioned otherwise. Moreover, Let $\mathbb{D}_{\mathrm{crys}}(T_{W})$ be the image of $\mathbb{N}(T_{W})/\pi\mathbb{N}(T_{W})$ in $\mathbb{D}_{\mathrm{crys}}(W).$ Then

(1)

$\mathbb{D}_{\mathrm{crys}}(T_{W})$ is filtered $\varphi$ -module over $\mathcal{O}_{E}$ ,
(2)

$\mathbb{D}_{\mathrm{crys}}(T_{W})=\mathcal{O}_{E}\cdot w_{1}\oplus\mathcal{O}_{E}\cdot w_{2}$ ,
(3)

the matrix of $\varphi$ with respect to the basis $w_{1},w_{2}$ is $A_{\varphi}$ .

For the representation $W$ , the eigenvalues of the $\varphi$ are $\alpha^{-1},\beta^{-1}$ . From now on we assume that $\alpha^{-1}$ and $\beta^{-1}$ are not integral powers of the prime $p$ . Since the Hodge-Tate weights of $W$ are $0$ and $k-1$ , we have the following theorem due to Berger:

Theorem 4.1 (Berger, [2, Theorem A.3]).

For the $G_{\mathbb{Q}_{p}}$ -stable $\mathcal{O}_{E}$ -lattice $T_{W}$ in $W$ , there exists a $\Lambda_{\mathcal{O}_{E}}(\Gamma)$ -module isomorphism

h^{1}_{\mathrm{Iw},T_{W}}\colon\mathbb{N}(T_{W})^{\psi=1}\to H^{1}_{\mathrm{Iw}}(\mathbb{Q}_{p},T_{W}).

Moreover, we can extend this isomorphism from $\Lambda_{\mathcal{O}_{E}}(\Gamma)$ -modules to $\Lambda_{E}(\Gamma)$ -modules

h^{1}_{\mathrm{Iw},W}\colon\mathbb{N}(W)^{\psi=1}\to H^{1}_{\mathrm{Iw}}(\mathbb{Q}_{p},W),

where $\mathbb{N}(W)=E\otimes\mathbb{N}(T_{W})$ and $H^{1}_{\mathrm{Iw}}(\mathbb{Q}_{p},W)=E\otimes H^{1}_{\mathrm{Iw}}(\mathbb{Q}_{p},T)$ .

4.2. Coleman maps

For the $p$ -adic representation $W$ and the $\mathcal{O}_{E}$ -lattice $T_{W}$ in $W$ , we deduce $\mathbb{N}(T_{W})\subset\varphi^{*}(\mathbb{N}(T_{W}))$ , since the Hodge-Tate weights of $W$ are non-negative, where $\varphi^{*}(\mathbb{N}(T_{W}))$ is $\mathbb{A}^{+}_{E}$ -submodule of $\mathbb{N}(T_{W})[\pi^{-1}]$ generated by $\varphi(\mathbb{N}(T_{W}))$ (See [21, Lemma 1.7]). Hence there exists a well-defined map $1-\varphi\colon\mathbb{N}(T_{W})\to\varphi^{*}(\mathbb{N}(T_{W}))$ which maps $\mathbb{N}(T_{W})^{\psi=1}$ to $(\varphi^{*}\mathbb{N}(T_{W})))^{\psi=0}$ .

Theorem 4.2 (Lei-Loeffler-Zerbes, Berger).

$(\varphi^{*}\mathbb{N}(T_{W}))^{\psi=0}$ is a free $\Lambda_{\mathcal{O}_{E}}(\Gamma)$ -module of rank $2$ . Moreover, for any basis $v_{1},v_{2}$ of $\mathbb{D}_{\mathrm{crys}}(T_{W})$ , there exists an $\mathcal{O}_{E}\otimes\mathbb{A}^{+}_{\mathbb{Q}_{p}}$ -basis $n_{1},n_{2}$ of $\mathbb{N}(T_{W})$ such that $n_{i}\equiv v_{i}\operatorname{mod}\pi$ and $(1+\pi)\varphi(n_{1}),(1+\pi)\varphi(n_{2})$ form a $\Lambda_{\mathcal{O}_{E}}(\Gamma)$ -basis of $(\varphi^{*}\mathbb{N}(T_{W}))^{\psi=0}$ .

Proof.

See[23, Lemma 3.15] for the proof for any crystalline representation of dimension $d$ . ∎

The above theorem gives an isomorphism of $\Lambda_{\mathcal{O}_{E}}(\Gamma)$ -modules

\mathfrak{J}\colon(\varphi^{*}\mathbb{N}(T_{W}))^{\psi=0}\to\Lambda_{\mathcal{O}_{E}}(\Gamma)^{\oplus 2}.

Definition 4.3 (The Coleman map).

We define the Coleman map

\operatorname{\underline{Col}}=(\operatorname{\underline{Col}}_{i})_{i=1}^{2}\colon\mathbb{N}(T_{W})^{\psi=1}\to\Lambda_{\mathcal{O}_{E}}(\Gamma)^{\oplus 2}

as the composition $\mathfrak{J}\circ(1-\varphi)$ .

This Coleman map $\operatorname{\underline{Col}}$ can be extended as a map from $\mathbb{N}(W)$ to get a $\Lambda_{E}(\Gamma)$ -module homomorphism

\operatorname{\underline{Col}}\colon\mathbb{N}(W)^{\psi=1}\to\Lambda_{E}(\Gamma)^{\oplus 2}.

From the above discussion, for the fixed basis $n^{\prime}_{1},n^{\prime}_{2}$ for $\mathbb{N}(T_{W})$ and basis $w_{1},w_{2}$ for $\mathbb{D}_{\mathrm{crys}}(T_{W})$ , we get a matrix $\underline{M}$ as follows: The elements $(1+\pi)(\varphi(n^{\prime}_{1})),(1+\pi)(\varphi(n^{\prime}_{2}))$ form a $\Lambda_{\mathcal{O}_{E}}(\Gamma)$ -basis of $(\varphi^{*}\mathbb{N}(T_{W}))^{\psi=0}$ . Furthermore the elements $(1+\pi)\otimes w_{1},(1+\pi)\otimes w_{2}$ form a basis of $(\mathbb{B}^{+}_{\mathrm{rig},E})^{\psi=0}\otimes\mathbb{D}_{\mathrm{crys}}(T_{W})$ as a $\mathcal{H}_{E}(\Gamma)$ -module. Since $n^{\prime}_{i}\equiv w_{i}\mod\pi$ for $i=1,2$ , there exists a unique $2\times 2$ matrix $\underline{M}\in M_{2,2}(\mathcal{H}_{E}(\Gamma))$ such that

(4.1)

\begin{bmatrix}(1+\pi)\varphi(n^{\prime}_{1})&(1+\pi)\varphi(n^{\prime}_{2})\end{bmatrix}=\begin{bmatrix}(1+\pi)\otimes w_{1}&(1+\pi)\otimes w_{2}\end{bmatrix}\underline{M}.

That is, $\underline{M}$ is a change of the basis matrix for the following homomorphism of $\mathcal{H}_{E}(\Gamma)$ -modules:

(\varphi\mathbb{N}(T_{W}))^{\psi=0}\hookrightarrow(\mathbb{B}^{+}_{\mathrm{rig},E})^{\psi=0}\otimes\mathbb{D}_{\mathrm{crys}}(T_{W}).

Remark 4.4.

More precisely, $\underline{M}\in M_{2,2}(\mathcal{H}(\Gamma_{1}))$ , since $n^{\prime}_{i}$ lie in $(1+\pi)\varphi(\mathbb{N}(T_{W}))\subset(1+\pi)\varphi(\mathbb{B}^{+}_{\mathrm{rig},E})\otimes\mathbb{D}_{\mathrm{crys}}(T_{W})$ .

This matrix $\underline{M}$ will play a crucial role in the upcoming sections. More precisely, we will show that $\underline{M}$ is a logarithmic matrix (in the sense of Sprung and Lei-Loeffler-Zerbes) that can be used in the decomposition of power series with unbounded denominators.

4.3. The big logarithm map, the $p$ -adic regulator and the relation between them

Recall that the eigenvalues of $\varphi$ are not integral powers of $p$ . Using this fact, we can construct the Perrin-Riou big logarithm as (see [29, Section 3.2.3], [2, Section II.5] for the details)

\Omega_{T_{W},k-1}\colon E\otimes(\mathbb{B}^{+}_{\mathrm{rig},\mathbb{Q}_{p}})^{\psi=0}\otimes\mathbb{D}_{\mathrm{crys}}(T_{W})\to\mathcal{H}_{E}(\Gamma)\otimes H^{1}_{\mathrm{Iw}}(\mathbb{Q}_{p},T_{W}).

Note that this map is a $\Lambda_{\mathcal{O}_{E}}(\Gamma)$ -module homomorphism. We can extend this to a $\Lambda_{E}(\Gamma)$ -module homomorphism

\Omega_{W,k-1}\colon E\otimes(\mathbb{B}^{+}_{\mathrm{rig},\mathbb{Q}_{p}})^{\psi=0}\otimes\mathbb{D}_{\mathrm{crys}}(W)\to\mathcal{H}_{E}(\Gamma)\otimes H^{1}_{\mathrm{Iw}}(\mathbb{Q}_{p},W).

This map interpolates the Bloch-Kato exponential map

\mathrm{exp}_{n,j}\colon\mathbb{Q}_{p,n}\otimes\mathbb{D}_{\mathrm{crys}}(W(j))\to H^{1}(\mathbb{Q}_{p,n},W(j)),

where $j$ is any integer.

For $i\in\mathbb{Z}$ , define $\ell_{i}=\dfrac{\log(1+X)}{\log_{p}(u)}-i$ . It is easy to see that $\ell_{i}\in\mathcal{H}_{E}(\Gamma)$ for all integers $i$ . Berger gave a description of $\Omega_{W,k-1}$ in the terms of $\ell_{i}$ ’s (see [2, Section II.1, Theorem II.13] for more details) as follows:

(4.2)

\Omega_{W,k-1}(z)=(\ell_{k-2}\circ\cdots\circ\ell_{0})(1-\varphi)^{-1}(\bar{z}),

where $z\in\mathcal{H}_{E}(\Gamma)\otimes\mathbb{D}_{\mathrm{crys}}(W)$ and $\bar{z}=(\mathfrak{M}\otimes 1)(z)$ .

Definition 4.5 (The $p$ -adic regulator).

The Perrin-Riou $p$ -adic regulator map $\mathcal{L}_{T_{W}}$ for the $G_{\mathbb{Q}_{p}}$ -stable $\mathcal{O}_{E}$ -lattice $T_{W}$ in $W$ is a $\Lambda_{\mathcal{O}_{E}}(\Gamma)$ -homomorphism defined as

\mathcal{L}_{T_{W}}\coloneqq(\mathfrak{M}^{-1}\otimes 1)\circ(1-\varphi)\circ(h^{1}_{\mathrm{Iw},T_{W}})^{-1}\colon H^{1}_{\mathrm{Iw}}(\mathbb{Q}_{p},T_{W})\to\mathcal{H}_{E}(\Gamma)\otimes\mathbb{D}_{\mathrm{crys}}(T_{W}).

The $p$ -adic regulator $\mathcal{L}_{T_{W}}$ can be extended to a $\Lambda_{E}(\Gamma)$ -homomorphism as

\mathcal{L}_{W}\colon H^{1}_{\mathrm{Iw}}(\mathbb{Q}_{p},W)\to\mathcal{H}_{E}(\Gamma)\otimes\mathbb{D}_{\mathrm{crys}}(W).

The $p$ -adic regulator $\mathcal{L}_{W}$ and the big logarithm $\Omega_{W,k-1}$ are related by the following lemma:

Lemma 4.6.

As maps on $H^{1}_{\mathrm{Iw}}(\mathbb{Q}_{p},W)$ , we have

\mathcal{L}_{W}=(\mathfrak{M}^{-1}\otimes 1)(\prod_{i=0}^{k-2}\ell_{i})(\Omega_{W,k-1})^{-1}.

In other words,

(4.3)

\Omega_{W,k-1}(\mathcal{L}_{W}(z))=(\prod_{i=0}^{k-2}\ell_{i})(z),

for all $z\in H^{1}_{\mathrm{Iw}}(\mathbb{Q}_{p},W)$ .

Proof.

See [21, Theorem 4.6]. ∎

The following lemma gives a relationship between $\mathcal{L}_{T_{W}}$ and the Coleman maps.

Lemma 4.7 (Lei-Loeffler-Zerbes [21]).

For $z\in\mathbb{N}(T_{W})^{\psi=0}$ , we have

(1-\varphi)(z)=\begin{bmatrix}(1+\pi)\otimes w_{1}&(1+\pi)\otimes w_{2}\end{bmatrix}\underline{M}\operatorname{\underline{Col}}(z).

Thus, we can rewrite $\mathcal{L}_{T_{W}}$ in terms of $(1+\pi)\otimes w_{1},(1+\pi)\otimes w_{2}$ as

\mathcal{L}_{T_{W}}(z)=\begin{bmatrix}(1+\pi)\otimes w_{1}&(1+\pi)\otimes w_{2}\end{bmatrix}\underline{M}\left(\operatorname{\underline{Col}}\circ(h^{1}_{\mathrm{Iw},T_{W}})^{-1}\right)(z),

where $z\in H^{1}_{\mathrm{Iw}}(\mathbb{Q}_{p},T_{W})$ .

Proof.

This can be proved using.

(1-\varphi)(z)=\begin{bmatrix}(1+\pi)\varphi(n_{1})&(1+\pi)\varphi(n_{2})\end{bmatrix}\cdot\operatorname{\underline{Col}}(z),

and the definitions of $\underline{M}$ and $\mathcal{L}_{T_{W}}$ . ∎

Now for $z\in H^{1}_{\mathrm{Iw}}(\mathbb{Q}_{p},T_{W})$ , $\mathcal{L}_{T_{W}}(z)$ is an element of $\mathcal{H}_{E}(\Gamma)\otimes\mathbb{D}_{\mathrm{crys}}(T_{W})$ . Hence, we can apply any character of $\Gamma$ to $\mathcal{L}_{T_{W}}(z)$ to get an element in $E\otimes\mathbb{D}_{\mathrm{crys}}(T_{W})$ .

Proposition 4.8.

Let $z\in H^{1}_{\mathrm{Iw}}(\mathbb{Q}_{p},T_{W})$ . Then for any integer $0\leq i\leq k-2$ , and for any Dirichlet character $\omega$ of conductor $p^{n}>1$ , we have

(4.4)		$\displaystyle(1-\varphi)^{-1}(1-p^{-1}\varphi^{-1})\chi^{i}(\mathcal{L}_{T_{W}}(z)\otimes t^{i}e_{-i})$	$\displaystyle\in\mathrm{Fil}^{0}(\mathbb{D}_{\mathrm{crys}}(T_{W}(-i))),$
(4.5)		$\displaystyle\varphi^{-n}(\chi^{i}\omega(\mathcal{L}_{T_{W}}(z)\otimes t^{i}e_{-i}))$	$\displaystyle\in\mathbb{Q}_{p,n}\otimes\mathrm{Fil}^{0}\mathbb{D}_{\mathrm{crys}}(T_{W}(-i)),$

where $\chi$ is the $p$ -adic cyclotomic character and $e_{-i}$ is the basis of $\mathbb{D}_{\mathrm{crys}}(\mathbb{Z}_{p}(-i))$ .

Proof.

We replace $V$ with $T_{W}$ in [21, Proposition 4.8] and the result follows. ∎

Lemma 4.9.

\det(\mathcal{L}_{W})=\prod_{i=0}^{k-2}\ell_{i}.

Proof.

Since the Hodge-Tate weights of $W$ are $0$ and $k-1$ , we have

\mathrm{dim}_{E}(\mathrm{Fil}^{i}\mathbb{D}_{\mathrm{crys}}(W))=1,

for all $-(k-2)\leq i\leq 0$ . Thus, replacing $V$ with $W$ , putting $d=2$ and $n_{i}=1$ for all integers $0\leq i\leq(k-2)$ in [21, Corollary 4.7], we get

\det(\mathcal{L}_{W})=\prod_{i=0}^{k-2}(\ell_{i})^{2-1}.

∎

4.4. The matrices of $\Omega_{W,k-2},\mathcal{L}_{W}$ and their connection with $\underline{M}$

For the rest of the article, fix an eigenbasis $\mathcal{B}_{\mathrm{eig}}=\{w_{\alpha},w_{\beta}\}$ of $\varphi$ for $\mathbb{D}_{\mathrm{crys}}(T_{W})$ , that is, $\mathcal{B}_{\mathrm{eig}}$ is a basis for $\mathbb{D}_{\mathrm{crys}}(T_{W})$ and $\varphi(w_{\alpha})=(\alpha)^{-1}w_{\alpha}$ and $\varphi(w_{\beta})=(\beta)^{-1}w_{\beta}$ . Thus, $(1+\pi)\otimes w_{\alpha},(1+\pi)\otimes w_{\beta}$ is a basis for $(\mathbb{B}^{+}_{\mathrm{rig},E})^{\psi=0}\otimes\mathbb{D}_{\mathrm{crys}}(T_{W})$ . We denote the basis $\{w_{1},w_{2}\}$ for $\mathbb{D}_{\mathrm{crys}}(T_{W})$ by $\mathcal{B}$ , which we have defined in subsection 3.1.

Recall that the matrix of $\varphi$ with respect to $\mathcal{B}$ is

A_{\varphi}=\begin{pmatrix}0&-1/vp^{k-1}\\[6.0pt] 1&a/vp^{k-1}\end{pmatrix},

and thus we get

\begin{pmatrix}\alpha^{-1}&0\\[6.0pt] 0&\beta^{-1}\end{pmatrix}=Q^{-1}A_{\varphi}Q,

where $Q=\begin{pmatrix}\alpha&-\beta\\[6.0pt] -vp^{k-1}&vp^{k-1}\end{pmatrix}$ .

