Non-cuspidal Bianchi modular forms and Katz $p$-adic $L$-functions

TL;DR

This paper constructs $p$-adic $L$-functions for non-cuspidal Bianchi modular forms over imaginary quadratic fields, introducing new notions and demonstrating factorization properties when $p$ splits.

Contribution

It introduces the concepts of $C$-cuspidality and partial Bianchi modular symbols to define $p$-adic $L$-functions for non-cuspidal forms, extending existing theories.

Findings

Constructed $p$-adic $L$-functions for non-cuspidal Bianchi forms.

Proved factorization of $p$-adic $L$-functions into Katz $p$-adic $L$-functions when $p$ splits.

Extended the theory of $p$-adic $L$-functions to non-cuspidal automorphic forms.

Abstract

Let $K$ be an imaginary quadratic field. In this article, we construct $p$-adic $L$-functions of non-cuspidal Bianchi modular forms by introducing the notions of $C$-cuspidality and partial Bianchi modular symbols. When $p$ splits in $K$, we focus on $p$-adic $L$-functions of non-cuspidal base change Bianchi modular forms, showing that they factor as products of two Katz $p$-adic $L$-functions.

Equations

$$\{\text{cuspidal}\}\subset \{C\text{-cuspidal}\}\subset \{\text{Bianchi modular forms}\}.$$

$$L_p^{\iota }(\mathcal{F},{\psi }_{p-\mathrm{fin}})=(\ast )\Lambda (\mathcal{F},\psi ),$$

$$L_p(f_{\phi /K}^p,{\psi }_{p-\mathrm{fin}})=(\ast )\Lambda (f_{\phi /K}^p,\psi ).$$

$$L_p(f_{\phi /K}^p,-)=\frac{L_p^{\iota }(f_{\phi /K}^p,-)}{\iota ({\Omega }_{f_{\phi /K}})}.$$

$$L_p(f_{\phi /K}^p,\kappa )=\frac{L_p({\phi }_{p-\mathrm{fin}}^c\kappa {\chi }_p)L_p({\phi }_{p-\mathrm{fin}}^c{\kappa }^c{\chi }_p)}{{\Omega }_p^{2k+2}},$$

$${\int }_{K\backslash {\mathbb{A}}_K}\mathcal{F}\left(\left(\begin{array}{cc}1& u\\ 0& 1\end{array}\right)g\right)du=0,$$

$$f^i$$

$$(z,t)⟼t^{-1}{F}^i(\begin{array}{cc}t& z\\ 0& 1\end{array}).$$

$$f^i(\gamma \cdot (z,t))={\phi }_{\mathfrak{n}}(d{)}^{-1}{f}^i(z,t){\rho }_{k+\ell +2}(J(\gamma ;(z,t))),$$

$$(f^i{|}_{\gamma })(z,t)=det(\gamma {)}^{-k/2}{\overline{det(\gamma )}}^{-\ell /2}{f}^i(\gamma \cdot (z,t)){\rho }_{k+\ell +2}^{-1}\left(J\left(\frac{\gamma }{\sqrt{det(\gamma )}};(z,t)\right)\right).$$

$$h(j)=\frac{k+\ell +2}{2}+\frac{j_2-j_1}{2},\text{and}\iota (j)=\left(\frac{1}{2}[j_1-(k+1)],\frac{1}{2}[j_2-(\ell +1)]\right).$$

$$\displaystyle\mathcal{F}\left[\left(\begin{matrix}t&z\\ 0&1\end{matrix}\right)\right]=|t|_{\mathbb{A}_{K}}\bigg{[}$$

$$\displaystyle+\sum_{\alpha\in K^{\times}}c(\alpha t\mathcal{D},\mathcal{F})W(\alpha t_{\infty})e_{K}(\alpha z)\bigg{]},$$

$$e_K=\left(\prod _{\mathfrak{q}\mathrm{prime}}(e_q\circ {\mathrm{Tr}}_{K_{\mathfrak{q}}/{\mathbb{Q}}_q})\right)\cdot (e_{\infty }\circ {\mathrm{Tr}}_{\mathbb{C}/\mathbb{R}}),$$

$$e_q\left(\sum _j{d}_j{q}^j\right)=e^{-2\pi i{\sum }_{j<0}{d}_j{q}^j}\mathrm{and}{e}_{\infty }(r)=e^{2\pi ir};$$

$$W(s):=\sum _{n=0}^{k+\ell +2}\left(\genfrac{}{}{0ex}{}{k+\ell +2}{n}\right){\left(\frac{s}{i|s|}\right)}^{\ell +1-n}{K}_{n-\ell -1}(4\pi |s|)X^{k+\ell +2-n}{Y}^n,$$

$$f^i\left((z,t);\left(\begin{array}{c}X\\ Y\end{array}\right)\right)=\sum _{n=0}^{k+\ell +2}{f}_n^i(z,t)X^{k+\ell +2-n}{Y}^n,$$

$$f_n^i(z,t)$$

$$+|t_i{|}_{f}t(\genfrac{}{}{0ex}{}{k+\ell +2}{n})\sum _{\alpha \in K^{\times }}\left[c(\alpha t_i\mathcal{D}){\left(\frac{\alpha }{i|\alpha |}\right)}^{\ell +1-n}{K}_{n-\ell -1}(4\pi |\alpha |t)e^{2\pi i(\alpha z+\overline{\alpha z})}\right]$$