We know that the $\Lambda_{\mathcal{O}_{E}}(\Gamma)$ -rank of $H^{1}_{\mathrm{Iw}}(\mathbb{Q}_{p},T_{W})$ is 2, since $\dim_{E}\mathbb{D}_{\mathrm{crys}}(W)=2$ and $T_{W}$ is an $\mathcal{O}_{E}$ -lattice in $W$ . See [29, Section 3.2] and [2, Proposition 2.7] for precise details. Thus, we may fix a $\Lambda_{\mathcal{O}_{E}}(\Gamma)$ -basis $\{z_{1},z_{2}\}$ for $H^{1}_{\mathrm{Iw}}(\mathbb{Q}_{p},T_{W})$ .

4.4.1. The matrix of $\Omega_{T_{W},k-1}$

Using the big logarithm map $\Omega_{T_{W},k-1}$ , we obtain the following equations

	$\displaystyle\Omega_{W,k-1}((1+\pi)\otimes w_{\alpha})$	$\displaystyle=a\cdot z_{1}+c\cdot z_{2},$
	$\displaystyle\Omega_{W,k-1}((1+\pi)\otimes w_{\beta})$	$\displaystyle=b\cdot z_{1}+d\cdot z_{2},$

where $a,b,c,d$ are elements in the distribution ring $\mathcal{H}_{E}(\Gamma)$ . In other words, we can write these equations as

\begin{bmatrix}\Omega_{W,k-1}((1+\pi)\otimes w_{\alpha})&\Omega_{W,k-1}((1+\pi)\otimes w_{\beta})\end{bmatrix}=\begin{bmatrix}z_{1}&z_{2}\end{bmatrix}\begin{pmatrix}a&b\\[6.0pt] c&d\end{pmatrix}.

Therefore, with respect to the basis $\{z_{1},z_{2}\}$ for $H^{1}_{\mathrm{Iw}}(\mathbb{Q}_{p},T_{W})$ and the basis $\mathcal{B}_{\mathrm{eig}}$ for $\mathbb{D}_{\mathrm{crys}}(T_{W})$ we can describe the matrix $M_{\Omega}$ of $\Omega_{T_{W},k-1}$ as,

(4.6)

M_{\Omega}=\begin{pmatrix}a&b\\[6.0pt] c&d\end{pmatrix}.

Recall that, if $F\in\mathcal{H}_{E,r}(\Gamma)$ , we say $F$ is $O(\log_{p}^{r})$ .

Lemma 4.10.

The elements $a,c$ are $O(\log_{p}^{v_{p}(\beta)})$ , whereas $b,d$ are $O(\log_{p}^{v_{p}(\alpha)})$ .

Proof.

From [29, Section 3.2.4], we note that for any $\mathcal{O}_{E}$ -lattice $T$ in a crystalline representation $V$ , if $f$ is an element of $(\mathbb{B}^{+}_{\mathrm{rig},E})^{\psi=0}\otimes\mathbb{D}_{\nu}(T(j))$ , where $\mathbb{D}_{\nu}(T(j))$ is a subspace of $\mathbb{D}_{\mathrm{crys}}(T(j))$ in which $\varphi$ has slope $\nu$ , then $\Omega_{T(J),h+j}(f)$ is $O(\log_{p}^{h+\nu})$ . In other words, $\Omega_{T(j),h+j}(f)$ lies in $\mathcal{H}_{h+\nu}(\Gamma)\otimes H^{1}_{\mathrm{Iw}}(\mathbb{Q}_{p},T(j))$ .

For the crystalline representation $W$ and the lattice $T_{W}$ in $W$ , we know that $\varphi(w_{\alpha})=\alpha^{-1}w_{\alpha}$ and $\varphi(w_{\beta})=\beta^{-1}w_{\beta}$ . Thus, $\Omega_{T_{W},k-1}((1+\pi)\otimes w_{\alpha})$ is $O(\log^{(k-1)-v_{p}(\alpha)}_{p})=O(\log^{v_{p}(\beta)}_{p})$ , since $v_{p}(\alpha)+v_{p}(\beta)=k-1$ . Similarly, $\Omega_{T_{W},k-1}((1+\pi)\otimes w_{\beta})$ is $O(\log^{v_{p}(\alpha)}_{p})$ .

But $\Omega_{T_{W},k-1}((1+\pi)\otimes w_{\alpha})=a\cdot z_{1}+c\cdot z_{2}\in\mathcal{H}_{E,v_{p}(\beta)}(\Gamma)\otimes H^{1}_{\mathrm{Iw}}(\mathbb{Q}_{p},T_{W})$ . Therefore we conclude that $a$ and $c$ have growth $O(\log_{p}^{v_{p}(\beta)})$ . In the same manner, $b$ and $d$ have growth $O(\log_{p}^{v_{p}(\alpha)})$ . ∎

4.4.2. The matrix of the $p$ -adic regulator $\mathcal{L}_{T_{W}}$

After applying the $p$ -adic regulator $\mathcal{L}_{T_{W}}$ on the $\Lambda_{\mathcal{O}_{E}}(\Gamma)$ -basis $z_{1},z_{2}$ of $H^{1}_{\mathrm{Iw}}(\mathbb{Q}_{p},T_{W})$ , we get

	$\displaystyle\mathcal{L}_{T_{W}}(z_{1})$	$\displaystyle=x_{1}\cdot w_{\alpha}+x_{3}\cdot w_{\beta},$
	$\displaystyle\mathcal{L}_{T_{W}}(z_{1})$	$\displaystyle=x_{2}\cdot w_{\alpha}+x_{4}\cdot w_{\beta}.$

We can rewrite these equations as

(4.7)

\begin{bmatrix}\mathcal{L}_{T_{W}}(z_{1})&\mathcal{L}_{T_{W}}(z_{2})\end{bmatrix}=\begin{bmatrix}w_{\alpha}&w_{\beta}\end{bmatrix}\begin{pmatrix}x_{1}&x_{2}\\[6.0pt] x_{3}&x_{4}\end{pmatrix}.

Hence, using the basis $\{z_{1},z_{2}\}$ for $H^{1}_{\mathrm{Iw}}(\mathbb{Q}_{p},T_{W})$ and the basis $\mathcal{B}_{\mathrm{eig}}$ for $\mathbb{D}_{\mathrm{crys}}(T_{W})$ , we get a matrix $[\mathcal{L}_{T_{W}}]_{\mathcal{B}_{\mathrm{eig}}}\in M_{2,2}(\mathcal{H}_{E}(\Gamma))$ of $\mathcal{L}_{T_{W}}$ as

(4.8)

[\mathcal{L}_{T_{W}}]_{\mathcal{B}_{\mathrm{eig}}}=\begin{pmatrix}x_{1}&x_{2}\\[6.0pt] x_{3}&x_{4}\end{pmatrix}.

Lemma 4.11.

We have the equation

(4.9)

[\mathcal{L}_{T_{W}}]_{\mathcal{B}_{\mathrm{eig}}}=\operatorname{adj}M_{\Omega},

where $\operatorname{adj}\underline{M}$ is the adjugate matrix of $\underline{M}$ . In particular, $x_{1},x_{2}$ are $O(\log_{p}^{v_{p}(\alpha)})$ and $x_{3},x_{4}$ are $O(\log_{p}^{v_{p}(\beta)})$ .

Proof.

From Lemma 4.6, we know that

\Omega_{W,k-1}\circ\mathcal{L}_{W}=\left(\prod_{i=0}^{k-2}\ell_{i}\right).

By restricting to the $\mathcal{O}_{E}$ -lattice $T_{W}$ in $W$ , we have, for any $z\in H^{1}_{\mathrm{Iw}}(\mathbb{Q}_{p},T_{W})$ ,

\displaystyle\Omega_{T_{W},k-1}(\mathcal{L}_{T_{W}}(z))=\left(\prod_{i=0}^{k-2}\ell_{i}\right)(z),

Thus, we use $\Lambda_{\mathcal{O}_{E}}(\Gamma)$ -basis $\{z_{1},z_{2}\}$ for $H^{1}_{\mathrm{Iw}}(\mathbb{Q}_{p},T_{W})$ , we get

(4.10)		$\displaystyle\Omega_{T_{W},k-1}(\mathcal{L}_{T_{W}}(z_{1}))$	$\displaystyle=\left(\prod_{i=0}^{k-2}\ell_{i}\right)(z_{1}),$
(4.11)		$\displaystyle\Omega_{T_{W},k-1}(\mathcal{L}_{T_{W}}(z_{2}))$	$\displaystyle=\left(\prod_{i=0}^{k-2}\ell_{i}\right)(z_{2}).$

In matrix form, we can rewrite the equations (4.10) and (4.11)

M_{\Omega}[\mathcal{L}_{T_{W}}]_{\mathcal{B}_{\mathrm{eig}}}=\left(\prod_{i=0}^{k-2}\ell_{i}\right)I_{2}.

From Lemma 4.9, we have $\det(\mathcal{L}_{T_{W}})=\prod_{i=0}^{k-2}\ell_{i}$ , hence we have $\det([\mathcal{L}_{T_{W}}]_{\mathcal{B}_{\mathrm{eig}}})=\prod_{i=0}^{k-2}\ell_{i}.$ Thus

M_{\Omega}[\mathcal{L}_{T_{W}}]_{\mathcal{B}_{\mathrm{eig}}}=\det(\mathcal{L}_{T_{W}})I_{2}.

Hence,

(4.12)

\begin{pmatrix}x_{1}&x_{2}\\[6.0pt] x_{3}&x_{4}\end{pmatrix}=[\mathcal{L}_{T_{W}}]_{\mathcal{B}_{\mathrm{eig}}}=\operatorname{adj}M_{\Omega}=\begin{pmatrix}d&-b\\[6.0pt] -c&a\end{pmatrix},

where $\operatorname{adj}M_{\Omega}$ is the adjugate matrix of the matrix $M_{\Omega}$ . Thus, from Lemma 4.10, we get $x_{1},x_{2}$ have growth $O(\log_{p}^{v_{p}(\alpha)})$ and $x_{3},x_{4}$ have growth $O(\log_{p}^{v_{p}(\beta)})$ . ∎

We use the basis $\mathcal{B}$ of $\mathbb{D}_{\mathrm{crys}}(T_{W})$ and the basis $\{z_{1},z_{2}\}$ for $H^{1}_{\mathrm{Iw}}(\mathbb{Q}_{p},T_{W})$ to get another matrix $[\mathcal{L}_{T_{W}}]_{\mathcal{B}}$ such that

\begin{bmatrix}\mathcal{L}_{T_{W}}(z_{1})&\mathcal{L}_{T_{W}}(z_{2}))\end{bmatrix}=\begin{bmatrix}w_{1}&w_{2}\end{bmatrix}[\mathcal{L}_{T_{W}}]_{\mathcal{B}}.

Since $\begin{bmatrix}w_{1}&w_{2}\end{bmatrix}=\begin{bmatrix}w_{\alpha}&w_{\beta}\end{bmatrix}Q^{-1}$ , we have

(4.13)

[\mathcal{L}_{T_{W}}]_{\mathcal{B}_{\mathrm{eig}}}=Q^{-1}[\mathcal{L}_{T_{W}}]_{\mathcal{B}}.

For any non-negative integer $n$ , we write $\omega_{n}(1+X)=(1+X)^{p^{n}}-1$ . Let $\Phi_{n}(1+X)=\omega_{n}(1+X)/\omega_{n-1}(1+X)$ be the $p^{n}$ -th cyclotomic polynomial for integers $n>1$ . Recall from Section 2, topological generators $\gamma_{0}$ of $\Gamma_{1}$ and $u$ of $1+p\mathbb{Z}_{p}$ such that $\chi(\gamma_{0})=u$ , where $\chi$ is the $p$ -adic cyclotomic character. For any integer $m\geq 1$ , we define

	$\displaystyle\Phi_{n,m}(1+X)$	$\displaystyle=\prod_{j=0}^{m-1}\Phi_{n}(u^{-j}(1+X)),$
	$\displaystyle\omega_{n,m}(1+X)$	$\displaystyle=\prod_{j=0}^{m-1}\omega_{n}(u^{-j}(1+X)),$
	$\displaystyle\delta_{m}(X)$	$\displaystyle=\prod_{j=0}^{m-1}(u^{-j}(1+X)-1),$
	$\displaystyle\log_{p,m}(1+X)$	$\displaystyle=\prod_{j=0}^{m-1}\log_{p}(u^{-j}(1+X)).$

Recall from Proposition 4.8, for any Dirichlet character $\omega$ of conductor $p^{n},n>1$ , we have

\varphi^{-n}(\chi^{i}\omega(\mathcal{L}_{T_{W}}(z)\otimes t^{i}e_{-i}))\in\mathbb{Q}_{p,n}\otimes\mathrm{Fil}^{0}\mathbb{D}_{\mathrm{crys}}(T_{W}(-i)),

for any $z\in H^{1}_{\mathrm{Iw}}(\mathbb{Q}_{p},T_{W})$ and $0\leq i\leq(k-2)$ .

Proposition 4.12.

The second row of the matrix $[\varphi^{-n}\mathcal{L}_{T_{W}}]_{\mathcal{B}}$ is divisible by $\Phi_{n-1,k-1}(\gamma_{0})$ over $\mathcal{H}_{E}(\Gamma)$ , for all integers $n>1$ .

Proof.

The Hodge-Tate weights of the crystalline representation $W$ are $0$ and $k-1$ . Thus, for $0\leq i\leq k-2$ , we have

\dim_{E}(\mathrm{Fil}^{0}\mathbb{D}_{\mathrm{crys}}(W(-i)))=\dim_{E}(\mathrm{Fil}^{-i}\mathbb{D}_{\mathrm{crys}}(W))=1.

We know that $\mathrm{Fil}^{-i}\mathbb{D}_{\mathrm{crys}}(T_{W})$ is one dimensional $\mathcal{O}_{E}$ -submodule of $\mathbb{D}_{\mathrm{crys}}(T_{W})$ generated by $w_{1}$ for all $0\leq i\leq k-2$ . Thus, $\mathrm{Fil}^{0}\mathbb{D}_{\mathrm{crys}}(T_{W})$ is generated by $w_{1}\otimes t^{i}e_{-i}$ .

Write

(4.14)

\varphi^{-n}(\mathcal{L}_{T_{W}}(z)\otimes t^{i}e_{-i})=F_{1,z}\cdot(w_{1}\otimes p^{-ni}t^{i}e_{-i})+F_{2,z}\cdot(w_{2}\otimes p^{-ni}t^{i}e_{-i}),

where $F_{1,z},F_{2,z}\in\mathcal{H}_{E}(\Gamma)$ and $z\in H^{1}_{\mathrm{Iw}}(\mathbb{Q}_{p},T_{W})$ .

Thus, (4.5) implies

(4.15)

F_{2,z}(\chi^{i}\omega)=0,

for all $0\leq i\leq k-2$ and for all Dirichlet character $\omega$ of conductor $p^{n}$ , where $n>1$ . Then [23, Theorem 5.4] implies $\Phi_{n-1,k-1}(\gamma_{0})$ divides $F_{2,z}$ .

Using the basis $z_{1},z_{2}$ for $H^{1}_{\mathrm{Iw}}(\mathbb{Q}_{p},T_{W})$ and the basis $\mathcal{B}$ for $\mathbb{D}_{\mathrm{crys}}(T_{W})$ , we get

\begin{bmatrix}\varphi^{-n}(\mathcal{L}_{T_{W}}(z_{1})\otimes t^{i}e_{-i})&\varphi^{-n}(\mathcal{L}_{T_{W}}(z_{2})\otimes t^{i}e_{-i})\end{bmatrix}=\begin{bmatrix}w_{1}\otimes t^{i}e_{-i}&w_{2}\otimes t^{i}e_{-i}\end{bmatrix}\begin{pmatrix}F_{1,z_{1}}&F_{1,z_{2}}\\[6.0pt] F_{2,z_{1}}&F_{2,z_{2}}\end{pmatrix}.

Then the matrix of $\varphi^{-n}\mathcal{L}_{T_{W}}$ with respect to basis $\mathcal{B}$ is

[\varphi^{-n}\mathcal{L}_{T_{W}}]_{\mathcal{B}}=\begin{pmatrix}F_{1,z_{1}}&F_{1,z_{2}}\\[6.0pt] F_{2,z_{1}}&F_{2,z_{2}}\end{pmatrix}.

Note that $[\varphi^{-n}\mathcal{L}_{T_{W}}]_{\mathcal{B}}\in M_{2,2}(\mathcal{H}(\Gamma))$ . Hence, using (4.15), we deduce that $\Phi_{n-1,k-1}(\gamma_{0})$ divides both $F_{2,z_{1}}$ and $F_{2,z_{2}}$ ∎

For $n>1$ , we can write

	$\displaystyle\begin{bmatrix}\varphi^{-n}(\mathcal{L}_{T_{W}}(z_{1}))&\varphi^{-n}(\mathcal{L}_{T_{W}}(z_{2}))\end{bmatrix}$	$\displaystyle=\begin{bmatrix}\varphi^{-n}(w_{\alpha})&\varphi^{-n}(w_{\beta})\end{bmatrix}[\mathcal{L}_{T_{W}}]_{\mathcal{B}_{\mathrm{eig}}},$
		$\displaystyle=\begin{bmatrix}w_{\alpha}&w_{\beta}\end{bmatrix}\begin{pmatrix}\alpha^{n}&0\\[6.0pt] 0&\beta^{n}\end{pmatrix}[\mathcal{L}_{T_{W}}]_{\mathcal{B}_{\mathrm{eig}}}.$

Thus, we get a matrix for $\varphi^{-n}\mathcal{L}_{T_{W}}$ with respect to the eigenbasis $\mathcal{B}_{\mathrm{eig}}$ for $\mathbb{D}_{\mathrm{crys}}(T_{W})$ and using (4.8), we get

(4.16)

[\varphi^{-n}\mathcal{L}_{T_{W}}]_{\mathcal{B}_{\mathrm{eig}}}=\begin{pmatrix}\alpha^{n}&0\\[6.0pt] 0&\beta^{n}\end{pmatrix}[\mathcal{L}_{T_{W}}]_{\mathcal{B}_{\mathrm{eig}}}=\begin{pmatrix}\alpha^{n}x_{1}&\alpha^{n}x_{2}\\[6.0pt] \beta^{n}x_{3}&\beta^{n}x_{4}\end{pmatrix}.

5. Logarithmic matrix $\underline{M}$ and the factorization of power series in one variable

In this section, we will first explore some properties of $\underline{M}$ which imply that $\underline{M}$ is a logarithmic matrix in the sense of Sprung and Lei-Loeffler-Zerbes. Next, we will use $\underline{M}$ to decompose power series with certain growth conditions into power series with bounded coefficients.

5.1. Properties of $\underline{M}$

For any $z\in H^{1}_{\mathrm{Iw}}(\mathbb{Q}_{p},T_{W})$ , we have

\mathcal{L}_{T_{W}}(z)=\begin{bmatrix}(1+\pi)\otimes w_{1}&(1+\pi)\otimes w_{2}\end{bmatrix}\underline{M}\begin{pmatrix}\operatorname{\underline{Col}}_{1}(z)\\[6.0pt] \operatorname{\underline{Col}}_{2}(z)\end{pmatrix}.

In matrix form, we write

\begin{bmatrix}\mathcal{L}_{T_{W}}(z_{1})&\mathcal{L}_{T_{W}}(z_{2})\end{bmatrix}=\begin{bmatrix}(1+\pi)\otimes w_{1}&(1+\pi)\otimes w_{2}\end{bmatrix}\underline{M}\begin{pmatrix}\operatorname{\underline{Col}}_{1}(z_{1})&\operatorname{\underline{Col}}_{1}(z_{2})\\[6.0pt] \operatorname{\underline{Col}}_{2}(z_{1})&\operatorname{\underline{Col}}_{2}(z_{2})\end{pmatrix}.

Thus,

(5.1)

[\mathcal{L}_{T_{W}}]_{\mathcal{B}}=\underline{M}\begin{pmatrix}\operatorname{\underline{Col}}_{1}(z_{1})&\operatorname{\underline{Col}}_{1}(z_{2})\\[6.0pt] \operatorname{\underline{Col}}_{2}(z_{1})&\operatorname{\underline{Col}}_{2}(Z_{2})\end{pmatrix}.

Similarly, we have

(5.2)

[\mathcal{L}_{T_{W}}]_{\mathcal{B}_{\mathrm{eig}}}=Q^{-1}\underline{M}\begin{pmatrix}\operatorname{\underline{Col}}_{1}(z_{1})&\operatorname{\underline{Col}}_{1}(z_{2})\\[6.0pt] \operatorname{\underline{Col}}_{2}(z_{1})&\operatorname{\underline{Col}}_{2}(Z_{2})\end{pmatrix},

since $[\mathcal{L}_{T_{W}}]_{\mathcal{B}_{\mathrm{eig}}}=Q^{-1}[\mathcal{L}_{T_{W}}]_{\mathcal{B}}.$

Proposition 5.1.

The elements in the first row of $Q^{-1}\underline{M}$ are inside $\mathcal{H}_{E,v_{p}(\alpha)}(\Gamma_{1})$ , while the elements in the second row are in the $\mathcal{H}_{E,v_{p}(\beta)}(\Gamma_{1})$ .

Proof.

Recall from Lemma 4.11,

[\mathcal{L}_{T_{W}}]_{\mathcal{B}_{\mathrm{eig}}}=\operatorname{adj}M_{\Omega}=\begin{pmatrix}x_{1}&x_{2}\\[6.0pt] x_{3}&x_{4}\end{pmatrix}.

Therefore, (5.2) implies

\begin{pmatrix}x_{1}&x_{2}\\[6.0pt] x_{3}&x_{4}\end{pmatrix}=Q^{-1}\underline{M}\begin{pmatrix}\operatorname{\underline{Col}}_{1}(z_{1})&\operatorname{\underline{Col}}_{1}(z_{2})\\[6.0pt] \operatorname{\underline{Col}}_{2}(z_{1})&\operatorname{\underline{Col}}_{2}(Z_{2})\end{pmatrix}.

But $\operatorname{\underline{Col}}_{i}(z_{j})$ are $O(1)$ for $i,j\in\{1,2\}$ , since they lie in the Iwasawa algebra $\Lambda_{E}(\Gamma)$ . Therefore, after writing $Q^{-1}\underline{M}=\begin{pmatrix}P_{1}&P_{2}\\[6.0pt] P_{3}&P_{4}\end{pmatrix}$ , we get

(5.3)

\begin{pmatrix}x_{1}&x_{2}\\[6.0pt] x_{3}&x_{4}\end{pmatrix}=\begin{pmatrix}P_{1}&P_{2}\\[6.0pt] P_{3}&P_{4}\end{pmatrix}\begin{pmatrix}\operatorname{\underline{Col}}_{1}(z_{1})&\operatorname{\underline{Col}}_{1}(z_{2})\\[6.0pt] \operatorname{\underline{Col}}_{2}(z_{1})&\operatorname{\underline{Col}}_{2}(Z_{2})\end{pmatrix}.

We take isotypic components on both sides with respect to the trivial character of $\Delta$ . After writing $[\mathrm{Col}^{\Delta}]$ for $\begin{pmatrix}\operatorname{\underline{Col}}^{\Delta}_{1}(z_{1})&\operatorname{\underline{Col}}^{\Delta}_{1}(z_{2})\\[6.0pt] \operatorname{\underline{Col}}^{\Delta}_{2}(z_{1})&\operatorname{\underline{Col}}^{\Delta}_{2}(Z_{2})\end{pmatrix}$ , equation (5.3) becomes

(5.4)

\begin{pmatrix}x_{1}^{\Delta}&x_{2}^{\Delta}\\[6.0pt] x_{3}^{\Delta}&x_{4}^{\Delta}\end{pmatrix}\operatorname{adj}[\mathrm{Col}^{\Delta}]=\begin{pmatrix}P_{1}&P_{2}\\[6.0pt] P_{3}&P_{4}\end{pmatrix}\det([\mathrm{Col}^{\Delta}]).

The result follows from the Lemma 4.11, since $\det([\mathrm{Col}^{\Delta}])$ is again $O(1)$ and $x_{1}^{\Delta},x_{2}^{\Delta}\in\mathcal{H}_{E,v_{p}(\alpha)}(\Gamma_{1})$ and $x_{3}^{\Delta},x_{4}^{\Delta}\in\mathcal{H}_{E,v_{p}(\beta)}(\Gamma_{1})$ . ∎

Lemma 5.2.

The second row of $A_{\varphi}^{-n}\underline{M}$ is divisible by the cyclotomic polynomial $\Phi_{n-1,k-1}(\gamma_{0})$ over $\mathcal{H}_{E}(\Gamma_{1})$ .

Proof.

We know that

\begin{bmatrix}\mathcal{L}_{T_{W}}(z_{1})&\mathcal{L}_{T_{W}}(z_{2})\end{bmatrix}=\begin{bmatrix}(1+\pi)\otimes w_{1}&(1+\pi)\otimes w_{2}\end{bmatrix}\underline{M}\begin{pmatrix}\operatorname{\underline{Col}}_{1}(z_{1})&\operatorname{\underline{Col}}_{1}(z_{2})\\[6.0pt] \operatorname{\underline{Col}}_{2}(z_{1})&\operatorname{\underline{Col}}_{2}(z_{2})\end{pmatrix}.

Let us denote $\begin{pmatrix}\operatorname{\underline{Col}}_{1}(z_{1})&\operatorname{\underline{Col}}_{1}(z_{2})\\[6.0pt] \operatorname{\underline{Col}}_{2}(z_{1})&\operatorname{\underline{Col}}_{2}(z_{2})\end{pmatrix}$ by $[\mathrm{Col}]$ . By an abuse of notation, we write $\varphi$ for $1\otimes\varphi$ .

After applying $\varphi^{-n}$ on both sides of the above equation, we obtain

	$\displaystyle\varphi^{-n}\left(\begin{bmatrix}\mathcal{L}_{T_{W}}(z_{1})&\mathcal{L}_{T_{W}}(z_{2})\end{bmatrix}\right)$	$\displaystyle=\begin{bmatrix}\varphi^{-n}(\mathcal{L}_{T_{W}}(z_{1}))&\varphi^{-n}(\mathcal{L}_{T_{W}}(z_{2}))\end{bmatrix},$
		$\displaystyle=\begin{bmatrix}(1+\pi)\otimes\varphi^{-n}(w_{1})&(1+\pi)\otimes\varphi^{-n}(w_{2})\end{bmatrix}\underline{M}[\mathrm{Col}],$
		$\displaystyle=\begin{bmatrix}(1+\pi)\otimes w_{1}&(1+\pi)\otimes w_{2}\end{bmatrix}A_{\varphi}^{-n}\underline{M}[\mathrm{Col}].$

Therefore, we get

[\varphi^{-n}\mathcal{L}_{T_{W}}]_{\mathcal{B}}=A^{-n}_{\varphi}\underline{M}[\mathrm{Col}].

After rearranging the above equation, we get

(5.5)

[\varphi^{-n}\mathcal{L}_{T_{W}}]_{\mathcal{B}}\operatorname{adj}[\mathrm{Col}]=A^{-n}_{\varphi}\underline{M}\det([\mathrm{Col}]),

From [21, Proposition 4.11, Theorem 4.12, Corollary 4.15], we can conclude that $\Phi_{n-1,k-1}(\gamma_{0})$ does not divide $\det([\mathrm{Col}])$ . Thus, the result follows from Proposition 4.12. ∎

Lemma 5.3.

The determinant of matrix $\underline{M}$ is $\dfrac{\log_{p,k-1}(\gamma_{0})}{\delta_{k-1}(\gamma_{0}-1)}$ upto a unit in $\Lambda_{E}(\Gamma_{1})$ .

Proof.

This is [21, Corollary 3.2]. See also [7, Lemma 2.7]. ∎

Thus, Proposition 5.1, Lemma 5.2, and Lemma 5.3 imply that the matrix $\underline{M}$ is a logarithmic matrix in the sense of Sprung and Lei-Loeffler-Zerbes.

5.2. Factorization using $\underline{M}$

Let $F,G$ be power series in $\mathcal{H}_{E}(\Gamma)$ . We write $F\sim G$ , if $F$ is $O(G)$ and $G$ is $O(F)$ .

Lemma 5.4.

We have $\det(\underline{M})\sim\log_{p}^{k-1}$ .

Proof.

From Lemma 5.3, we get $\det(\underline{M})=*\dfrac{\log_{p,k-1}(\gamma_{0})}{\delta_{k-1}(\gamma_{0}-1)}$ , where $*$ is a unit in $\Lambda_{E}(\Gamma)$ . Hence the result follows from the definition of $\log^{k-1}_{p}(\gamma_{0})$ and the fact that $\delta_{k-1}(\gamma_{0}-1)$ is polynomial and hence $O(1)$ . ∎

Theorem 5.5.

For $\lambda\in\{\alpha,\beta\}$ , let $F_{\lambda}\in\mathcal{H}_{E,v_{p}(\lambda)}(\Gamma)$ , such that for any integer $0\leq j\leq k-2$ and for any Dirichlet character $\omega$ of conductor $p^{n}$ we have $F_{\lambda}(\chi^{j}\omega)=\lambda^{-n}C_{\omega,j}$ , where $C_{\omega,j}\in\overline{\mathbb{Q}_{p}}$ that is independent of $\lambda$ . Then, there exist $F_{\flat},F_{\sharp}\in\Lambda_{E}(\Gamma)$ such that

(5.6)

\begin{pmatrix}F_{\alpha}\\[6.0pt] F_{\beta}\end{pmatrix}=Q^{-1}\underline{M}\begin{pmatrix}F_{\sharp}\\[6.0pt] F_{\flat}\end{pmatrix}.

Proof.

From Lemma 5.2, we know that the second row of $A_{\varphi}^{-n}\underline{M}$ is divisible by $\Phi_{n-1,k-1}(\gamma_{0})$ . Hence we can write

A_{\varphi}^{-n}\underline{M}=\begin{pmatrix}a&b\\[6.0pt] c\cdot\Phi_{n-1,k-1}(\gamma_{0})&d\cdot\Phi_{n-1,k-1}(\gamma_{0})\end{pmatrix},

where $a,b,c,d$ are power series.

Recall, $Q^{-1}=\frac{1}{\det Q}\begin{pmatrix}vp^{k-1}&\beta\\[6.0pt] vp^{k-1}&\alpha\end{pmatrix},$ and $\operatorname{adj}Q^{-1}\underline{M}=\begin{pmatrix}P_{4}&-P_{2}\\[6.0pt] -P_{3}&P_{1}\end{pmatrix}$ . Note that $\det Q\neq 0$ since $\alpha\neq\beta$ .

Thus for every positive integer $n$ , we have

	$\displaystyle Q^{-1}\underline{M}$	$\displaystyle=Q^{-1}A^{n}_{\varphi}QQ^{-1}A_{\varphi}^{-n}\underline{M},$
		$\displaystyle=\begin{pmatrix}\frac{1}{\alpha^{n}}&0\\[12.0pt] 0&\frac{1}{\beta^{n}}\end{pmatrix}\dfrac{1}{\det Q}\begin{pmatrix}vp^{k-1}&\beta\\[12.0pt] vp^{k-1}&\alpha\end{pmatrix}\begin{pmatrix}a&b\\[12.0pt] c\cdot\Phi_{n-1,k-1}(\gamma_{0})&d\cdot\Phi_{n-1,k-1}(\gamma_{0})\end{pmatrix},$
		$\displaystyle=\frac{1}{\det Q}\begin{pmatrix}\frac{1}{\alpha^{n}}&0\\[12.0pt] 0&\frac{1}{\beta^{n}}\end{pmatrix}\begin{pmatrix}a\cdot vp^{k-1}+\beta\cdot c\cdot\Phi_{n-1,k-1}(\gamma_{0})&b\cdot vp^{k-1}+\beta\cdot d\cdot\Phi_{n-1,k-1}(\gamma_{0})\\[12.0pt] a\cdot vp^{k-1}+\alpha\cdot c\cdot\Phi_{n-1,k-1}(\gamma_{0})&b\cdot vp^{k-1}+\alpha\cdot d\cdot\Phi_{n-1,k-1}(\gamma_{0})\end{pmatrix},$
		$\displaystyle=\frac{1}{\det Q}\begin{pmatrix}\frac{1}{\alpha^{n}}(a\cdot vp^{k-1}+\beta\cdot c\cdot\Phi_{n-1,k-1}(\gamma_{0}))&\frac{1}{\alpha^{n}}(b\cdot vp^{k-1}+\beta\cdot d\cdot\Phi_{n-1,k-1}(\gamma_{0}))\\[12.0pt] \frac{1}{\beta^{n}}(a\cdot vp^{k-1}+\alpha\cdot c\cdot\Phi_{n-1,k-1}(\gamma_{0}))&\frac{1}{\beta^{n}}(b\cdot vp^{k-1}+\alpha\cdot d\cdot\Phi_{n-1,k-1}(\gamma_{0}))\end{pmatrix}.$

Hence

(5.7)

\operatorname{adj}Q^{-1}\underline{M}=\frac{1}{\det Q}\begin{pmatrix}\dfrac{1}{\beta^{n}}(b\cdot vp^{k-1}+\alpha\cdot d\cdot\Phi_{n-1,k-1}(\gamma_{0}))&\dfrac{-1}{\alpha^{n}}(b\cdot vp^{k-1}+\beta\cdot d\cdot\Phi_{n-1,k-1}(\gamma_{0}))\\[12.0pt] \dfrac{-1}{\beta^{n}}(a\cdot vp^{k-1}+\alpha\cdot c\cdot\Phi_{n-1,k-1}(\gamma_{0}))&\dfrac{1}{\alpha^{n}}(a\cdot vp^{k-1}+\beta\cdot c\cdot\Phi_{n-1,k-1}(\gamma_{0}))\end{pmatrix}.

For integer $0\leq j\leq(k-2)$ , and Dirichlet character $\omega$ of conductor $p^{n}$ , equation (5.7) implies

(\operatorname{adj}Q^{-1}\underline{M})(\chi^{j}\omega)=\begin{pmatrix}\dfrac{1}{\beta^{n}}*&\dfrac{-1}{\alpha^{n}}*\\[12.0pt] \dfrac{-1}{\beta^{n}}*^{\prime}&\dfrac{1}{\alpha^{n}}*^{\prime}\end{pmatrix},

where $*,*^{\prime}$ are in $\overline{\mathbb{Q}_{p}}$ .

Thus, if we write $F_{1}=P_{4}F_{\alpha}-P_{2}F_{\beta},$ then, from (5.7), we get

	$\displaystyle F_{1}(\chi^{j}\omega)$	$\displaystyle=P_{4}(\chi^{j}\omega)F_{\alpha}(\chi^{j}\omega)-P_{2}(\chi^{j}\omega)F_{\beta}(\chi^{j}\omega),$
		$\displaystyle=\frac{1}{\beta^{n}}\frac{C_{\omega,j}}{\alpha^{n}}-\frac{1}{\alpha^{n}}\frac{C_{\omega,j}}{\beta^{n}},$
		$\displaystyle=0.$

Similarly, if we write $F_{2}=-P_{3}F_{\alpha}+P_{1}F_{\beta}$ , then $F_{2}(\chi^{j}\omega)=0$ .

Hence, for every positive integer $n$ , the zeros of $\Phi_{n-1,k-1}$ are also zeros of $F_{1}$ and $F_{2}$ . In other words, the roots of $\det(Q^{-1}\underline{M})=(\text{some constant})\dfrac{\log_{p,k-1}(\gamma_{0})}{\delta_{k-1}(\gamma_{0}-1)}$ are also the roots of $F_{1}$ , and $F_{2}$ . Therefore, $\det Q^{-1}\underline{M}$ divides both $F_{1},F_{2}$ in $\mathcal{H}_{E}(\Gamma)$ .

Note that $P_{4}F_{\alpha}$ is $O(\log_{p}^{k-1})$ , since $P_{4}$ is $O(\log_{p}^{v_{p}(\beta)})$ and $F_{\alpha}$ is $O(\log_{p}^{v_{p}(\alpha)})$ . Similarly, $P_{2}F_{\beta},P_{3}F_{\alpha},$ and $P_{1}F_{\beta}$ are $O(\log_{p}^{k-1})$ .