$$C_i:={\Gamma }_i(\mathfrak{m})\infty \cup {\Gamma }_i(\mathfrak{m})0.$$

$$\{\text{cuspidal}\}\subset \{C\text{-cuspidal}\}\subset \{\text{Bianchi modular forms}\}.$$

$$\mathcal{F}{|}_{T_{\mathfrak{q}}}=(f^1,...,f^h){|}_{T_{\mathfrak{q}}}=\left(f^{j_1}{|}_{\left[{\Gamma }_{j_1}(\mathfrak{n})(\begin{array}{cc}1& 0\\ 0& {\alpha }_1\end{array}){\Gamma }_1(\mathfrak{n})\right]},...,f^{j_h}{|}_{\left[{\Gamma }_{j_h}(\mathfrak{n})(\begin{array}{cc}1& 0\\ 0& {\alpha }_h\end{array}){\Gamma }_h(\mathfrak{n})\right]}\right).$$

$${\alpha }_i=\frac{y_{\mathfrak{q}}{y}_i}{t_{j_i}}.$$

$$(\begin{array}{cc}1& 0\\ 0& {\alpha }_i\end{array})\cdot \frac{x}{y}=\frac{x}{{\alpha }_{i}y}=\frac{t_{j_i}x}{y_{\mathfrak{q}}{y}_{i}y}$$

$$\gamma {\sigma }_{s_i}\cdot \infty ={\gamma }_{j_i}(\begin{array}{cc}1& 0\\ 0& {\alpha }_i\end{array}){\gamma }_i{\sigma }_{s_i}\cdot \infty ={\gamma }_{j_i}(\begin{array}{cc}1& 0\\ 0& {\alpha }_i\end{array}){\gamma }_i\cdot s_i={\gamma }_{j_i}(\begin{array}{cc}1& 0\\ 0& {\alpha }_i\end{array})\cdot s_i^{\prime }={\gamma }_{j_i}\cdot s_{j_i}=s_{j_i}^{\prime }\in C_{j_i}.$$

$$L(\mathcal{F},\psi ,s):=\sum _{\begin{array}{c}0\ne \mathfrak{a}\subset {\mathcal{O}}_K\\ (\mathfrak{a},\mathfrak{f})=1\end{array}}c(\mathfrak{a},\mathcal{F})\psi (\mathfrak{a})N(\mathfrak{a}{)}^{-s}(s\in \mathbb{C})$$

$$L^i(\mathcal{F},\psi ,s)=L(f^i,\psi ,s):=w^{-1}\sum _{\alpha \in K^{\times }}c(\alpha \delta I_i,\mathcal{F})\psi (\alpha \delta I_i)N(\alpha \delta I_i{)}^{-s}$$

$$L(\mathcal{F},\psi ,s)=L^1(\mathcal{F},\psi ,s)+\cdot \cdot \cdot +L^h(\mathcal{F},\psi ,s).$$

$$\tau (\psi )=\sum _{\begin{array}{c}[a]\in {\mathfrak{f}}^{-1}/{\mathcal{O}}_K\\ ((a)\mathfrak{f},\mathfrak{f})=1\end{array}}\psi (a\mathfrak{f}){\psi }_{\infty }(a/\delta )e^{2\pi i{\mathrm{Tr}}_{K/\mathbb{Q}}(a/\delta )}.$$

$$\frac{1}{\tau ({\psi }^{-1})}\sum _{\begin{array}{c}[a]\in {\mathfrak{f}}^{-1}/{\mathcal{O}}_K\\ ((a)\mathfrak{f},\mathfrak{f})=1\end{array}}\psi (a\mathfrak{f}{)}^{-1}{\psi }_{\infty }(a/\delta {)}^{-1}{e}^{2\pi i{\mathrm{Tr}}_{K/\mathbb{Q}}(ac/\delta )}=\{\begin{array}{ll}{\psi }_{\mathfrak{f}}(c{)}^{-1}& :((c)\mathfrak{f},\mathfrak{f})=1,\\ 0& :\mathrm{otherwise}.\end{array}$$

$$\tau (\psi )=\psi (x_{\mathfrak{f}}){\psi }_{\mathfrak{f}}(x_{\mathfrak{f}}{)}^{-1}{\psi }_{\infty }(\delta {)}^{-1}\sum _{\begin{array}{c}[a]\in {\mathfrak{f}}^{-1}/{\mathcal{O}}_K\\ ((a)\mathfrak{f},\mathfrak{f})=1\end{array}}{\psi }_{\mathfrak{f}}(a{)}^{-1}{e}^{2\pi i{\mathrm{Tr}}_{K/\mathbb{Q}}(a/\delta )}.$$

$$c_{s,u}^i(a)=2{\left(\genfrac{}{}{0ex}{}{k+\ell +2}{\ell +s-u+1}\right)}^{-1}(-1{)}^{\ell -u+1}{\int }_0^{\infty }{t}^{s+u}{f}_{\ell +s-u+1}^i(a,t)dt.$$

Full text

When $p$ splits in $K$, we deepen in the construction of $p$-adic $L$-functions of non-cuspidal base change Bianchi modular forms, showing that they factor as products of two Katz $p$-adic $L$-functions.