Let

(5.8)

\begin{matrix}F_{\sharp}=\dfrac{P_{4}F_{\alpha}-P_{2}F_{\beta}}{\det Q^{-1}\underline{M}}&\text{and}&F_{\flat}=\dfrac{-P_{3}F_{\alpha}+P_{1}F_{\beta}}{\det Q^{-1}\underline{M}}.\end{matrix}

Then $F_{\sharp}$ and $F_{\flat}$ have bounded coefficients since the numerators of both of them are $O(\log_{p}^{k-1})$ , denominators of both of them are $\det(Q^{-1}\underline{M})$ and by Lemma 5.4 $\det(Q^{-1}\underline{M})\sim\log_{p}^{k-1}$ . Hence $F_{\sharp}$ and $F_{\flat}$ are $O(1)$ (i.e. bounded). Therefore, we can conclude that $F_{\sharp}$ and $F_{\flat}$ are in $\Lambda_{E}(\Gamma)$ . This completes the proof.∎

6. Preliminaries about ray class groups, Hecke characters, and the two variable distribution algebras

For the rest of the article, we fix a quadratic imaginary field $K$ and a prime $p\geq 3$ which splits in $K$ as $p\mathcal{O}_{K}=\mathfrak{p}\overline{\mathfrak{p}}$ . Let $h_{K}\geq 1$ be the class number of $K$ .

Assumption 6.1.

Throughout the article, we assume that $p\not\mid h_{K}$ .

6.1. Ray class groups and ray class fields

Let $K_{\infty}$ be the unique $\mathbb{Z}^{2}_{p}$ extension of $K$ . If $\mathcal{I}$ is an ideal of $K$ , we write $G_{\mathcal{I}}$ for the ray class group $K$ modulo $\mathcal{I}$ . We define

\text{Cl}(K,p^{\infty})=G_{p^{\infty}}=\varprojlim_{n}G_{(p)^{n}},\hskip 3.61371ptG_{\mathfrak{p}^{\infty}}=\varprojlim_{n}G_{\mathfrak{p}^{n}},\hskip 3.61371ptG_{\overline{\mathfrak{p}}^{\infty}}=\varprojlim_{n}G_{\overline{\mathfrak{p}}^{n}}.

These are the Galois groups of the ray class fields $K(p^{\infty}),K(\mathfrak{p}^{\infty})$ and $K(\overline{\mathfrak{p}}^{\infty})$ respectively. For $\mathfrak{q}\in\{\mathfrak{p},\overline{\mathfrak{p}}\}$ , let $H(\mathfrak{q}^{\infty})$ be the subfield of $G_{\mathfrak{q}^{\infty}}$ such that $\operatorname{Gal}(H(\mathfrak{q}^{\infty})/K)\cong\mathbb{Z}_{p}$ . Note $H(\mathfrak{q}^{\infty})\subset K_{\infty}$ .

Remark 6.2.

We have an isomorphism $G_{p^{\infty}}\cong\Delta_{K}\times\mathbb{Z}_{p}\times\mathbb{Z}_{p}\cong\Delta\times\overline{\langle\gamma_{\mathfrak{p}}\rangle}\times\overline{\langle\gamma_{\overline{\mathfrak{p}}}\rangle}$ , where $\Delta_{K}$ is a finite abelian group, $\gamma_{\mathfrak{p}}$ and $\gamma_{\overline{\mathfrak{p}}}$ topologically generate $\mathbb{Z}_{p}$ parts of $G_{\mathfrak{p}^{\infty}}$ and $G_{\overline{\mathfrak{p}}^{\infty}}$ respectively.

Remark 6.3.

By the assumption $p\not\mid h_{K}$ , there exists a unique prime in $K_{\infty}$ above $\mathfrak{p}$ and a unique prime above $\overline{\mathfrak{p}}$ . By an abuse of notation, we will also denote by $\mathfrak{p}$ and $\overline{\mathfrak{p}}$ by the unique prime above $\mathfrak{p}$ and $\overline{\mathfrak{p}}$ respectively in $K_{\infty}$ . Therefore, for $\mathfrak{q}\in\{\mathfrak{p},\overline{\mathfrak{p}}\}$ , $\operatorname{Gal}(H(\mathfrak{q}^{\infty})_{\mathfrak{q}}/K_{\mathfrak{q}})=\overline{\langle\gamma_{\mathfrak{q}}\rangle}\cong\operatorname{Gal}(H(\mathfrak{q}^{\infty})/K)\cong\mathbb{Z}_{p}$ .

Since $p$ splits in $K$ , the local field $K_{\mathfrak{q}}$ is isomorphic to $\mathbb{Q}_{p}$ , for $\mathfrak{q}\in\{\mathfrak{p},\overline{\mathfrak{p}}\}$ . Thus, $1+\pi_{\mathfrak{q}}\mathcal{O}_{K_{\mathfrak{q}}}\cong 1+p\mathbb{Z}_{p}$ , where $\pi_{\mathfrak{q}}$ is a uniformizer of $\mathcal{O}_{K_{\mathfrak{q}}}$ . Recall the topological generator $u\in 1+p\mathbb{Z}_{p}$ such that $\chi(\gamma_{0})=u$ , where $\gamma_{0}$ generates $\Gamma_{1}\cong\mathbb{Z}_{p}$ , and $\chi$ is $p$ -adic cyclotomic character. Thus, we may set $u_{\mathfrak{p}}=u_{\overline{\mathfrak{p}}}=u$ , where $u_{\mathfrak{q}}$ is a topological generator of $1+\pi_{\mathfrak{q}}\mathcal{O}_{K_{\mathfrak{q}}}$ . From now on, we fix this $u$ .

By local class field theory, there exists a group isomorphism (Artin map)

\operatorname{Art}_{\mathfrak{q}}:\mathcal{O}_{K_{\mathfrak{q}}}^{\times}\to\operatorname{Gal}(H(\mathfrak{q}^{\infty})_{\mathfrak{q}}/K_{\mathfrak{q}})\cong\operatorname{Gal}(H(\mathfrak{q}^{\infty})/K),

such that

\operatorname{Art}_{\mathfrak{q}}(u_{\mathfrak{q}})=\operatorname{Art}_{\mathfrak{q}}(u)=\gamma_{\mathfrak{q}},

where $\gamma_{\mathfrak{q}}$ is a topological generator of $\operatorname{Gal}(H(\mathfrak{q}^{\infty})_{\mathfrak{q}}/K_{\mathfrak{q}})$ . By an abuse of notations, let $\gamma_{\mathfrak{q}}$ be a topological generator of $\operatorname{Gal}(H(\mathfrak{q}^{\infty})/K)$ and $\operatorname{Art}_{\mathfrak{q}}(u)=\gamma_{\mathfrak{q}}$ .

6.2. Hecke characters as the characters on the ray class groups

Let $\mathbb{A}_{K}^{\times}$ denote the group of ideles of $K$ and write $\mathbb{A}_{K}^{\times}=\mathbb{A}_{\infty}^{\times}\times\mathbb{A}_{f}^{\times}$ , where $\mathbb{A}_{\infty}^{\times}$ is the infinite part and $\mathbb{A}_{f}^{\times}$ is the finite part. We can embed $K^{\times}$ into $\mathbb{A}_{K}^{\times}$ diagonally. Fix embeddings $i_{\infty}:\overline{\mathbb{Q}}\hookrightarrow\mathbb{C}$ and $i_{p}:\overline{\mathbb{Q}}\hookrightarrow\mathbb{C}_{p}$ .

Definition 6.4 (Hecke characters).

(1)

A Hecke character $\Xi$ of $K$ is a continuous homomorphism $\Xi:\mathbb{A}_{K}^{\times}\to\mathbb{C}^{\times}$ that is trivial on $K^{\times}$ . In other words, a Hecke character of $K$ is a continuous homomorphism $\Xi:K^{\times}\backslash\mathbb{A}_{K}^{\times}\to\mathbb{C}^{\times}$ .
(2)

We say a Hecke character $\Xi$ is algebraic if for each embedding $\kappa:K\hookrightarrow\mathbb{C}$ , there exists $n_{\kappa}\in\mathbb{Z}$ such that $\Xi(x)=\prod_{\kappa}(\kappa(x))^{-n_{\kappa}}$ for each $x$ in the connected component of the identity in $K_{\infty}^{\times}$ .
(3)

Let $\Xi:K^{\times}\backslash\mathbb{A}_{K}^{\times}\to\mathbb{C}^{\times}$ be an algebraic Hecke character of $K$ . We say that $\Xi$ has infinity type $(q,r)\in\mathbb{Z}^{2}$ if $\Xi_{\infty}(z)=z^{q}\overline{z}^{r}$ , where for each place $v$ of $K$ , we let $\Xi_{v}:K_{v}^{\times}\to\mathbb{C}^{\times}$ be the $v$ -component of $\Xi$ .

From now on, all the Hecke characters mentioned in this article are algebraic Hecke characters. We can view Hecke characters as $p$ -adic characters:

Definition 6.5 ( $p$ -adic avatar of an algebraic Hecke character).

Let $\Xi$ be a Hecke character of $K$ of conductor $\mathfrak{p}^{n_{\mathfrak{p}}}\overline{\mathfrak{p}}^{n_{\overline{\mathfrak{p}}}}$ and infinity type $(a,b)$ . The $p$ -adic avatar of $\Xi$ is defined as

\widehat{\Xi}(x)\coloneqq x_{\mathfrak{p}}^{a}x_{\overline{\mathfrak{p}}}^{b}i_{p}(\Xi(x_{fin})).

By the class field theory, the correspondence $\Xi\mapsto\widehat{\Xi}$ establishes a bijection between the set of algebraic Hecke characters of $K$ of conductor dividing $(p^{\infty})$ and the set of locally algebraic $\overline{\mathbb{Q}_{p}}$ -valued characters of $G_{p^{\infty}}$ .

Now we combine this $p$ -adic avatar of the Hecke character and the global Artin reciprocity map $\operatorname{rec}$ to define a Galois character on the ray class group $G_{p^{\infty}}$ .

Let $\Xi$ be a Hecke character of $K$ of conductor $\mathfrak{p}^{n_{\mathfrak{p}}}\overline{\mathfrak{p}}^{n_{\overline{\mathfrak{p}}}}$ and infinity type $(a,b)$ .

Definition 6.6.

The Galois character of $\Xi$ is $\widetilde{\Xi}:G^{ab}_{K}\to\mathbb{C}_{p}^{\times}$ given by

\widetilde{\Xi}=\widehat{\Xi}\circ\operatorname{rec}^{-1},

where $\operatorname{rec}$ is the global Artin isomorphism.

Note that $G_{p^{\infty}}\subset G^{ab}_{K}$ and Remark 6.3 implies $\operatorname{rec}^{-1}(\gamma_{\mathfrak{q}})=\operatorname{Art}_{\mathfrak{q}}^{-1}(\gamma_{\mathfrak{q}})$ .

By an abuse of notation, let $\widetilde{\Xi}$ denote $\widetilde{\Xi}\mid_{G_{p^{\infty}}}$ . Define $\widetilde{\Xi}_{\mathfrak{q}}\coloneqq\widetilde{\Xi}|_{\overline{\langle\gamma_{\mathfrak{q}}\rangle}}$ . Hence, if $\Xi$ is a Hecke character of the infinity type $(a,b)$ and conductor $\mathfrak{p}^{n_{\mathfrak{p}}}\overline{\mathfrak{p}}^{n_{\overline{\mathfrak{p}}}}$ , then

\widetilde{\Xi}_{\mathfrak{p}}(\gamma_{\mathfrak{p}})=u^{a}\zeta_{\mathfrak{p}},

and

\widetilde{\Xi}_{\overline{\mathfrak{p}}}(\gamma_{\overline{\mathfrak{p}}})=u^{b}\zeta_{\overline{\mathfrak{p}}},

where $\zeta_{\mathfrak{q}}$ is a $(p^{n_{\mathfrak{q}}-1})$ -th root of unity. In other words, for any $\sigma\in\Delta_{K},$

(6.1)

\widetilde{\Xi}(\sigma\times\gamma_{\mathfrak{p}}\times\gamma_{\overline{\mathfrak{p}}})=\widetilde{\Xi}_{\mathfrak{p}}(\gamma_{\mathfrak{p}})\cdot\widetilde{\Xi}_{\overline{\mathfrak{p}}}(\gamma_{\overline{\mathfrak{p}}})\times\text{some root of unity}.

From the above discussion, we get

\widecheck{\Xi}\coloneqq\widetilde{\Xi}|_{\overline{\langle\gamma_{\mathfrak{p}}\rangle}\times\overline{\langle\gamma_{\overline{\mathfrak{p}}}\rangle}}=\widetilde{\Xi}_{\mathfrak{p}}\cdot\widetilde{\Xi}_{\overline{\mathfrak{p}}}.

Moreover, for any Hecke character $\omega_{1}$ of infinity type $(a,0)$ and conductor $\mathfrak{p}^{n_{\mathfrak{p}}}$ and Hecke character $\omega_{2}$ of infinity type $(0,b),$ and conductor $\overline{\mathfrak{p}}^{n_{\overline{\mathfrak{p}}}}$ , then the product $\widetilde{\omega_{1}}_{\mathfrak{p}}\cdot\widetilde{\omega_{2}}_{\overline{\mathfrak{p}}}$ is a character on $\overline{\langle\gamma_{\mathfrak{p}}\rangle}\times\overline{\langle\gamma_{\overline{\mathfrak{p}}}\rangle}$ .

6.3. Two variable distribution modules

We will extend $\mathcal{H}_{E}(\Gamma_{1})$ from previous sections. Let

\mathcal{H}_{\mathbb{C}_{p},r}(\Gamma_{1})=\{f(\gamma_{0}-1)\colon f(X)=\sum_{n\geq 0}a_{n}X^{n}\in\mathbb{C}_{p}[[X]],\sup_{n}(n^{-r}|a_{n}|_{p})<\infty\}.

and

\mathcal{H}_{\mathbb{C}_{p}}(\Gamma_{1})=\bigcup_{r\geq 0}\mathcal{H}_{\mathbb{C}_{p},r}(\Gamma_{1}).

Note that $\mathcal{H}_{E}(\Gamma_{1})=E[[\gamma_{0}-1]]\bigcap\mathcal{H}_{\mathbb{C}_{p}}(\Gamma_{1})$ , for any field extension $E$ of $\mathbb{Q}_{p}$ .

Define a map $\tau_{\mathfrak{q}}=\operatorname{Art}_{\mathfrak{q}}\circ\chi$ between $\Gamma_{1}$ and $\Gamma_{\mathfrak{q}}\coloneqq\operatorname{Gal}(H(\mathfrak{q}^{\infty})/K)$ which sends $\gamma_{0}$ to $\gamma_{\mathfrak{q}}$ . This change of variable map can be extended to ring isomorphism

	$\displaystyle\tau_{\mathfrak{q}}\colon\mathcal{H}_{\mathbb{C}_{p}}(\Gamma_{1})$	$\displaystyle\to\mathcal{H}_{\mathbb{C}_{p}}(\Gamma_{\mathfrak{q}}),$
	$\displaystyle f(\gamma_{0}-1)$	$\displaystyle\mapsto f(\gamma_{\mathfrak{q}}-1).$

Similarly, for any field extension $E$ of $\mathbb{Q}_{p}$ , we again have an isomorphism

\tau_{\mathfrak{q}}:\mathcal{H}_{E}(\Gamma_{1})\to\mathcal{H}_{E}(\Gamma_{\mathfrak{q}}).

For any finite extension $E$ of $\mathbb{Q}_{p}$ , let

\Lambda_{E}(G_{p^{\infty}})=E[\Delta_{K}]\otimes\Lambda_{E}(\Gamma_{\mathfrak{p}})\widehat{\otimes}\Lambda_{E}(\Gamma_{\overline{\mathfrak{p}}}),

where $\Lambda_{E}(\Gamma_{\mathfrak{q}})$ is the Iwasawa algebra $E\otimes\mathcal{O}_{E}[[\Gamma_{\mathfrak{q}}]]$ for $\mathfrak{q}\in\{\mathfrak{p},\overline{\mathfrak{p}}\}$ . The two-variable Iwasawa algebra $\Lambda_{E}(G_{p^{\infty}})$ is isomorphic to the power series ring

E[\Delta_{K}]\otimes\mathcal{O}_{E}[[T_{1},T_{2}]],

by identifying $\gamma_{\mathfrak{p}}-1$ with $T_{1}$ and $\gamma_{\overline{\mathfrak{p}}}-1$ with $T_{2}$ .

We define the

\mathcal{H}_{\mathbb{C}_{p},r,s}\coloneqq\mathcal{H}_{\mathbb{C}_{p},r}(\Gamma_{\mathfrak{p}})\widehat{\otimes}\mathcal{H}_{\mathbb{C}_{p},s}(\Gamma_{\overline{\mathfrak{p}}}),

and

\mathcal{H}_{E,r,s}\coloneqq\mathcal{H}_{\mathbb{C}_{p},r,s}\cap E[[\gamma_{\mathfrak{p}}-1,\gamma_{\overline{\mathfrak{p}}}-1]],

for any extension $E$ of $\mathbb{Q}_{p}$ . We also define

\mathcal{H}_{\mathbb{C}_{p},r,s}(G_{p^{\infty}})\coloneqq\mathbb{C}_{p}[\Delta_{K}]\otimes\mathcal{H}_{\mathbb{C}_{p},r,s},

and

\mathcal{H}_{E,r,s}(G_{p^{\infty}})\coloneqq E[\Delta_{K}]\otimes\mathcal{H}_{E,r,s},

for any extension $E$ of $\mathbb{Q}_{p}$ and $\Delta$ is the finite abelian group appearing in the Galois group $G_{p^{\infty}}$ .

Note that

\Lambda_{E}(G_{p^{\infty}})=\mathcal{H}_{E,0,0}(G_{p^{\infty}}).

Moreover, $\mathcal{H}_{E,r,s}$ is a $\Lambda_{E}(\Gamma_{\mathfrak{p}})\widehat{\otimes}\Lambda_{E}(\Gamma_{\overline{\mathfrak{p}}})$ -module, and $\mathcal{H}_{E,r,s}(G_{p^{\infty}})$ is a $\Lambda_{E}(G_{p^{\infty}})$ -module.

For $F\in\mathcal{H}_{E,r,s}$ , and a Hecke character $\Xi$ of the infinity type $(a,b)$ and conductor $\mathfrak{p}^{n_{\mathfrak{p}}}\overline{\mathfrak{p}}^{n_{\overline{\mathfrak{p}}}}$ , define

(6.2)		$\displaystyle F^{(\widetilde{\Xi}_{\mathfrak{p}})}$	$\displaystyle\coloneqq F(u^{a}\zeta_{\mathfrak{p}}-1,\gamma_{\overline{\mathfrak{p}}}-1),$
(6.3)		$\displaystyle F^{(\widetilde{\Xi}_{\overline{\mathfrak{p}}})}$	$\displaystyle\coloneqq F(\gamma_{\mathfrak{p}}-1,u^{b}\zeta_{\overline{\mathfrak{p}}}-1),$

where $\zeta_{\mathfrak{q}}$ is a primitive $p^{(n_{\mathfrak{q}}-1)}$ -th root of unity for $\mathfrak{q}\in\{\mathfrak{p},\overline{\mathfrak{p}}\}$ .

Lemma 6.7.

Let $E$ be a finite extension of $\mathbb{Q}_{p}$ , $F\in\mathcal{H}_{E,r,s}$ be a power series and let $\Xi$ be a Hecke character of $K$ of infinity type $(a,b)$ and conductor $\mathfrak{p}^{n_{\mathfrak{p}}}.\overline{\mathfrak{p}}^{n_{\overline{\mathfrak{p}}}}$ . Then, $F^{(\widetilde{\Xi}_{\mathfrak{p}})}\in\mathcal{H}_{E(\widetilde{\Xi}_{\mathfrak{p}}(\gamma_{\mathfrak{p}})),s}(\Gamma_{\overline{\mathfrak{p}}})$ and $F^{(\widetilde{\Xi}_{\overline{\mathfrak{p}}})}\in\mathcal{H}_{E(\widetilde{\Xi}_{\overline{\mathfrak{p}}}(\gamma_{\overline{\mathfrak{p}}})),r}(\Gamma_{\mathfrak{p}})$ , where $E(\widetilde{\Xi}_{\mathfrak{q}}(\gamma_{\mathfrak{q}}))$ is finite extension of E by adjoining values of $\widetilde{\Xi}_{\mathfrak{q}}(\gamma_{\mathfrak{q}})$ for $\mathfrak{q}\in\{\mathfrak{p},\overline{\mathfrak{p}}\}$ .

Proof.

The power series $F$ belongs to the power series ring $\mathcal{H}_{E,r,s}$ which is completed tensor product of $\mathcal{H}_{E,r}(\Gamma_{\mathfrak{p}})$ and $\mathcal{H}_{E,s}(\Gamma_{\overline{\mathfrak{p}}})$ . Thus, if we substitute $\gamma_{\mathfrak{p}}$ by $u^{a}\zeta_{\mathfrak{p}}$ , then $F^{(\widetilde{\Xi}_{\mathfrak{p}})}$ will be a power series with growth $s$ and the coefficients of $F^{(\widetilde{\Xi}_{\mathfrak{p}})}$ will be in $E(\widetilde{\Xi}_{\mathfrak{p}}(\gamma_{\mathfrak{p}}))$ .

Similarly, substituting $\gamma_{\overline{\mathfrak{p}}}$ by $u^{b}\zeta_{\overline{\mathfrak{p}}}$ , then $F^{(\widetilde{\Xi}_{\overline{\mathfrak{p}}})}\in\mathcal{H}_{E(\widetilde{\Xi}_{\overline{\mathfrak{p}}}(\overline{\mathfrak{p}})),s}(\Gamma_{\mathfrak{p}})$ . ∎

Isotypical components. Let $\eta:\Delta_{K}\to\mathbb{Z}_{p}^{\times}$ (or $\eta:\Delta_{K}\to\overline{\mathbb{Q}_{p}}^{\times}$ )be a character, where $\Delta_{K}$ is a finite abelian subgroup appearing in $G_{p^{\infty}}$ . Write $e_{\eta}=\frac{1}{|\Delta_{K}|}\sum_{\sigma\in\Delta}\eta^{-1}(\sigma)\sigma$ . For $*\in\{\mathbb{C}_{p},E\}$ , if $F\in\mathcal{H}_{*,r,s}(G_{p^{\infty}})$ , write $F^{\eta}=e_{\eta}(F)$ for its image in $e_{\eta}(\mathcal{H}_{*,r,s}(G_{p^{\infty}}))\cong\mathcal{H}_{*,r,s}$ . Note that this isomorphism is a ring isomorphism If $\eta=1$ , we simply write $F^{\Delta_{K}}$ instead of $F^{\eta}$ . Note that $F^{\Delta_{K}}\in\mathcal{H}_{*,r,s}$ .

7. Bianchi modular forms and their $p$ -adic $L$ -functions

In this section, we will briefly recall the definition of Bianchi modular forms, their $L$ -functions, and the $p$ -adic $L$ -functions constructed by Williams in [35].

7.1. Bianchi modular forms

We define Bianchi modular forms adelically. Fix a quadratic imaginary field $K$ .

Definition 7.1 (Level).

For any integral ideal $\mathcal{N}$ of $\mathcal{O}_{K}$ , we let

\Omega_{1}(\mathcal{N})\coloneqq\left\{\begin{pmatrix}a&b\\ c&d\end{pmatrix}\in\text{GL}_{2}(\widehat{\mathcal{O}_{K}})\colon c\in\mathcal{N}\widehat{\mathcal{O}_{K}},a,d\in 1+\mathcal{N}\widehat{\mathcal{O}_{K}}\right\},

where $\widehat{\mathcal{O}_{K}}=\mathcal{O}_{K}\otimes\widehat{\mathbb{Z}}$ .

Henceforth, we will always take $k\geq 2$ to be an integer.

For any ring $R$ , $V_{2k-2}(R)$ denote the ring of homogeneous polynomials over $R$ in two variables of degree $2k-2$ .

Definition 7.2 (Bianchi modular cusp forms).

We say a function

\mathcal{F}:\text{GL}_{2}(\mathbb{A}_{K})\to V_{2k-2}(\mathbb{C})

is a cusp form of weight $(k,k)$ and level $\mathcal{N}$ (i.e. level $\Omega_{1}(\mathcal{N})$ ) if it satisfies

(1)

For all $u\in\text{SU}_{2}(\mathbb{C})$ and $z\in\mathbb{C}^{\times}$ ,

$\mathcal{F}(zgu)=\mathcal{F}(g)\rho(u,z),$

where $\rho:\text{SU}_{2}(\mathbb{C})\times\mathbb{C}^{\times}\to\text{GL}(V_{2k-2}(\mathbb{C}))$ is an antihomomorphism.
(2)

The function $\mathcal{F}$ is right-invariant under the group $\Omega_{1}(\mathcal{N})$ .
(3)

The function $\mathcal{F}$ is left-invariant under the group $\text{GL}_{2}(K)$ .
(4)

The function $\mathcal{F}$ is an eigenfunction of a Casimir operator $\partial$ ,
(5)

The function $\mathcal{F}$ satisfies the cuspidal condition for all $g\in\text{GL}_{2}(\mathbb{A}_{K})$ , that is, we have

$\int_{K\backslash\mathbb{A}_{K}}\mathcal{F}(ug)\,du=0.$

Remarks 7.3.

(1)

In a result by Harder, he showed that if $\mathcal{F}$ is Bianchi modular cusp form of weight $(k_{1},k_{2})$ , then $k_{1}=k_{2}$ . See [12].
(2)

A cusp form $\mathcal{F}$ descends to give a collection of $h$ automorphic forms $\mathcal{F}^{i}:\mathcal{H}_{3}\to V_{2k-2}(\mathbb{C})$ , where $h$ is the class number of $K$ , and $\mathcal{H}_{3}\coloneqq\text{GL}_{2}(\mathbb{C})/[\text{SU}_{2}(\mathbb{C})Z(\text{GL}_{2}(\mathbb{C}))]$ is the hyperbolic $3$ -space.
(3)

For Fourier expansion of Bianchi modular forms, see [35, Section 1.2].

Definition 7.4 (Twisted $L$ -function of a Bianchi modular form).

Let $\mathcal{F}$ be a Bianchi modular cusp form of any weight and level $\mathcal{N}$ . Let $\Xi$ be a Hecke character of conductor $\mathfrak{f}$ . The twisted $L$ -function of $\mathcal{F}$ is defined as

L(\mathcal{F},\Xi,s)=\sum_{\begin{subarray}{c}0\neq\mathfrak{m}\subset\mathcal{O}_{K},\\ (\mathfrak{f},\mathfrak{m})=1\end{subarray}}c(\mathfrak{m},\mathcal{F})\Xi(\mathfrak{m})N(\mathfrak{m})^{-s},

where $c(\mathfrak{m},\mathcal{F})$ is the $\mathfrak{m}$ -th Fourier coefficient of $\mathcal{F}$ and $s\in\mathbb{C}$ in some suitable right-half plane.

We renormalize this $L$ -function, using Deligne’s $\Gamma$ -factor. We define

\Lambda(\mathcal{F},\Xi,s)=\dfrac{\Gamma(q+s)\Gamma(r+s)}{(2\pi\iota)^{q+s}(2\pi\iota)^{r+s}}L(\mathcal{F},\Xi,s),

where $(q,r)$ is the infinity type of $\Xi$ .

7.2. $p$ -adic $L$ -function of Bianchi modular forms

For the quadratic imaginary field $K/\mathbb{Q}$ with class number $h_{K}$ and discriminant $-D$ , let $p$ be an odd prime splitting in $K$ as $(p)=\mathfrak{p}\overline{\mathfrak{p}}$ . Let $e_{\mathfrak{q}}$ denote the ramification index of the prime $\mathfrak{q}$ in $K$ .

Theorem 7.5 (Williams, [35, Theorem 7.4]).

Let $\mathcal{F}$ be a Bianchi modular cusp form of weight $(k,k)$ and level $\mathcal{N}$ such that $(p)\mid\mathcal{N}$ . Let $a_{\mathfrak{p}}$ denote the $U_{\mathfrak{p}}$ -eigenvalues of $\mathcal{F}$ where $v_{p}(a_{\mathfrak{p}})<(k+1)/e_{\mathfrak{p}}$ for all $\mathfrak{p}\mid p$ . For any ideal $\mathfrak{f}$ , we define the operator $U_{\mathfrak{f}}$ as

U_{\mathfrak{f}}\coloneqq\prod_{\mathfrak{p}^{n}\mid\mid\mathfrak{f}}U_{\mathfrak{p}}^{n}.

Then there exists a locally analytic distribution $L_{p,\mathcal{F}}$ on the ray class group $G_{p^{\infty}}$ such that for any Hecke character $\Xi$ of infinity type $(0,0)\leq(a,b)\leq(k-2,k-2)$ and conductor $\mathfrak{f}$ , we have

(7.1)

L_{p,\mathcal{F}}(\widetilde{\Xi})=(\text{explicit factor})\dfrac{1}{a_{\mathfrak{f}}}\dfrac{\Lambda(\mathcal{F},\Xi)}{\Omega_{\mathcal{F}}},

where $a_{\mathfrak{f}}$ is $U_{\mathfrak{f}}$ -eigenvalue of $\mathcal{F}$ , $\Lambda(\mathcal{F},\Xi)$ is renormalized $L$ -series, and $\Omega_{\mathcal{F}}$ is a complex period.

The distribution $L_{p}$ is $(v_{p}(\lambda_{\mathfrak{p}}))_{\mathfrak{p}\mid p}$ -admissible and is unique with these interpolation and growth properties.

Remark 7.6.

In [35, Section 6.3], Williams has defined the admissibility for locally analytic distributions on $\mathcal{O}_{K}\otimes\mathbb{Z}_{p}$ . But using methods from [24], we can extend the notion of admissibility for locally analytic distributions on $\mathcal{O}_{K}\otimes\mathbb{Z}_{p}$ to locally analytic distributions on ray class group $G_{p^{\infty}}$ . See [35, Section 7.4] for more details.

Remark 7.7.

For real numbers $r,s\geq 0$ and let $E$ be a finite extension of $\mathbb{Q}_{p}$ , we define $D^{(r,s)}(G_{p^{\infty}},E)$ to be the set of distributions $\mu$ of $G_{p^{\infty}}$ such that for fixed integers $m,n\geq 0$ ,

\inf_{g\in G_{p^{\infty}}}v_{p}(\mu(1_{g\langle\gamma_{\mathfrak{p}}\rangle^{p^{m}}\langle\gamma_{\overline{\mathfrak{p}}}\rangle^{p^{n}}}))\geq R-um-vn,

for some constant $R\in\mathbb{R}$ which only depends on $\mu$ .

Since $p$ splits in the quadratic imaginary field $K$ , we can identify $D^{(u,v)}(G_{p^{\infty}},E)$ with $\mathcal{H}_{E,u,v}(G_{p^{\infty}})$ . See [19, Section 2.1] for the details.

Therefore if $p$ splits in $K$ as $(p)=\mathfrak{p}\overline{\mathfrak{p}}$ , and $\mathcal{F}$ is a Bianchi modular form which satisfies the conditions of the above theorem, then $L_{p,\mathcal{F}}$ is a $(v_{p}(a_{\mathfrak{p}}),v_{p}(a_{\overline{\mathfrak{p}}}))$ -admissible locally analytic distribution on $G_{p^{\infty}}$ , that is, by [35] and [24],

(7.2)

L_{p,\mathcal{F}}\in D^{(v_{p}(a_{\mathfrak{p}}),v_{p}(a_{\overline{\mathfrak{p}}}))}(G_{p^{\infty}},F),

where $F$ is a finite extension of $\mathbb{Q}_{p}$ containing $a_{\mathfrak{p}}$ and $a_{\overline{\mathfrak{p}}}$ . By Remark 7.7, we conclude

L_{p,\mathcal{F}}\in\mathcal{H}_{F,v_{p}(a_{\mathfrak{p}}),v_{p}(a_{\overline{\mathfrak{p}}})}(G_{p^{\infty}}).

8. Factorisation of $p$ -adic $L$ -functions of Bianchi modular forms

In this section, we first define $p$ -stabilizations of Bianchi modular forms. Next, we modify our logarithmic matrix $\underline{M}$ and prove the main factorization theorem of two-variable $p$ -adic $L$ -functions.

8.1. $p$ -stabilization of Bianchi modular forms

We begin with a Bianchi modular form $\mathcal{F}$ of weight $(k,k)$ and level $\mathfrak{n}$ such that $(p\mathcal{O}_{K},\mathfrak{n})=1$ . We further assume $\mathcal{F}$ is a Hecke eigenform and for $\mathfrak{q}\in\{\mathfrak{p},\mathfrak{p}\}$ , we have $T_{\mathfrak{q}}\mathcal{F}=a_{\mathfrak{q}}\mathcal{F}$ . Note that the norm of $\mathfrak{q}$ is $p$ . Consider the Hecke polynomial $X^{2}-a_{\mathfrak{q}}X+p^{k-1}=(X-\alpha_{\mathfrak{q}})(X-\beta_{\mathfrak{q}})$ .

There are four $p$ -stabilisations $\mathcal{F}^{\alpha_{\mathfrak{p}},\alpha_{\overline{\mathfrak{p}}}},\mathcal{F}^{\alpha_{\mathfrak{p}},\beta_{\overline{\mathfrak{p}}}},\mathcal{F}^{\beta_{\mathfrak{p}},\alpha_{\overline{\mathfrak{p}}}},$ and $\mathcal{F}^{\beta_{\mathfrak{p}},\beta_{\overline{\mathfrak{p}}}}$ of level $p\mathfrak{n}$ , such that for $*\in\{\alpha_{\mathfrak{p}},\beta_{\overline{\mathfrak{p}}}\}$ and $\dagger\in\{\alpha_{\overline{\mathfrak{p}}},\beta_{\overline{\mathfrak{p}}}\}$ , we have

(8.1)		$\displaystyle U_{\mathfrak{p}}(\mathcal{F}^{*,\dagger})$	$\displaystyle=\mathcal{F}^{,\dagger},$
(8.2)		$\displaystyle U_{\overline{\mathfrak{p}}}(\mathcal{F}^{*,\dagger})$	$\displaystyle=\dagger\mathcal{F}^{*,\dagger}.$

For more details about $p$ -stabilizations, refer to [27, Section 3.3].

Assumption 8.1.

Throughout the article, we assume

(1)

$\mathcal{F}$ is non-ordinary at both the primes $\mathfrak{p}$ and $\overline{\mathfrak{p}}$ i.e. $v_{p}(a_{\mathfrak{p}})$ and $v_{p}(a_{\overline{\mathfrak{p}}})>0$ .
(2)

$v_{p}(a_{\mathfrak{q}})>\left\lfloor\dfrac{k-2}{p-1}\right\rfloor$ .

Let $E$ be a finite extension of $\mathbb{Q}_{p}$ which contains $\alpha_{\mathfrak{p}},\alpha_{\overline{\mathfrak{p}}},\beta_{\mathfrak{p}}$ and $\beta_{\overline{\mathfrak{p}}}$ .