1. Introduction

Let $p$ be a fixed prime number. $P$-adic $L$-functions are fundamental objects in modern number theory, that serve as a bridge between algebraic number theory, arithmetic geometry, and analysis.

In [PS11], Pollack and Stevens gave a construction of the $p$-adic $L$-function of cuspidal modular forms using the theory of overconvergent modular symbols and after that, there have been several works generalizing their ideas (see for example [BS18], [Wil17], [BW19], for the context of $\mathrm{GL}_{2}$). For non-cuspidal modular forms, Bellaïche and Dasgupta introduced the notions of $C$-cuspidality and partial modular symbols in [BD15], and constructed the $p$-adic $L$-function of Eisenstein series.

Let $K$ be an imaginary quadratic field, and consider the automorphic forms for $\mathrm{GL}_{2}$ over $K$, better known as Bianchi modular forms. In this situation, Williams constructed in [Wil17] the $p$-adic $L$-function in the cuspidal case by developing the Bianchi modular symbols. In this article, we construct the $p$-adic $L$-function of non-cuspidal Bianchi modular forms by adapting and combining the methods of Williams and Bellaïche-Dasgupta. Moreover, we show an important connection between Katz $p$-adic $L$-functions and our construction in the case of non-cuspidal base change Bianchi modular forms.

1.1. $C$-cuspidal Bianchi modular forms and partial symbols

For our study of non-cuspidal Bianchi modular forms, in section 2.3 we generalize to the Bianchi setting the notion of $C$-cuspidality given in [BD15]. Such property is related with the vanishing of constant terms of Fourier expansions at suitable cusps and for Bianchi modular forms with level at $p$ we have:

[TABLE]

For cuspidal Bianchi modular forms, there exists an integral formula for their complex $L$-function. For $C$-cuspidal Bianchi modular forms, we prove an analogous formula in Proposition 2.18.

In section 3, we introduce algebraic analogues of $C$-cuspidal Bianchi modular forms, called partial Bianchi modular symbols, that are easier to study $p$-adically. To define them, we generalize the classical partial modular symbols introduced in [BD15] and adapt the Bianchi modular symbols of parallel weight used in [Wil17]. In Proposition 3.9, we prove that we can attach, in a Hecke-equivariant way, a partial Bianchi modular symbol to a $C$-cuspidal Bianchi modular form.

To link partial Bianchi modular symbols with spaces of $p$-adic distributions, we introduce the overconvergent partial Bianchi modular symbols in section 4.2 and generalizing [Wil17], we obtain a classicality result in Proposition 4.3.

In section 4.4, we construct the $p$-adic $L$-function of a $C$-cuspidal Bianchi eigenform $\mathcal{F}$ of weight $(k,\ell)$ and level $K_{0}(\mathfrak{n})$, with $U_{\mathfrak{p}}$-eigenvalues $\lambda_{\mathfrak{p}}$ having suitable $p$-adic valuation (see Definition 4.5). For this, we first attach to $\mathcal{F}$ a complex-valued partial Bianchi modular eigensymbol $\phi_{\mathcal{F}}$ using Proposition 3.9. Then, by fixing an isomorphism $\iota$ between $\mathbb{C}$ and $\overline{\mathbb{Q}}_{p}$, we view $p$-adically the values of $\phi_{\mathcal{F}}$ and then lift it uniquely to an overconvergent partial Bianchi modular eigensymbol $\Psi_{\mathcal{F}}$ using Proposition 4.3. After taking the Mellin transform of $\Psi_{\mathcal{F}}$, we obtain (see Theorem 4.12 for the precise statement):

Theorem 1.1**.**

There exists a locally analytic distribution $L_{p}^{\iota}(\mathcal{F},-)$ on the ray class group $\mathrm{Cl}_{K}(p^{\infty})$, such that for any Hecke character $\psi$ of $K$ of conductor $\mathfrak{f}|(p^{\infty})$ and infinity type $0\leqslant(q,r)\leqslant(k,\ell)$, we have

[TABLE]

where $\psi_{p-\mathrm{fin}}$ is the $p$-adic avatar of $\psi$, $(*)$ is an explicit factor depending on $\psi$ and $\Lambda(\mathcal{F},-)$ is the normalized $L$-function of $\mathcal{F}$.

The distribution $L_{p}^{\iota}(\mathcal{F},-)$ satisfies suitable growth conditions (see section 4.3) and hence is unique.

By Remark 4.13, we can construct the $p$-adic $L$-function of certain non-cuspidal Bianchi modular forms by turning them into $C$-cuspidal forms and applying the previous theorem.

1.2. Non-cuspidal base change and Katz $p$-adic $L$-functions

A case of historical interest where we can apply the methods in section 1.1 to construct $p$-adic $L$-functions is the non-cuspidal base change situation.

Suppose that $p$ splits in $K$ and let $\varphi$ be a Hecke character of $K$ with conductor $\mathfrak{M}$ coprime to $p$ and infinity type $(-k-1,0)$ with $k\geqslant 0$. Denote by $f_{\varphi}$ the elliptic CM modular form induced by $\varphi$ and let $f_{\varphi/K}$ be the base change to $K$ of $f_{\varphi}$, which is known to be a non-cuspidal Bianchi modular form of weight $(k,k)$. In Proposition 5.3, we prove that the ordinary $p$-stabilization of $f_{\varphi/K}$, which we denote by $f_{\varphi/K}^{p}$, is a $C$-cuspidal Bianchi modular form. In particular, by Theorem 1.1, we can obtain its $p$-adic $L$-function $L_{p}^{\iota}(f_{\varphi/K}^{p},-)$.