8.2. Modified logarithmic matrices

Recall the ring isomorphism $\tau_{\mathfrak{q}}:\mathcal{H}_{F}(\Gamma_{1})\to\mathcal{H}_{F}(\Gamma_{\mathfrak{q}})$ , for $\mathfrak{q}\in\{\mathfrak{p},\overline{\mathfrak{p}}\}$ . Consider the change of variable map between matrices induced by $\tau_{\mathfrak{q}}$ :

\operatorname{Mat}_{\mathfrak{q}}:M_{2,2}(\mathcal{H}_{F}(\Gamma_{1}))\to M_{2,2}(\mathcal{H}_{F}(\Gamma_{\mathfrak{q}})),

such that

\operatorname{Mat}_{\mathfrak{q}}\left(\begin{pmatrix}A&B\\[6.0pt] C&D\end{pmatrix}\right)=\begin{pmatrix}\tau_{\mathfrak{q}}(A)&\tau_{\mathfrak{q}}(B)\\[6.0pt] \tau_{\mathfrak{q}}(C)&\tau_{\mathfrak{q}}(D)\end{pmatrix}.

Note that this map is also a ring isomorphism.

Let $\mathcal{F}$ be a Bianchi modular form of level $\mathfrak{n}$ coprime to $p$ and let $a_{\mathfrak{q}}$ and $\beta_{\mathfrak{q}}$ be the $T_{\mathfrak{q}}$ -eigenvalues, for $\mathfrak{q}\in\{\mathfrak{p},\overline{\mathfrak{p}}\}$ . Let

	$\displaystyle A_{\varphi,\mathfrak{q}}$	$\displaystyle=\begin{pmatrix}0&\dfrac{-1}{p^{k-1}}\\[12.0pt] 1&\dfrac{a_{\mathfrak{q}}}{p^{k-1}},\end{pmatrix},$
	$\displaystyle Q_{\mathfrak{q}}$	$\displaystyle=\begin{pmatrix}\alpha_{\mathfrak{q}}&-\beta_{\mathfrak{q}}\\[6.0pt] -p^{k-1}&p^{k-1}\end{pmatrix},$
	$\displaystyle\underline{M_{\mathfrak{q}}}$	$\displaystyle=\operatorname{Mat}_{\mathfrak{q}}(\underline{M}).$

Theorem 8.2.

For $\mathfrak{q}\in\{\mathfrak{p},\overline{\mathfrak{p}}\}$ ,

(1)

The elements in the first row of $Q_{\mathfrak{q}}^{-1}\underline{M_{\mathfrak{q}}}$ are inside $\mathcal{H}_{E,v_{p}(\alpha_{\mathfrak{q}})}(\Gamma_{\mathfrak{q}})$ , while the elements in the second row are in $\mathcal{H}_{E,v_{p}(\beta_{\mathfrak{q}})}(\Gamma_{\mathfrak{q}})$ .
(2)

The second row of the matrix $A_{\varphi,\mathfrak{q}}^{-n}\underline{M_{\mathfrak{q}}}$ is divisible by the cyclotomic polynomial $\Phi_{n-1,k-1}(\gamma_{\mathfrak{q}})$ .
(3)

The determinant of $\underline{M_{\mathfrak{q}}}$ is $\dfrac{\log_{p,k-1}(\gamma_{\mathfrak{q}})}{\delta_{k-1}(\gamma_{\mathfrak{q}}-1)}$ upto a unit in $\Lambda_{E}(\Gamma_{\mathfrak{q}})$ .

Proof.

(1) follows from Proposition 5.1 and the definitions of $Q_{\mathfrak{q}}$ and $\underline{M_{\mathfrak{q}}}$ . (2) follows from Lemma 5.2.

For (3), we know $\underline{M_{\mathfrak{q}}}=\operatorname{Mat}_{\mathfrak{q}}(\underline{M})$ . Thus,

	$\displaystyle\det(\underline{M_{\mathfrak{q}}})$	$\displaystyle=\det(\operatorname{Mat}_{\mathfrak{q}}(\underline{M})),$
		$\displaystyle=\tau_{\mathfrak{q}}(\det(\underline{M})),$

since $\tau_{\mathfrak{q}}$ is ring isomorphism. Hence the result follows from Lemma 5.3. ∎

For the rest of the article, we write

(8.3)

Q_{\mathfrak{q}}^{-1}\underline{M_{\mathfrak{q}}}=\begin{pmatrix}P_{1,\mathfrak{q}}&P_{2,\mathfrak{q}}\\[6.0pt] P_{3,\mathfrak{q}}&P_{4,\mathfrak{q}}\end{pmatrix}.

8.3. The main theorem and its proof

In this section, we generalize the results of [19] and use them to decompose the two variable $p$ -adic $L$ -functions of Bianchi modular forms.

For the Bianchi cusp form $\mathcal{F}$ of level $\mathfrak{n}$ which is not divisible by $(p)$ , let $a_{\mathfrak{q}}$ be the $T_{\mathfrak{q}}$ -eigenvalue of $\mathcal{F}$ for $\mathfrak{q}\in\{\mathfrak{p},\overline{\mathfrak{p}}\}$ . Recall that we have four $p$ -stabilizations $\mathcal{F}^{\alpha_{\mathfrak{p}},\alpha_{\overline{\mathfrak{p}}}},\mathcal{F}^{\alpha_{\mathfrak{p}},\beta_{\overline{\mathfrak{p}}}},\mathcal{F}^{\beta_{\mathfrak{p}},\alpha_{\overline{\mathfrak{p}}}}$ , and $\mathcal{F}^{\beta_{\mathfrak{p}},\beta_{\overline{\mathfrak{p}}}}$ of level $p\mathcal{N}$ , where $\alpha_{\mathfrak{q}},\beta_{\mathfrak{q}}$ are the roots of Hecke polynomial $X^{2}-a_{\mathfrak{q}}X+p^{k-1}$ , for $\mathfrak{q}\in\{\mathfrak{p},\overline{\mathfrak{p}}\}$ .

Therefore, from Theorem 7.5 and equation (7.2), for $*\in\{\alpha_{\mathfrak{p}},\beta_{\mathfrak{p}}\}$ and $\bullet\in\{\alpha_{\overline{\mathfrak{p}}},\beta_{\overline{\mathfrak{p}}}\}$ , we have

L_{*,\bullet}\coloneqq L_{p,\mathcal{F}^{*,\bullet}}\in\mathcal{H}_{E,v_{p}(*),v_{p}(\bullet)}(G_{p^{\infty}}).

Moreover, for any Hecke character $\Xi$ of infinity type $(0,0)\leq(a,b)\leq(k-2,k-2)$ and conductor $\mathfrak{p}^{n_{\mathfrak{p}}}\overline{\mathfrak{p}}^{n_{\overline{\mathfrak{p}}}}$ with $n_{\mathfrak{p}},n_{\overline{\mathfrak{p}}}\geq 1$ , we have

(8.4)	$\displaystyle L_{\alpha_{\mathfrak{p}},\alpha_{\overline{\mathfrak{p}}}}(\widetilde{\Xi})$	$\displaystyle=\alpha_{\mathfrak{p}}^{-n_{\mathfrak{p}}}\alpha_{\overline{\mathfrak{p}}}^{-n_{\overline{\mathfrak{p}}}}C_{a,b,\widetilde{\Xi}},$
(8.5)	$\displaystyle L_{\alpha_{\mathfrak{p}},\beta_{\overline{\mathfrak{p}}}}(\widetilde{\Xi})$	$\displaystyle=\alpha_{\mathfrak{p}}^{-n_{\mathfrak{p}}}\beta_{\overline{\mathfrak{p}}}^{-n_{\overline{\mathfrak{p}}}}C_{a,b,\widetilde{\Xi}},$
(8.6)	$\displaystyle L_{\beta_{\mathfrak{p}},\alpha_{\overline{\mathfrak{p}}}}(\widetilde{\Xi})$	$\displaystyle=\beta_{\mathfrak{p}}^{-n_{\mathfrak{p}}}\alpha_{\overline{\mathfrak{p}}}^{-n_{\overline{\mathfrak{p}}}}C_{a,b,\widetilde{\Xi}},$
(8.7)	$\displaystyle L_{\beta_{\mathfrak{p}},\beta_{\overline{\mathfrak{p}}}}(\widetilde{\Xi})$	$\displaystyle=\beta_{\mathfrak{p}}^{-n_{\mathfrak{p}}}\beta_{\overline{\mathfrak{p}}}^{-n_{\overline{\mathfrak{p}}}}C_{a,b,\widetilde{\Xi}},$

where $C_{a,b,\widetilde{\Xi}}\in\overline{\mathbb{Q}_{p}}$ is a constant independent of $\alpha_{\mathfrak{p}},\beta_{\mathfrak{p}},\alpha_{\overline{\mathfrak{p}}},\beta_{\overline{\mathfrak{p}}}$ .

The main theorem is as follows:

Theorem 8.3.

There exist two variable power series with bounded coefficients, that is, there exist $L_{\sharp,\sharp},L_{\sharp,\flat},L_{\flat,\sharp},L_{\flat,\flat}\in\Lambda_{E}(G_{p^{\infty}})$ such that

(8.8)

\begin{pmatrix}L_{\alpha_{\mathfrak{p}},\alpha_{\overline{\mathfrak{p}}}}&L_{\beta_{\mathfrak{p}},\alpha_{\overline{\mathfrak{p}}}}\\[6.0pt] L_{\alpha_{\mathfrak{p}},\beta_{\overline{\mathfrak{p}}}}&L_{\beta_{\mathfrak{p}},\beta_{\overline{\mathfrak{p}}}}\end{pmatrix}=Q^{-1}_{\overline{\mathfrak{p}}}\underline{M_{\overline{\mathfrak{p}}}}\begin{pmatrix}L_{\sharp,\sharp}&L_{\flat,\sharp}\\[6.0pt] L_{\sharp,\flat}&L_{\flat,\flat}\end{pmatrix}(Q^{-1}_{\mathfrak{p}}\underline{M_{\mathfrak{p}}})^{T}.

Remark 8.4.

Theorem 8.3 is analogus to [19, Theorem 2.2] and [7, Equation (24)].

We first factorize through the variable $\gamma_{\mathfrak{p}}$ and then through $\gamma_{\overline{\mathfrak{p}}}$ . In other words, we will first decompose the matrix $\begin{pmatrix}L_{\alpha_{\mathfrak{p}},\alpha_{\overline{\mathfrak{p}}}}&L_{\beta_{\mathfrak{p}},\alpha_{\overline{\mathfrak{p}}}}\\[6.0pt] L_{\alpha_{\mathfrak{p}},\beta_{\overline{\mathfrak{p}}}}&L_{\beta_{\mathfrak{p}},\beta_{\overline{\mathfrak{p}}}}\end{pmatrix}$ in terms of matrix, say $C$ , and $\underline{M_{\mathfrak{p}}}$ . Then we decompose $C$ as a product of matrices $\begin{pmatrix}L_{\sharp,\sharp}&L_{\flat,\sharp}\\[6.0pt] L_{\sharp,\flat}&L_{\flat,\flat}\end{pmatrix}$ and $\underline{M_{\overline{\mathfrak{p}}}}$ .

First, we need the following classical result:

Lemma 8.5.

Let $\gamma$ be a topological generator of $\mathbb{Z}_{p}$ . Let $s,h$ be non-negative integers and assume $s<h$ . If $F\in E[[\gamma-1]]$ is $O(\log_{p}^{s})$ and vanishes at all characters of type $\chi^{i}\omega$ for all $0\leq i\leq h-1$ , where $\chi$ is any character which sends $\gamma$ to another topological generator $u\in 1+p\mathbb{Z}_{p}$ (for example the cyclotomic character) and $\omega$ is any Dirichlet character of conductor $p^{n},n\geq 1$ . Then $F$ is identically $0$ .

Proof.

See [1, Lemme II.2.5] and [33, Lemma 2.10].

We give a sketch here. $F$ is $o(log_{p}^{h})$ , since $F$ is $O(\log_{p}^{s})$ and $s<h$ . Suppose $F$ is not $0$ and has infinitely many zeros, then

F=\log_{p,h}(\gamma)\cdot G,

where $G\neq 0$ is another power series. Note that $\log_{p,h}(\gamma)$ is not $o(\log_{p}^{h})$ and therefore $F$ is not $o(\log_{p}^{h})$ . This is a contradiction. ∎

Recall that $E$ is a finite extension of $\mathbb{Q}_{p}$ containing $\alpha_{\mathfrak{p}},\beta_{\mathfrak{p}},\alpha_{\overline{\mathfrak{p}}}$ and $\beta_{\overline{\mathfrak{p}}}$ . Let $S_{\mathfrak{p}}$ be the set of all Hecke characters on the ray class group $G_{p^{\infty}}$ with infinity type $(0,0)\leq(a,0)\leq(k-2,0)$ and conductor $\mathfrak{p}^{n_{\mathfrak{p}}}$ , for $n_{\mathfrak{p}}>1$ . Similarly, let $S_{\overline{\mathfrak{p}}}$ be the set of all Hecke characters on the ray class group $G_{p^{\infty}}$ with infinity type $(0,0)\leq(0,b)\leq(0,k-2)$ and conductor $\overline{\mathfrak{p}}^{n_{\overline{\mathfrak{p}}}}$ , for $n_{\overline{\mathfrak{p}}}>1$ .

Proposition 8.6.

There exist $L_{\sharp,\alpha_{\overline{\mathfrak{p}}}},L_{\flat,\alpha_{\overline{\mathfrak{p}}}}\in\mathcal{H}_{E,0,v_{p}(\alpha_{\overline{\mathfrak{p}}})}(G_{p^{\infty}})$ and $L_{\sharp,\beta_{\overline{\mathfrak{p}}}},L_{\flat,\beta_{\overline{\mathfrak{p}}}}\in\mathcal{H}_{E,0,v_{p}(\beta_{\overline{\mathfrak{p}}})}(G_{p^{\infty}})$ such that

\begin{pmatrix}L_{\alpha_{\mathfrak{p}},\alpha_{\overline{\mathfrak{p}}}}&L_{\beta_{\mathfrak{p}},\alpha_{\overline{\mathfrak{p}}}}\\[6.0pt] L_{\alpha_{\mathfrak{p}},\beta_{\overline{\mathfrak{p}}}}&L_{\beta_{\mathfrak{p}},\beta_{\overline{\mathfrak{p}}}}\end{pmatrix}=\begin{pmatrix}L_{\sharp,\alpha_{\overline{\mathfrak{p}}}}&L_{\flat,\alpha_{\overline{\mathfrak{p}}}}\\[6.0pt] L_{\sharp,\beta_{\overline{\mathfrak{p}}}}&L_{\flat,\beta_{\overline{\mathfrak{p}}}}\end{pmatrix}(Q^{-1}_{\mathfrak{p}}\underline{M_{\mathfrak{p}}})^{T}.

Proof.

This is a generalization of [19, Proposition 2.3]. Recall that $L_{*,\bullet}$ are locally analytic distributions on the ray class group $G_{p^{\infty}}\cong\Delta_{K}\times\overline{\langle\gamma_{\mathfrak{p}}\rangle}\times\overline{\langle\gamma_{\overline{\mathfrak{p}}}\rangle}$ . For any character $\eta:\Delta_{K}\to\overline{\mathbb{Z}_{p}}^{\times}$ , we will prove that for $L_{\alpha_{\mathfrak{p}},\alpha_{\overline{\mathfrak{p}}}}^{\eta},L_{\beta_{\mathfrak{p}},\alpha_{\overline{\mathfrak{p}}}}^{\eta}\in\mathcal{H}_{E,v_{p}(\alpha_{\mathfrak{p}}),v_{p}(\alpha_{\overline{\mathfrak{p}}})},$ there exist $L_{\sharp,\alpha_{\overline{\mathfrak{p}}}}^{\eta},L_{\flat,\alpha_{\overline{\mathfrak{p}}}}^{\eta}\in\mathcal{H}_{E,0,v_{p}(\alpha_{\overline{\mathfrak{p}}})}$ such that

(8.9)

\begin{pmatrix}L_{\alpha_{\mathfrak{p}},\alpha_{\overline{\mathfrak{p}}}}^{\eta}\\[6.0pt] L_{\beta_{\mathfrak{p}},\alpha_{\overline{\mathfrak{p}}}}^{\eta}\end{pmatrix}=Q^{-1}_{\mathfrak{p}}\underline{M_{\mathfrak{p}}}\begin{pmatrix}L_{\sharp,\alpha_{\overline{\mathfrak{p}}}}^{\eta}\\[6.0pt] L_{\flat,\alpha_{\overline{\mathfrak{p}}}}^{\eta}\end{pmatrix}.

and therefore,

\begin{pmatrix}L_{\alpha_{\mathfrak{p}},\alpha_{\overline{\mathfrak{p}}}}\\[6.0pt] L_{\beta_{\mathfrak{p}},\alpha_{\overline{\mathfrak{p}}}}\end{pmatrix}=Q^{-1}_{\mathfrak{p}}\underline{M_{\mathfrak{p}}}\begin{pmatrix}L_{\sharp,\alpha_{\overline{\mathfrak{p}}}}\\[6.0pt] L_{\flat,\alpha_{\overline{\mathfrak{p}}}}\end{pmatrix}.

Fix $\eta=1$ . The proof for other characters is similar.

Let $\Xi$ be any Hecke character of $K$ of infinity type $(0,0)\leq(a,b)\leq(k-2,k-2)$ and conductor $\mathfrak{p}^{n_{\mathfrak{p}}}\overline{\mathfrak{p}}^{n_{\overline{\mathfrak{p}}}}$ , where $n_{\mathfrak{p}},n_{\overline{\mathfrak{p}}}\geq 1$ . Using equations (8.4) and (8.6), we get

(8.10)		$\displaystyle L_{\alpha_{\mathfrak{p}},\alpha_{\overline{\mathfrak{p}}}}^{\Delta_{K}}(\widecheck{\Xi})$	$\displaystyle=\alpha_{\mathfrak{p}}^{-n_{\mathfrak{p}}}(\alpha_{\overline{\mathfrak{p}}}^{-n_{\overline{\mathfrak{p}}}}\cdot C_{a,b,\widetilde{\Xi}}),$
(8.11)		$\displaystyle L_{\beta_{\mathfrak{p}},\alpha_{\overline{\mathfrak{p}}}}^{\Delta_{K}}(\widecheck{\Xi})$	$\displaystyle=\beta_{\mathfrak{p}}^{-n_{\mathfrak{p}}}(\alpha_{\overline{\mathfrak{p}}}^{-n_{\overline{\mathfrak{p}}}}\cdot C_{a,b,\widetilde{\Xi}}),$

where $\widecheck{\Xi}=\widetilde{\Xi}|_{\overline{\langle\gamma_{\mathfrak{p}}\rangle}\times\overline{\langle\gamma_{\overline{\mathfrak{p}}}\rangle}}=\widetilde{\Xi}_{\mathfrak{p}}\cdot\widetilde{\Xi}_{\overline{\mathfrak{p}}}$ . Denote $\alpha^{-n_{\overline{\mathfrak{p}}}}C_{a,b,\widetilde{\Xi}}$ by $D$ . Using (6.3), we can rewrite equations (8.10) and (8.11) as

(8.12)		$\displaystyle L_{\alpha_{\mathfrak{p}},\alpha_{\overline{\mathfrak{p}}}}^{\Delta_{K}(\widetilde{\Xi}_{\overline{\mathfrak{p}}})}(\widetilde{\Xi}_{\mathfrak{p}})$	$\displaystyle=\alpha_{\mathfrak{p}}^{-n_{\mathfrak{p}}}D,$
(8.13)		$\displaystyle L_{\beta_{\mathfrak{p}},\alpha_{\overline{\mathfrak{p}}}}^{\Delta_{K}(\widetilde{\Xi}_{\overline{\mathfrak{p}}})}(\widetilde{\Xi}_{\mathfrak{p}})$	$\displaystyle=\beta_{\mathfrak{p}}^{-n_{\mathfrak{p}}}D.$

Note that $L_{\alpha_{\mathfrak{p}},\alpha_{\overline{\mathfrak{p}}}}^{{\Delta_{K}(\widetilde{\Xi}_{\overline{\mathfrak{p}}})}}\in\mathcal{H}_{E^{\prime},v_{p}(\alpha_{\mathfrak{p}})}(\Gamma_{\mathfrak{p}})$ and $L_{\beta_{\mathfrak{p}},\alpha_{\overline{\mathfrak{p}}}}^{{\Delta_{K}(\widetilde{\Xi}_{\overline{\mathfrak{p}}})}}\in\mathcal{H}_{E^{\prime},v_{p}(\beta_{\mathfrak{p}})}(\Gamma_{\mathfrak{p}})$ , where $E^{\prime}$ is some field extension of $E$ .

Let

G_{1}=P_{4,\mathfrak{p}}L_{\alpha_{\mathfrak{p}},\alpha_{\overline{\mathfrak{p}}}}^{\Delta_{K}(\widetilde{\Xi}_{\overline{\mathfrak{p}}})}-P_{2,\mathfrak{p}}L_{\beta_{\mathfrak{p}},\alpha_{\overline{\mathfrak{p}}}}^{\Delta_{K}(\widetilde{\Xi}_{\overline{\mathfrak{p}}})}

and

G_{2}=-P_{3,\mathfrak{p}}L_{\alpha_{\mathfrak{p}},\alpha_{\overline{\mathfrak{p}}}}^{\Delta_{K}(\widetilde{\Xi}_{\overline{\mathfrak{p}}})}+P_{1,\mathfrak{p}}L_{\beta_{\mathfrak{p}},\alpha_{\overline{\mathfrak{p}}}}^{\Delta_{K}(\widetilde{\Xi}_{\overline{\mathfrak{p}}})}.

Then, from Theorem 8.2, $G_{1},G_{2}\in\mathcal{H}_{E^{\prime},k-1}$ and $G_{1}(\widetilde{\Xi}_{\mathfrak{p}})=G_{2}(\widetilde{\Xi}_{\mathfrak{p}})=0$ . Hence, Theorem 5.5 implies that $\det(Q_{\mathfrak{p}}^{-1}\underline{M_{\mathfrak{p}}})$ divides both $G_{1}$ and $G_{2}$ in $\mathcal{H}_{E^{\prime},k-1}(\Gamma_{\mathfrak{p}})$ .

Thus,

(8.14)

P_{4,\mathfrak{p}}(\widecheck{\Xi})L_{\alpha_{\mathfrak{p}},\alpha_{\overline{\mathfrak{p}}}}^{\Delta_{K}}(\widecheck{\Xi})-P_{2,\mathfrak{p}}(\widecheck{\Xi})L_{\alpha_{\mathfrak{p}},\alpha_{\overline{\mathfrak{p}}}}^{\Delta_{K}}(\check{\Xi})=-P_{3,\mathfrak{p}}(\widecheck{\Xi})L_{\alpha_{\mathfrak{p}},\alpha_{\overline{\mathfrak{p}}}}^{\Delta_{K}}(\widecheck{\Xi})+P_{1,\mathfrak{p}}(\widecheck{\Xi})L_{\beta_{\mathfrak{p}},\alpha_{\overline{\mathfrak{p}}}}^{\Delta_{K}}(\widecheck{\Xi})=0,

for any Hecke character $\Xi$ of infinity type $(0,0)\leq(a,b)\leq(k-2,k-2)$ and conductor $\mathfrak{p}^{n_{\mathfrak{p}}}\overline{\mathfrak{p}}^{n_{\overline{\mathfrak{p}}}}$ .

Thus, for any Hecke characters $\omega_{1}\in S_{\mathfrak{p}}$ , $\omega_{2}\in S_{\overline{\mathfrak{p}}}$ , (8.14) implies

(P_{4,\mathfrak{p}}L_{\alpha_{\mathfrak{p}},\alpha_{\overline{\mathfrak{p}}}}^{\Delta_{K}}-P_{2,\mathfrak{p}}L_{\beta_{\mathfrak{p}},\alpha_{\overline{\mathfrak{p}}}}^{\Delta_{K}})(\widetilde{\omega_{1}}_{\mathfrak{p}}\widetilde{\omega_{2}}_{\overline{\mathfrak{p}}})=0,

which we rewrite as

(8.15)

(P_{4,\mathfrak{p}}L_{\alpha_{\mathfrak{p}},\alpha_{\overline{\mathfrak{p}}}}^{\Delta_{K}}-P_{2,\mathfrak{p}}L_{\beta_{\mathfrak{p}},\alpha_{\overline{\mathfrak{p}}}}^{\Delta_{K}})^{(\widetilde{\omega_{1}}_{\mathfrak{p}})}(\widetilde{\omega_{2}}_{\overline{\mathfrak{p}}})=0.

Similarly.

(8.16)

(-P_{3,\mathfrak{p}}L_{\alpha_{\mathfrak{p}},\alpha_{\overline{\mathfrak{p}}}}^{\Delta_{K}}+P_{1,\mathfrak{p}}L_{\beta_{\mathfrak{p}},\alpha_{\overline{\mathfrak{p}}}}^{\Delta_{K}})^{(\widetilde{\omega_{1}}_{\mathfrak{p}})}(\widetilde{\omega_{2}}_{\overline{\mathfrak{p}}})=0.

Hence, the distributions $(P_{4,\mathfrak{p}}L_{\alpha_{\mathfrak{p}},\alpha_{\overline{\mathfrak{p}}}}^{\Delta_{K}}-P_{2,\mathfrak{p}}L_{\beta_{\mathfrak{p}},\alpha_{\overline{\mathfrak{p}}}}^{\Delta_{K}})^{(\widetilde{\omega_{1}}_{\mathfrak{p}})}$ and $(-P_{3,\mathfrak{p}}L_{\alpha_{\mathfrak{p}},\alpha_{\overline{\mathfrak{p}}}}^{\Delta_{K}}+P_{1,\mathfrak{p}}L_{\beta_{\mathfrak{p}},\alpha_{\overline{\mathfrak{p}}}}^{\Delta_{K}})^{(\widetilde{\omega_{1}}_{\mathfrak{p}})}$ vanish at all characters $\omega_{2}\in S_{\overline{\mathfrak{p}}}$ . Moreover, these two distributions belong to $\mathcal{H}_{v_{p}(\alpha_{\overline{\mathfrak{p}}}),E^{\prime}}(\Gamma_{\overline{\mathfrak{p}}})$ and $v_{p}(\alpha_{\overline{\mathfrak{p}}})<k-1$ . Therefore, Lemma 8.5 implies

	$\displaystyle(P_{4,\mathfrak{p}}L_{\alpha_{\mathfrak{p}},\alpha_{\overline{\mathfrak{p}}}}^{\Delta_{K}}-P_{2,\mathfrak{p}}L_{\beta_{\mathfrak{p}},\alpha_{\overline{\mathfrak{p}}}}^{\Delta_{K}})^{(\widetilde{\omega_{1}}_{\mathfrak{p}})}$	$\displaystyle=0,$
	$\displaystyle(-P_{3,\mathfrak{p}}L_{\alpha_{\mathfrak{p}},\alpha_{\overline{\mathfrak{p}}}}^{\Delta_{K}}+P_{1,\mathfrak{p}}L_{\beta_{\mathfrak{p}},\alpha_{\overline{\mathfrak{p}}}}^{\Delta_{K}})^{(\widetilde{\omega_{1}}_{\mathfrak{p}})}$	$\displaystyle=0.$

Hence, using Theorem 5.5 again, we conclude that $\det(Q_{\mathfrak{p}}^{-1}\underline{M_{\mathfrak{p}}})$ divide $(P_{4,\mathfrak{p}}L_{\alpha_{\mathfrak{p}},\alpha_{\overline{\mathfrak{p}}}}^{\Delta_{K}}-P_{2,\mathfrak{p}}L_{\beta_{\mathfrak{p}},\alpha_{\overline{\mathfrak{p}}}}^{\Delta_{K}})$ and $(-P_{3,\mathfrak{p}}L_{\alpha_{\mathfrak{p}},\alpha_{\overline{\mathfrak{p}}}}^{\Delta_{K}}+P_{1,\mathfrak{p}}L_{\beta_{\mathfrak{p}},\alpha_{\overline{\mathfrak{p}}}}^{\Delta_{K}})$ over the two-variable distribution algebra $\mathcal{H}_{E,k-1,v_{p}(\alpha_{\overline{\mathfrak{p}}})}$ .

Write

	$\displaystyle L_{\sharp,\alpha_{\overline{\mathfrak{p}}}}^{\Delta_{K}}$	$\displaystyle=\dfrac{P_{4,\mathfrak{p}}L_{\alpha_{\mathfrak{p}},\alpha_{\overline{\mathfrak{p}}}}^{\Delta_{K}}-P_{2,\mathfrak{p}}L_{\beta_{\mathfrak{p}},\alpha_{\overline{\mathfrak{p}}}}^{\Delta_{K}}}{\det(Q_{\mathfrak{p}}^{-1}\underline{M_{\mathfrak{p}}})},$
	$\displaystyle L_{\flat,\alpha_{\overline{\mathfrak{p}}}}^{\Delta_{K}}$	$\displaystyle=\dfrac{-P_{3,\mathfrak{p}}L_{\alpha_{\mathfrak{p}},\alpha_{\overline{\mathfrak{p}}}}^{\Delta_{K}}+P_{1,\mathfrak{p}}L_{\beta_{\mathfrak{p}},\alpha_{\overline{\mathfrak{p}}}}^{\Delta_{K}}}{\det(Q_{\mathfrak{p}}^{-1}\underline{M_{\mathfrak{p}}})}.$

Then, Theorem 5.5 implies $L_{\sharp,\alpha_{\overline{\mathfrak{p}}}}^{\Delta_{K}}$ and $L_{\flat,\alpha_{\overline{\mathfrak{p}}}}^{\Delta_{K}}$ lie in $\mathcal{H}_{E,0,v_{p}(\alpha_{\overline{\mathfrak{p}}})}$ .

We then write

	$\displaystyle L_{\sharp,\alpha_{\overline{\mathfrak{p}}}}$	$\displaystyle=\dfrac{P_{4,\mathfrak{p}}L_{\alpha_{\mathfrak{p}},\alpha_{\overline{\mathfrak{p}}}}-P_{2,\mathfrak{p}}L_{\beta_{\mathfrak{p}},\alpha_{\overline{\mathfrak{p}}}}}{\det(Q_{\mathfrak{p}}^{-1}\underline{M_{\mathfrak{p}}})},$
	$\displaystyle L_{\flat,\alpha_{\overline{\mathfrak{p}}}}$	$\displaystyle=\dfrac{-P_{3,\mathfrak{p}}L_{\alpha_{\mathfrak{p}},\alpha_{\overline{\mathfrak{p}}}}+P_{1,\mathfrak{p}}L_{\beta_{\mathfrak{p}},\alpha_{\overline{\mathfrak{p}}}}}{\det(Q_{\mathfrak{p}}^{-1}\underline{M_{\mathfrak{p}}})}.$

Since $\det(Q_{\mathfrak{p}}^{-1}\underline{M_{\mathfrak{p}}})$ divides each isotypic component of the two distributions in the numerators, $L_{\sharp,\alpha_{\overline{\mathfrak{p}}}}$ and $L_{\flat,\alpha_{\overline{\mathfrak{p}}}}$ are elements in $\mathcal{H}_{E,0,v_{p}(\alpha_{\overline{\mathfrak{p}}})}(G_{p^{\infty}})$ .

The proof for

\begin{pmatrix}L_{\alpha_{\mathfrak{p}},\beta_{\overline{\mathfrak{p}}}}\\[6.0pt] L_{\beta_{\mathfrak{p}},\beta_{\overline{\mathfrak{p}}}}\end{pmatrix}=Q^{-1}_{\mathfrak{p}}\underline{M_{\mathfrak{p}}}\begin{pmatrix}L_{\sharp,\beta_{\overline{\mathfrak{p}}}}\\[6.0pt] L_{\flat,\beta_{\overline{\mathfrak{p}}}}\end{pmatrix}.

is identical. ∎

Recall $\mathcal{H}_{r,E}\coloneqq\{f(X)=\sum_{n\geq 0}a_{n}X^{n}\in E[[X]]\colon\sup_{n}(n^{-r}|a_{n}|_{p})<\infty\}$ . Then, we can identify $\mathcal{H}_{E,r,s}\coloneqq\mathcal{H}_{E,r}(\Gamma_{\mathfrak{p}})\widehat{\otimes}\mathcal{H}_{E,s}(\Gamma_{\overline{\mathfrak{p}}})$ with $\mathcal{H}_{E,r}\widehat{\otimes}\mathcal{H}_{E,s}$ by identifying $X=\gamma_{\mathfrak{p}}-1$ and $Y=\gamma_{\overline{\mathfrak{p}}}-1$ . We define the operator $\partial_{\mathfrak{p}}$ to be the partial derivative $\dfrac{\partial}{\partial X}$ . The next proposition is a generalization of [19, Lemma 2.4].

Proposition 8.7.

Let $\Xi$ be any character Hecke character of $K$ of the infinity type $(0,0)\leq(a,b)\leq(k-2,k-2)$ and conductor $\mathfrak{p}^{n_{\mathfrak{p}}}\overline{\mathfrak{p}}^{n_{\overline{\mathfrak{p}}}}$ with $n_{\mathfrak{p}},n_{\overline{\mathfrak{p}}}>0$ . Then, there exist constants $A_{a,b,\Xi},B_{a,b,\Xi}\in\overline{\mathbb{Q}_{p}}$ such that

\begin{matrix}\partial_{\mathfrak{p}}L_{\alpha_{\mathfrak{p}},\alpha_{\overline{\mathfrak{p}}}}(\widetilde{\Xi})=\alpha_{\overline{\mathfrak{p}}}^{-n_{\overline{\mathfrak{p}}}}A_{a,b,\widetilde{\Xi}},&\partial_{\mathfrak{p}}L_{\alpha_{\mathfrak{p}},\beta_{\overline{\mathfrak{p}}}}(\widetilde{\Xi})=\beta_{\overline{\mathfrak{p}}}^{-n_{\overline{\mathfrak{p}}}}A_{a,b,\widetilde{\Xi}},\\[6.0pt] \partial_{\mathfrak{p}}L_{\beta_{\mathfrak{p}},\alpha_{\overline{\mathfrak{p}}}}(\widetilde{\Xi})=\alpha_{\overline{\mathfrak{p}}}^{-n_{\overline{\mathfrak{p}}}}B_{a,b,\widetilde{\Xi}},&\partial_{\mathfrak{p}}L_{\beta_{\mathfrak{p}},\beta_{\overline{\mathfrak{p}}}}(\widetilde{\Xi})=\beta_{\overline{\mathfrak{p}}}^{-n_{\overline{\mathfrak{p}}}}B_{a,b,\widetilde{\Xi}}.\end{matrix}

Proof.

We will only show that

\displaystyle\alpha_{\overline{\mathfrak{p}}}^{n_{\overline{\mathfrak{p}}}}\partial_{\mathfrak{p}}L_{\alpha_{\mathfrak{p}},\alpha_{\overline{\mathfrak{p}}}}(\widetilde{\Xi})

\displaystyle=\beta_{\overline{\mathfrak{p}}}^{n_{\overline{\mathfrak{p}}}}\partial_{\mathfrak{p}}L_{\alpha_{\mathfrak{p}},\beta_{\overline{\mathfrak{p}}}}(\widetilde{\Xi}),

for any Hecke character $\Xi$ of of the infinity type $(0,0)\leq(a,b)\leq(k-2,k-2)$ and conductor $\mathfrak{p}^{n_{\mathfrak{p}}}\overline{\mathfrak{p}}^{n_{\overline{\mathfrak{p}}}}$ with $n_{\mathfrak{p}},n_{\overline{\mathfrak{p}}}>0$ .