Part of our interest in the Bianchi modular form $f_{\varphi/K}^{p}$, relies on the fact that we can deepen the construction of its $p$-adic $L$-function. More precisely, we can avoid the use of an isomorphism between $\mathbb{C}$ and $\overline{\mathbb{Q}}_{p}$ and additionally, we can factorize its $p$-adic $L$-function as the product of two Katz $p$-adic $L$-functions.

In section 5.2, we study the $L$-function of $f_{\varphi/K}$ and prove that it factors as the product of two Hecke $L$-functions in Lemma 5.4. Using such factorization, combined with an algebraicity result of Hecke $L$-functions, we show the existence of a complex period $\Omega_{f_{\varphi/K}}$, which allows us to prove algebraicity of critical $L$-values of $f_{\varphi/K}$ in Proposition 5.7.

Let $\phi_{f_{\varphi/K}}$ be the complex-valued partial Bianchi modular symbol attached to $f_{\varphi/K}^{p}$ in Proposition 3.9, then by defining $\phi^{\prime}_{f_{\varphi/K}}:=\phi_{f_{\varphi/K}}/\Omega_{f_{\varphi/K}}$ we prove in Proposition 5.10 that $\phi^{\prime}_{f_{\varphi/K}}$ has algebraic values and consequently, we can view it as having $p$-adic values. Finally, using Proposition 4.3, we can lift $\phi^{\prime}_{f_{\varphi/K}}$ to an overconvergent Bianchi modular symbol and taking its Mellin transform we obtain the following (see Theorem 5.11):

Theorem 1.2**.**

There exists a unique measure $L_{p}(f_{\varphi/K}^{p},-)$ on the ray class group $\mathrm{Cl}_{K}(p^{\infty})$ such that for any Hecke character $\psi$ of $K$ of conductor $\mathfrak{f}|p^{\infty}$ and infinity type $0\leqslant(q,r)\leqslant(k,k)$, we have

[TABLE]

Remark 1.3*.*

The measures $L_{p}(f_{\varphi/K}^{p},-)$ and $L_{p}^{\iota}(f_{\varphi/K}^{p},-)$ obtained from Theorems 1.2 and 1.1 respectively can be related, in Proposition 5.12, we prove

[TABLE]

The factorization of the $L$-function of $f_{\varphi/K}$ in Lemma 5.4 translates to the $p$-adic side. In fact, consider the $p$-adic $L$-function of $f_{\varphi/K}$ of Theorem 1.2 and the $p$-adic $L$-function $L_{p}(-)$ constructed by Katz in [Kat78] and generalized by Hida and Tilouine in [HT93]. We obtain the following (see Theorem 5.14 for more details):

Theorem 1.4**.**

Under the hypothesis of Theorem 1.2, for all $p$-adic characters $\kappa$ we have

[TABLE]

where $\chi_{p}$ is the $p$-adic avatar of the adelic norm character and $\Omega_{p}$ is the $p$-adic period in Theorem 5.13.

Acknowledgements

I would like to thank my PhD advisor Daniel Barrera for suggesting this topic to me, as well as for the many conversations we have had on the subject. Special thanks to Chris Williams for kindly answering a lot of questions. I also thank Samit Dasgupta, Antonio Cauchi, Guhan Venkat and Eduardo Friedman for helpful comments that led to an improvement of this article. This work was funded by the National Agency for Research and Development (ANID)/Scholarship Program/BECA DOCTORADO NACIONAL/2018 - 21180506. Likewise, some research visits were supported by the project FONDECYT 11201025.

2. $C$-cuspidal Bianchi modular forms

In this section, we recall basic properties of Bianchi modular forms and define a suitable vanishing condition on its constant terms called $C$-cuspidality. In section 2.4 we study $L$-functions of $C$-cuspidal forms.