Fix a Hecke character $\omega_{1}\in S_{\mathfrak{p}}$ . Then, for any Hecke character $\omega_{2}\in S_{\overline{\mathfrak{p}}}$ , (8.4) and (8.5) imply

(8.17)

\alpha_{\overline{\mathfrak{p}}}^{n_{\overline{\mathfrak{p}}}}L_{\alpha_{\mathfrak{p}},\alpha_{\overline{\mathfrak{p}}}}^{\Delta_{K}}(\widetilde{\omega_{1,\mathfrak{p}}}\widetilde{\omega_{2,\overline{\mathfrak{p}}}})=\beta_{\overline{\mathfrak{p}}}^{n_{\overline{\mathfrak{p}}}}L_{\alpha_{\mathfrak{p}},\beta_{\overline{\mathfrak{p}}}}^{\Delta_{K}}(\widetilde{\omega_{1,\mathfrak{p}}}\widetilde{\omega_{2,\overline{\mathfrak{p}}}}),

where $L_{\alpha_{\mathfrak{p}},\alpha_{\overline{\mathfrak{p}}}}^{\Delta_{K}},L_{\alpha_{\mathfrak{p}},\beta_{\overline{\mathfrak{p}}}}^{\Delta_{K}}$ are isotypic components of $L_{\alpha_{\mathfrak{p}},\alpha_{\overline{\mathfrak{p}}}},L_{\alpha_{\mathfrak{p}},\beta_{\overline{\mathfrak{p}}}}$ respectively with respect to the trivial character of $\Delta_{K}$ . Using (6.3), we rewrite (8.17) as

(8.18)

\alpha_{\overline{\mathfrak{p}}}^{n_{\overline{\mathfrak{p}}}}L_{\alpha_{\mathfrak{p}},\alpha_{\overline{\mathfrak{p}}}}^{\Delta_{K}(\widetilde{\omega_{2,\overline{\mathfrak{p}}}})}(\widetilde{\omega_{1,\mathfrak{p}}})=\beta_{\overline{\mathfrak{p}}}^{n_{\overline{\mathfrak{p}}}}L_{\alpha_{\mathfrak{p}},\beta_{\overline{\mathfrak{p}}}}^{\Delta_{K}(\widetilde{\omega_{2,\overline{\mathfrak{p}}}})}(\widetilde{\omega_{1,\mathfrak{p}}}).

From Lemma 6.7, we know that $L_{\alpha_{\mathfrak{p}},\alpha_{\overline{\mathfrak{p}}}}^{\Delta_{K}(\widetilde{\omega_{2,\overline{\mathfrak{p}}}})},L_{\alpha_{\mathfrak{p}},\beta_{\overline{\mathfrak{p}}}}^{\Delta_{K}(\widetilde{\omega_{2,\overline{\mathfrak{p}}}})}\in\mathcal{H}_{E^{\prime},v_{p}(\alpha_{\mathfrak{p}})}(\Gamma_{\mathfrak{p}})$ for some extension $E^{\prime}$ of $E$ . As $v_{p}(\alpha_{\mathfrak{p}})<k-1$ , using Lemma 8.5 we have

\alpha_{\overline{\mathfrak{p}}}^{n_{\overline{\mathfrak{p}}}}L_{\alpha_{\mathfrak{p}},\alpha_{\overline{\mathfrak{p}}}}^{\Delta_{K}(\widetilde{\omega_{2,\overline{\mathfrak{p}}}})}=\beta_{\overline{\mathfrak{p}}}^{n_{\overline{\mathfrak{p}}}}L_{\alpha_{\mathfrak{p}},\beta_{\overline{\mathfrak{p}}}}^{\Delta_{K}(\widetilde{\omega_{2,\overline{\mathfrak{p}}}})}.

Hence, their partial derivatives also agree, i.e.

(8.19)

\alpha_{\overline{\mathfrak{p}}}^{n_{\overline{\mathfrak{p}}}}\partial_{\mathfrak{p}}L_{\alpha_{\mathfrak{p}},\alpha_{\overline{\mathfrak{p}}}}^{\Delta_{K}(\widetilde{\omega_{2,\overline{\mathfrak{p}}}})}=\beta_{\overline{\mathfrak{p}}}^{n_{\overline{\mathfrak{p}}}}\partial_{\mathfrak{p}}L_{\alpha_{\mathfrak{p}},\beta_{\overline{\mathfrak{p}}}}^{\Delta_{K}(\widetilde{\omega_{2,\overline{\mathfrak{p}}}})}.

But, for any power series $F\in\mathcal{H}_{E,v_{p}(\alpha_{\mathfrak{p}}),s}$ ,

\partial_{\mathfrak{p}}(F^{(\widetilde{\omega_{2,\overline{\mathfrak{p}}}})})(\widetilde{\omega_{1,\mathfrak{p}}})=(\partial_{\mathfrak{p}}F)(\widetilde{\omega_{1,\mathfrak{p}}}\widetilde{\omega_{2,\overline{\mathfrak{p}}}}),

for all Hecke characters $\omega_{1}\in S_{\mathfrak{p}}$ .

Therefore for any Hecke character $\Xi$ ,

(8.20)

\alpha_{\overline{\mathfrak{p}}}^{n_{\overline{\mathfrak{p}}}}\partial_{\mathfrak{p}}L_{\alpha_{\mathfrak{p}},\alpha_{\overline{\mathfrak{p}}}}^{\Delta_{K}}(\widecheck{\Xi})=\beta_{\overline{\mathfrak{p}}}^{n_{\overline{\mathfrak{p}}}}\partial_{\mathfrak{p}}L_{\alpha_{\mathfrak{p}},\beta_{\overline{\mathfrak{p}}}}^{\Delta_{K}}(\widecheck{\Xi}),

Since equation (8.20) is true for any isotypic component, we have

(8.21)

\alpha_{\overline{\mathfrak{p}}}^{n_{\overline{\mathfrak{p}}}}\partial_{\mathfrak{p}}L_{\alpha_{\mathfrak{p}},\alpha_{\overline{\mathfrak{p}}}}(\widetilde{\Xi})=\beta_{\overline{\mathfrak{p}}}^{n_{\overline{\mathfrak{p}}}}\partial_{\mathfrak{p}}L_{\alpha_{\mathfrak{p}},\beta_{\overline{\mathfrak{p}}}}(\widetilde{\Xi}).

∎

Proposition 8.8.

There exist $L_{\sharp,\sharp},L_{\sharp,\flat},L_{\flat,\sharp},L_{\flat,\flat}\in\Lambda_{E}(G_{p^{\infty}})$ such that

\begin{pmatrix}L_{\sharp,\alpha_{\overline{\mathfrak{p}}}}&L_{\flat,\alpha_{\overline{\mathfrak{p}}}}\\[6.0pt] L_{\sharp,\beta_{\overline{\mathfrak{p}}}}&L_{\flat,\beta_{\overline{\mathfrak{p}}}}\end{pmatrix}=(Q_{\overline{\mathfrak{p}}}^{-1}\underline{M_{\overline{\mathfrak{p}}}})\begin{pmatrix}L_{\sharp,\sharp}&L_{\flat,\sharp}\\[6.0pt] L_{\sharp,\flat}&L_{\flat,\flat}\end{pmatrix}.

Proof.

The proof is similar to the proof of [19, Proposition 2.5]. We will prove that

\begin{pmatrix}L_{\sharp,\alpha_{\overline{\mathfrak{p}}}}\\[6.0pt] L_{\sharp,\beta_{\overline{\mathfrak{p}}}}\end{pmatrix}=(Q^{-1}_{\overline{\mathfrak{p}}}\underline{M_{\overline{\mathfrak{p}}}})\begin{pmatrix}L_{\sharp,\sharp}\\[6.0pt] L_{\sharp,\flat}\end{pmatrix}.

The proof for the other set of power series is similar.

Let $\omega_{1}\in S_{\mathfrak{p}}$ and $\omega_{2}\in S_{\overline{\mathfrak{p}}}$ and $\Xi=\omega_{1}.\omega_{2}$ . Recall, from Proposition 8.6, for $*\in\{\alpha_{\overline{\mathfrak{p}}},\beta_{\overline{\mathfrak{p}}}\}$ , we have

(8.22)

L_{\sharp,*}=\dfrac{P_{4,\mathfrak{p}}L_{\alpha_{\mathfrak{p}},*}-P_{2,\mathfrak{p}}L_{\beta_{\mathfrak{p}},*}}{\det(Q^{-1}_{\mathfrak{p}}\underline{M_{\mathfrak{p}}})}.

Thus,

(8.23)

L_{\sharp,*}\det(Q^{-1}_{\mathfrak{p}}\underline{M_{\mathfrak{p}}})=P_{4,\mathfrak{p}}L_{\alpha_{\mathfrak{p}},*}-P_{2,\mathfrak{p}}L_{\beta_{\mathfrak{p}},*}.

From Theorem 8.2, we know that $\det(\underline{M_{\mathfrak{p}}})$ is equal to, upto a $p$ -adic unit, $\dfrac{\log_{p,k-1}(\gamma_{\mathfrak{p}})}{\delta_{k-1}(\gamma_{\mathfrak{p}})}$ . Therefore, $\det(Q^{-1}_{\mathfrak{p}}\underline{M_{\mathfrak{p}}})(\widetilde{\omega_{1,\mathfrak{p}}})=0$ . Hence, $(\partial_{\mathfrak{p}}\det(Q^{-1}_{\mathfrak{p}}\underline{M_{\mathfrak{p}}}))(\widetilde{\omega_{1,\mathfrak{p}}})\neq 0$ .

For the rest of the proof, we will use isotypic components corresponding to the trivial character of $\Delta_{K}$ . From (8.23), we get,

\partial_{\mathfrak{p}}L^{\Delta_{K}}_{\sharp,*}\det(Q^{-1}_{\mathfrak{p}}\underline{M_{\mathfrak{p}}})+L^{\Delta_{K}}_{\sharp,*}\partial_{\mathfrak{p}}\det(Q^{-1}_{\mathfrak{p}}\underline{M_{\mathfrak{p}}})=\partial_{\mathfrak{p}}P_{4,\mathfrak{p}}L^{\Delta_{K}}_{\alpha_{\mathfrak{p}},*}+P_{4,\mathfrak{p}}\partial_{\mathfrak{p}}L^{\Delta_{K}}_{\alpha_{\mathfrak{p}},*}-(\partial_{\mathfrak{p}}P_{2,\mathfrak{p}}L^{\Delta_{K}}_{\beta_{\mathfrak{p}},*}+P_{2,\mathfrak{p}}\partial_{\mathfrak{p}}L^{\Delta_{K}}_{\beta_{\mathfrak{p}},*}).

We evaluate the above equation at $\widecheck{\Xi}=\widetilde{\Xi}|_{\overline{\langle\gamma_{\mathfrak{p}}\rangle}\times\overline{\langle\gamma_{\overline{\mathfrak{p}}}\rangle}}$ , where $\Xi=\omega_{1}\omega_{2}$ and apply Proposition 8.7 together with the equations (8.4) to (8.7) to get

(8.24)

L^{\Delta_{K}}_{\sharp,*}(\widecheck{\Xi}).(\partial_{\mathfrak{p}}\det(Q^{-1}_{\mathfrak{p}}\underline{M_{\mathfrak{p}}}))(\widecheck{\Xi})=(*)^{-n_{\overline{\mathfrak{p}}}}\cdot M_{\widetilde{\Xi}},

where $M_{\widetilde{\Xi}}$ is the constant

(\partial_{\mathfrak{p}}P_{4,\mathfrak{p}})(\widecheck{\Xi})(\alpha_{\mathfrak{p}}^{-n_{\mathfrak{p}}}C_{a,b,\widetilde{\Xi}})+P_{4,\mathfrak{p}}(\widecheck{\Xi})A_{a,b,\widetilde{\Xi}}-(\partial_{\mathfrak{p}}P_{2,\mathfrak{p}})(\widecheck{\Xi})(\beta_{\mathfrak{p}}^{-n_{\mathfrak{p}}}C_{a,b,\widetilde{\Xi}})-P_{2,\mathfrak{p}}(\widecheck{\Xi})B_{a,b,\widetilde{\Xi}}.

In other words, we have

	$\displaystyle L_{\sharp,\alpha_{\overline{\mathfrak{p}}}}^{\Delta_{K}}(\widecheck{\Xi})$	$\displaystyle=\alpha_{\overline{\mathfrak{p}}}^{-n_{\overline{\mathfrak{p}}}}\dfrac{M_{\widetilde{\Xi}}}{(\partial_{\mathfrak{p}}\det(Q^{-1}_{\mathfrak{p}}\underline{M_{\mathfrak{p}}}))(\widecheck{\Xi})},$
	$\displaystyle L_{\sharp,\beta_{\overline{\mathfrak{p}}}}^{\Delta_{K}}(\widecheck{\Xi})$	$\displaystyle=\beta_{\overline{\mathfrak{p}}}^{-n_{\overline{\mathfrak{p}}}}\dfrac{M_{\widetilde{\Xi}}}{(\partial_{\mathfrak{p}}\det(Q^{-1}_{\mathfrak{p}}\underline{M_{\mathfrak{p}}}))(\widecheck{\Xi})}.$

Since $\Xi=\omega_{1}\omega_{2}$ , we can rewrite the above equations as

	$\displaystyle L_{\sharp,\alpha_{\overline{\mathfrak{p}}}}^{\Delta_{K}(\widetilde{\omega_{1,\mathfrak{p}}})}(\widetilde{\omega_{2,\overline{\mathfrak{p}}}})$	$\displaystyle=\alpha_{\overline{\mathfrak{p}}}^{-n_{\overline{\mathfrak{p}}}}\dfrac{M_{\widetilde{\Xi}}}{(\partial_{\mathfrak{p}}\det(Q^{-1}_{\mathfrak{p}}\underline{M_{\mathfrak{p}}}))(\widecheck{\Xi})},$
	$\displaystyle L_{\sharp,\beta_{\overline{\mathfrak{p}}}}^{\Delta_{K}(\widetilde{\omega_{1,\mathfrak{p}}})}(\widetilde{\omega_{2,\overline{\mathfrak{p}}}})$	$\displaystyle=\beta_{\overline{\mathfrak{p}}}^{-n_{\overline{\mathfrak{p}}}}\dfrac{M_{\widetilde{\Xi}}}{(\partial_{\mathfrak{p}}\det(Q^{-1}_{\mathfrak{p}}\underline{M_{\mathfrak{p}}})).(\widecheck{\Xi})}.$

Thus, after using Theorem 5.5 and the proof of Proposition 8.6 (we need to change $\mathfrak{p}$ with $\overline{\mathfrak{p}}$ ), we deduce the desired result. ∎

Proof of Theorem 8.3.

Combining the factorizations obtained in Proposition 8.6 and Proposition 8.8, we deduce the result. ∎

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Signed pp-adic LL-functions of Bianchi modular forms

Abstract.

Key words and phrases:

1. Introduction

Remark 1.1.

Theorem A (Theorem 8.3).

1.1. Plan of the article

Theorem B (Theorem 5.5).

Remark 1.2.

Remark 1.3.

Acknowledgements

2. Setup and notations

Assumption 2.1.

Iwasawa algebras

Power series rings

Fontaine’s rings

Mellin transform

3. Crystalline representations and Wach modules

3.1. Crystalline representations

3.1.1. Crystalline representations as filtered φ\varphi-modules

Theorem 3.1 (Colmez-Fontaine[9], Berger-Li-Zhu[4]).

Proof.

3.2. Wach modules

3.2.1. Famillies of Wach modules

Lemma 3.2.

Proof.

Lemma 3.3.

Proof.

Theorem 3.4.

Proof.

4. Perrin-Riou’s big logarithm map, Coleman maps, and the pp-adic regulator

4.1. Iwasawa cohomology and Wach modules

Theorem 4.1 (Berger, [2, Theorem A.3]).

4.2. Coleman maps

Theorem 4.2 (Lei-Loeffler-Zerbes, Berger).

Proof.

Definition 4.3 (The Coleman map).

Remark 4.4.

4.3. The big logarithm map, the pp-adic regulator and the relation between them

Definition 4.5 (The pp-adic regulator).

Lemma 4.6.

Proof.

Lemma 4.7 (Lei-Loeffler-Zerbes [21]).

Proof.

Proposition 4.8.

Proof.

Lemma 4.9.

Proof.

4.4. The matrices of ΩW,k−2,ℒW\Omega_{W,k-2},\mathcal{L}_{W} and their connection with M¯\underline{M}

4.4.1. The matrix of ΩTW,k−1\Omega_{T_{W},k-1}

Lemma 4.10.

Proof.

4.4.2. The matrix of the pp-adic regulator ℒTW\mathcal{L}_{T_{W}}

Lemma 4.11.

Proof.

Proposition 4.12.

Proof.

5. Logarithmic matrix M¯\underline{M} and the factorization of power series in one variable

5.1. Properties of M¯\underline{M}

Proposition 5.1.

Proof.

Lemma 5.2.

Proof.

Lemma 5.3.

Proof.

5.2. Factorization using M¯\underline{M}

Lemma 5.4.

Proof.

Theorem 5.5.

Proof.

6. Preliminaries about ray class groups, Hecke characters, and the two variable distribution algebras

Assumption 6.1.

6.1. Ray class groups and ray class fields

Remark 6.2.

Remark 6.3.

6.2. Hecke characters as the characters on the ray class groups

Definition 6.4 (Hecke characters).

Definition 6.5 (pp-adic avatar of an algebraic Hecke character).

Definition 6.6.

6.3. Two variable distribution modules

Signed $p$ -adic $L$ -functions of Bianchi modular forms

3.1.1. Crystalline representations as filtered $\varphi$ -modules

4. Perrin-Riou’s big logarithm map, Coleman maps, and the $p$ -adic regulator

4.3. The big logarithm map, the $p$ -adic regulator and the relation between them

Definition 4.5 (The $p$ -adic regulator).

4.4. The matrices of $\Omega_{W,k-2},\mathcal{L}_{W}$ and their connection with $\underline{M}$

4.4.1. The matrix of $\Omega_{T_{W},k-1}$

4.4.2. The matrix of the $p$ -adic regulator $\mathcal{L}_{T_{W}}$

5. Logarithmic matrix $\underline{M}$ and the factorization of power series in one variable

5.1. Properties of $\underline{M}$

5.2. Factorization using $\underline{M}$

Definition 6.5 ( $p$ -adic avatar of an algebraic Hecke character).

7. Bianchi modular forms and their $p$ -adic $L$ -functions

Definition 7.4 (Twisted $L$ -function of a Bianchi modular form).

7.2. $p$ -adic $L$ -function of Bianchi modular forms

8. Factorisation of $p$ -adic $L$ -functions of Bianchi modular forms

8.1. $p$ -stabilization of Bianchi modular forms