2.1. Notations

Let $p$ be a rational prime and fix throughout the paper embeddings $\iota_{\infty}:\overline{\mathbb{Q}}\hookrightarrow\mathbb{C}$ and $\iota_{p}:\overline{\mathbb{Q}}\hookrightarrow\overline{\mathbb{Q}}_{p}$, note that $\iota_{p}$ fixes a $p$-adic valuation $v_{p}$ on $\overline{\mathbb{Q}}_{p}$. Let $K$ be an imaginary quadratic field with discriminant $-D$, ring of integers $\mathcal{O}_{K}$ and different ideal $\mathcal{D}$ generated by $\delta=\sqrt{-D}$. Let $\mathfrak{n}=(p)\mathfrak{m}$ be an ideal of $\mathcal{O}_{K}$ with $\mathfrak{m}$ coprime to $(p)$. Denote the two embeddings of $K$ into $\mathbb{C}$ by $id$ and $c$ and henceforth write $(k,\ell)$ for $k\cdot id+\ell\cdot c\in\mathbb{Z}[{id,c}]$, with $k,\ell\geqslant 0$. Note that we are implicitly viewing $K\hookrightarrow\overline{\mathbb{Q}}$ under $\iota_{\infty}^{-1}\circ id$. Denote by $K_{\mathfrak{q}}$ the completion of $K$ with respect to the prime $\mathfrak{q}$ of $K$, $\mathcal{O}_{\mathfrak{q}}$ the ring of integers of $K_{\mathfrak{q}}$ and fix a uniformizer $\pi_{\mathfrak{q}}$ at $\mathfrak{q}$. Denote the adele ring of $K$ by $\mathbb{A}_{K}=\mathbb{C}\times\mathbb{A}_{K}^{f}$ where $\mathbb{A}_{K}^{f}$ are the finite adeles. Furthermore, denote the class group of $K$ by $\mathrm{Cl}(K)$ and the class number of $K$ by $h$, and -once and for all- fix a set of representatives $I_{1},...,I_{h}$ for $\mathrm{Cl}(K)$, with $I_{1}=\mathcal{O}_{K}$ and each $I_{i}$ for $2\leqslant i\leqslant h$ integral and prime, with each $I_{i}$ coprime to $\mathfrak{n}$ and $\mathcal{D}$. Let $V_{n}(R)$ denote the space of homogeneous polynomials over a ring $R$ in two variables of degree $n\geqslant 0$. Note that $V_{n}(\mathbb{C})$ is an irreducible complex right representation of $\mathrm{SU}_{2}(\mathbb{C})$, denote it by $\rho_{n}$.

For a general Hecke character $\psi$ of $K$ we denote by $\psi_{\infty}$, $\psi_{f}$ and $\psi_{\mathfrak{q}}$ the restriction of $\psi$ to $\mathbb{C}^{\times}$, $\mathbb{A}_{K}^{f,\times}$ and $K_{\mathfrak{q}}^{\times}$ respectively.

2.2. Bianchi modular forms

We write $K_{0}(\mathfrak{n})$ for the open compact subgroup of $\mathrm{GL_{2}}(\mathcal{O}_{K}\otimes_{\mathbb{Z}}\widehat{\mathbb{Z}})$ of matrices $(\begin{smallmatrix}*&*\\ 0&*\end{smallmatrix})$ modulo $\mathfrak{n}$. Let $\varphi$ be a Hecke character, with infinity type $(-k,-\ell)$ and conductor dividing $\mathfrak{n}$. For $u_{f}=(\begin{smallmatrix}a&b\\ c&d\end{smallmatrix})\in K_{0}(\mathfrak{n})$ we set $\varphi_{\mathfrak{n}}(u_{f})=\varphi_{\mathfrak{n}}(d)=\prod_{\mathfrak{q}|\mathfrak{n}}\varphi_{\mathfrak{q}}(d_{\mathfrak{q}})$.

Definition 2.1**.**

We say a function $\mathcal{F}:\mathrm{GL_{2}}(\mathbb{A}_{K})\rightarrow V_{k+\ell+2}(\mathbb{C})$ is a Bianchi modular form of weight $(k,\ell)$, level $K_{0}(\mathfrak{n})$ and central action $\varphi$ if it satisfies:

(i) $\mathcal{F}$ is left-invariant under $\mathrm{GL_{2}}(K)$;

(ii) $\mathcal{F}(zg)=\varphi(z)\mathcal{F}(g)$ for $z\in\mathbb{A}_{K}^{\times}\cong Z(\mathrm{GL_{2}}(\mathbb{A}_{K}))$, where $Z(G)$ denote the center of the group $G$;

(iii) $\mathcal{F}(gu)=\varphi_{\mathfrak{n}}(u_{f})\mathcal{F}(g)\rho_{k+\ell+2}(u_{\infty})$ for $u=u_{f}\cdot u_{\infty}\in K_{0}(\mathfrak{n})\times\mathrm{SU_{2}}(\mathbb{C})$;

(iv) $\mathcal{F}$ is an eigenfunction of the operators $D_{\sigma}$, for $\sigma\in\{id,c\}$, where $D_{\sigma}/4$ denotes a component of the Casimir operator in the Lie algebra $\mathfrak{sl}_{2}(\mathbb{C})\otimes_{\mathbb{R}}\mathbb{C}$, and where we consider $\mathcal{F}(g_{\infty}g_{f})$ as a function of $g_{\infty}\in\mathrm{GL_{2}}(\mathbb{C}).$

The space of such functions will be denoted by $\mathcal{M}_{(k,\ell)}(K_{0}(\mathfrak{n}),\varphi)$. We say $\mathcal{F}$ is a cuspidal Bianchi modular form if also satisfies:

(v) for all $g\in\mathrm{GL_{2}}(\mathbb{A}_{K})$

[TABLE]

where $du$ is the Lebesgue measure on $\mathbb{A}_{K}$. The space of such functions will be denoted by $S_{(k,\ell)}(K_{0}(\mathfrak{n}),\varphi)$.

Remark 2.2*.*

From [Hid94, Cor 2.2]) we have $S_{(k,\ell)}(K_{0}(\mathfrak{n}),\varphi)=0$ if $k\neq\ell$ i.e., all non-trivial cuspidal Bianchi modular forms have parallel weight $(k,k)$.

Recall the set of representatives $I_{1},...,I_{h}$ for $\mathrm{Cl}(K)$ fixed in section 2.1 and denote by $\pi_{i}$ their corresponding fixed uniformizer. Set $g_{i}=(\begin{smallmatrix}1&0\\ 0&t_{i}\end{smallmatrix})$ where $t_{1}=1$ and for each $i\geqslant 2$, define $t_{i}=(1,...,1,\pi_{i},1,...)\in\mathbb{A}_{K}^{\times}$. Since $\mathrm{GL_{2}}(\mathbb{A}_{K})=\coprod_{i=1}^{h}\mathrm{GL_{2}}(K)\cdot g_{i}\cdot[\mathrm{GL_{2}}(\mathbb{C})\times K_{0}(\mathfrak{n})]$, a Bianchi modular form $\mathcal{F}\in\mathcal{M}_{(k,\ell)}(K_{0}(\mathfrak{n}),\varphi)$, descends to a collection of $h$ functions $F^{i}:\mathrm{GL_{2}}(\mathbb{C})\longrightarrow V_{k+\ell+2}(\mathbb{C})$ via $F(g):=\mathcal{F}(g_{i}g)$.

Let $\mathcal{H}_{3}:=\mathbb{C}\times\mathbb{R}_{>0}$ be the hyperbolic 3-space. Since $\mathrm{GL_{2}}(\mathbb{C})=Z(\mathrm{GL_{2}}(\mathbb{C}))\cdot\mathrm{B}\cdot\mathrm{SU_{2}}(\mathbb{C})$ where $\mathrm{B}=\left\{(\begin{smallmatrix}t&z\\ 0&1\end{smallmatrix}):z\in\mathbb{C},t\in\mathbb{R}_{>0}\right\}\cong\mathcal{H}_{3}$, we can descend further using ii) and iii) in Definition 2.1 to obtain $h$ functions

[TABLE]

Let $\gamma=(\begin{smallmatrix}a&b\\ c&d\end{smallmatrix})\in\Gamma_{i}(\mathfrak{n}):=\mathrm{SL_{2}}(K)\cap g_{i}K_{0}(\mathfrak{n})g_{i}^{-1}\mathrm{GL_{2}}(\mathbb{C})$, then each $f^{i}$ satisfies the automorphy condition

[TABLE]

where $\gamma\cdot(z,t)=\left(\frac{(az+b)\overline{(cz+d)}+a\overline{c}|t|^{2}}{|cz+d|^{2}+|ct|^{2}},\frac{|ad-bc|t}{|cz+d|^{2}+|ct|^{2}}\right)$ and $J(\gamma;(z,t)):=(\begin{smallmatrix}cz+d&\overline{ct}\\ -ct&\overline{cz+d}\end{smallmatrix})$. Thus $f^{i}\in\mathcal{M}_{(k,\ell)}(\Gamma_{i}(\mathfrak{n}),\varphi_{\mathfrak{n}}^{-1})$, the space of Bianchi modular forms on $\mathcal{H}_{3}$ satisfying the automorphy condition (2.2). If $\mathcal{F}\in S_{(k,k)}(K_{0}(\mathfrak{n}),\varphi)$ is cuspidal then we say $f^{i}$ is a cuspidal Bianchi modular form and the space of such forms is denoted by $S_{(k,k)}(\Gamma_{i}(\mathfrak{n}),\varphi_{\mathfrak{n}}^{-1})$.

Definition 2.3**.**

Let $\gamma\in\mathrm{GL_{2}}(\mathbb{C})$, for $f^{i}\in\mathcal{M}_{(k,\ell)}(\Gamma_{i}(\mathfrak{n}),\varphi_{\mathfrak{n}}^{-1})$, define the function $f^{i}|_{\gamma}:\mathcal{H}_{3}\rightarrow V_{k+\ell+2}(\mathbb{C})$ by

[TABLE]

Remark 2.4*.*

Note $f^{i}\in\mathcal{M}_{(k,\ell)}(\Gamma_{i}(\mathfrak{n}),\varphi_{\mathfrak{n}}^{-1})$ satisfies:

i) $f^{i}|_{g}(0,1)=F^{i}(g)$ for $g\in\mathrm{GL}_{2}(\mathbb{C})$, and for $g=(\begin{smallmatrix}t&z\\ 0&1\end{smallmatrix})\in\mathrm{B}$ we obtain (2.1).

ii) $\left(f^{i}|_{\gamma}\right)(z,t)=\varphi_{\mathfrak{n}}(d)^{-1}f^{i}(z,t)$ for $\gamma=(\begin{smallmatrix}a&b\\ c&d\end{smallmatrix})\in\Gamma_{i}(\mathfrak{n})$.

2.3. Fourier expansion and cuspidal conditions

Recall that $k,\ell\geqslant 0$, consider the set $J_{(k,\ell)}=\{(\pm(k+1),\pm(\ell+1))\}$ and for $j=(j_{1},j_{2})\in J_{(k,\ell)}$ define

[TABLE]

Let $\mathcal{F}$ be a Bianchi modular form of weight $(k,\ell)$, level $K_{0}(\mathfrak{n})$ and central action $\varphi$, then it has a Fourier expansion given by (see [Hid94, Thm 6.7]):

[TABLE]

where

• i)

The Fourier coefficients $c(\cdot,\mathcal{F})$, and $c_{j}(\cdot,\mathcal{F})$ for each $j\in J_{(k,\ell)}$, are functions on the fractional ideals of $K$ that vanish outside the integral ideals.

• ii)

$e_{K}$ is an additive character of $K\backslash\mathbb{A}_{K}$ defined by

[TABLE]

for

[TABLE]

and

• iii)

$W:\mathbb{C}^{\times}\rightarrow V_{k+\ell+2}(\mathbb{C})$ is the Whittaker function

[TABLE]

where $K_{n}(x)$ is a Bessel function.

Remark 2.5*.*

i) Note that $W(s)$ is not symmetric in $k$ and $\ell$, this comes from the definition of the Whittaker function in [Hid94, (6.1)] after fixing the weight $(k,\ell)$.

ii) Let $\mathcal{F}=\sum_{n=0}^{k+\ell+2}\mathcal{F}_{n}X^{k+\ell+2-n}Y^{n}$ be a Bianchi modular form, then by (2.4), the constant term in the Fourier expansion of $\mathcal{F}_{n}$ is trivial if $n\notin\{h(j)|j\in J_{(k,\ell)}\}=\{0,k+1,\ell+1,k+\ell+2\}$.

The Fourier expansion of $\mathcal{F}$ descends to $\mathcal{H}_{3}$ by

[TABLE]

where

[TABLE]

with $j=(j_{1},j_{2})\in J_{(k,\ell)}$ as above and $\delta_{n,h(j)}=1$ if $n=h(j)$ and $\delta_{n,h(j)}=0$ otherwise.

Note that to ease notation, we have written $c_{j}(t_{i}\mathcal{D})$ (resp. $c(\alpha t_{i}\mathcal{D})$) instead $c_{j}(t_{i}\mathcal{D},\mathcal{F})$ (resp. $c(\alpha t_{i}\mathcal{D},\mathcal{F})$).

For each $i=1,...,h$, equation (2.3) may be thought of as the Fourier expansion of $f^{i}$ at the cusp of infinity, which by Remark 2.5, satisfies that the constant term in the Fourier expansion of $f^{i}_{n}$ is trivial if $n\notin\{0,k+1,\ell+1,k+\ell+2\}$.

We must consider Fourier expansions at all the “$K$-rational” cusps $\mathbb{P}^{1}(K)=K\cup\{\infty\}$, for this, let $\sigma\in\mathrm{GL}_{2}(K)$ sending $\infty$ to the cusp $s$. For each $i=1,...,h$, since $f^{i}\in\mathcal{M}_{(k,\ell)}(\Gamma_{i}(\mathfrak{n}),\varphi_{\mathfrak{n}}^{-1})$ then $f^{i}|_{\sigma}\in\mathcal{M}_{(k,\ell)}(\sigma^{-1}\Gamma_{i}(\mathfrak{n})\sigma,\varphi_{\mathfrak{n}}^{-1})$ and hence $f^{i}|_{\sigma}$ has a Fourier expansion as in (2.3).

Definition 2.6**.**

We say that $f^{i}$ vanishes at the cusp $s$ if $(f^{i}|_{\sigma})_{n}$ has trivial constant term for $0\leqslant n\leqslant k+\ell+2$, and quasi-vanishes at the cusp $s$ if $(f^{i}|_{\sigma})_{n}$ has trivial constant term for $1\leqslant n\leqslant k+\ell+1$.

Remark 2.7*.*

i) The property of vanishing and quasi-vanishing at the cusp $s$ are well-defined, i.e., are independent of the choice of $\sigma$; for the vanishing case see [Byg98, §6.2.2] and note that the same argument works for quasi-vanishing.

ii) Let $\mathcal{F}\in S_{(k,k)}(K_{0}(\mathfrak{n}),\varphi)$ be a cuspidal Bianchi modular form, then the cuspidal condition v) in Definition 2.1 is equivalent to the vanishing of $f^{i}$ at all cusps for each $0\leqslant i\leqslant h$ (see [Zha93, Prop 3.2]).

Recall that $\mathfrak{n}=(p)\mathfrak{m}$ with $\mathfrak{m}$ coprime to $(p)$ and define for each $i=1,...,h$ the set of cusps

[TABLE]

Since $\Gamma_{i}(\mathfrak{m})=\{(\begin{smallmatrix}a&b\\ c&d\end{smallmatrix})\in\mathrm{SL}_{2}(K):b\in I_{i},\;c\in\mathfrak{m}I_{i}^{-1}\}$, we have $C_{i}\subset\mathbb{P}^{1}(K)$ contains $\infty$ and elements $\frac{x}{y}\in K$ with $x\in I_{i}$ and either $y\in\mathfrak{m}$ or $y\in(\mathcal{O}_{K}/\mathfrak{m})^{\times}$.

Definition 2.8**.**

We say that $f^{i}$ is $C_{i}$-cuspidal if quasi-vanishes at all cusps in $C_{i}$.

The previous definition of $C_{i}$-cuspidality differs from the one given in [BD15] for modular forms. We are not asking for the vanishing of $f^{i}$ at the cusps $C_{i}$, instead, we just need quasi-vanishing, i.e., we do not care about the vanishing of the functions $f^{i}_{0}$ and $f^{i}_{k+\ell+2}$ at the cusps $C_{i}$. The motivation for considering quasi-vanishing instead of vanishing will become clear in Proposition 2.18.

To state $C_{i}$-cuspidality for all $i$ as a property of $\mathcal{F}$, we write $C:=(C_{1},...,C_{h})$ and introduce the notion of $C$-cuspidality.

Definition 2.9**.**

We say that $\mathcal{F}$ is C-cuspidal if $f^{i}$ is $C_{i}$-cuspidal for $i=1,...,h$.

Remark 2.10*.*

Note that for Bianchi modular forms with level at $p$ we have

[TABLE]

For $\mathfrak{q}\subset\mathcal{O}_{K}$ prime, consider our fixed uniformizer $\pi_{\mathfrak{q}}$ of $K_{\mathfrak{q}}$ as an element in $\mathbb{A}_{K}^{f,\times}$ trivial at every place $\neq\mathfrak{q}$. For each $\mathfrak{q}$ there is a Hecke operator acting on $\mathcal{M}_{(k,\ell)}(K_{0}(\mathfrak{n}),\varphi)$ given by the double coset $[K_{0}(\mathfrak{n})(\begin{smallmatrix}\pi_{\mathfrak{q}}&0\\ 0&1\end{smallmatrix})K_{0}(\mathfrak{n})]$. These operators are denoted by $T_{\mathfrak{q}}$ if $\mathfrak{q}\nmid\mathfrak{n}$ and $U_{\mathfrak{q}}$ otherwise, and are all independent of choices of representatives. An eigenform is a Bianchi modular form that is a simultaneous eigenvector for all the Hecke operators.

We can describe the action of Hecke operators on each Bianchi modular form $f^{i}$ for principal ideals, but when the ideals are not principal, we need to use the whole collection $(f^{1},...,f^{h})$.

Recall our fixed representatives $I_{1},...,I_{h}$ for the class group and consider the prime ideal $\mathfrak{q}\nmid\mathfrak{n}$, then for each $i\in\{1,...,h\}$ there is a unique $j_{i}\in\{1,...,h\}$ such that $\mathfrak{q}I_{i}=(\alpha_{i})I_{j_{i}}$, for $\alpha_{i}\in K$. Then $T_{\mathfrak{q}}$ acts on each component of $(f^{1},...,f^{h})$ by double cosets by

[TABLE]

Definition 2.11**.**

Let $\mathcal{H}_{\mathfrak{n},p}$ denote the $\mathbb{Q}$-algebra generated by the Hecke operators $\{T_{\mathfrak{q}}:(\mathfrak{q},\mathfrak{n})=1\}$ and $\{U_{\mathfrak{p}}:\mathfrak{p}|p\}$.

Proposition 2.12**.**

Let $\mathcal{F}\in\mathcal{M}_{(k,\ell)}(K_{0}(\mathfrak{n}),\varphi)$ be a $C$-cuspidal Bianchi modular form, with $\mathfrak{n}=(p)\mathfrak{m}$ and $(\mathfrak{m},(p))=1$; then $\mathcal{H}_{\mathfrak{n},p}$ acts on $C$-cuspidal forms.

Proof.

By (2.6) we have to show that for all prime $\mathfrak{q}\nmid\mathfrak{m}$ with $\mathfrak{q}I_{i}=(\alpha_{i})I_{j_{i}}$ the function $f^{j_{i}}|{\left[\Gamma_{j_{i}}(\mathfrak{n})(\begin{smallmatrix}1&0\\ 0&\alpha_{i}\end{smallmatrix})\Gamma_{i}(\mathfrak{n})\right]}$ is $C_{i}$-cuspidal. For this, take $s_{i}\in C_{i}$, $\sigma_{s_{i}}\in\mathrm{GL}_{2}(K)$ such that $\sigma_{s_{i}}\cdot\infty=s_{i}$ and $\gamma\in\Gamma_{j_{i}}(\mathfrak{n})(\begin{smallmatrix}1&0\\ 0&\alpha_{i}\end{smallmatrix})\Gamma_{i}(\mathfrak{n})$. We have to show that the constant term of $(f^{j_{i}}|_{\gamma\sigma_{s_{i}}})_{n}$ vanishes for $1\leqslant n\leqslant k+\ell+1$.

We first observe $2$ facts:

1. $\Gamma_{i}(\mathfrak{n})$ stabilizes $C_{i}$: note $\Gamma_{i}(\mathfrak{m})$ clearly stabilizes $C_{i}:=\Gamma_{i}(\mathfrak{m})\infty\cup\Gamma_{i}(\mathfrak{m})0$ and hence its subgroup $\Gamma_{i}(\mathfrak{n})$.

2. $(\begin{smallmatrix}1&0\\ 0&\alpha_{i}\end{smallmatrix})\cdot c_{i}\in C_{j_{i}}$ for all $c_{i}\in C_{i}$: since $(\mathfrak{q},\mathfrak{m})=1$ and $(I_{i},\mathfrak{m})=1$, there exists $y_{\mathfrak{q}}\in\mathfrak{q}$ and $y_{i}\in I_{i}$, such that $y_{\mathfrak{q}},y_{i}\in(\mathcal{O}_{K}/\mathfrak{m})^{\times}$. By the identity $\mathfrak{q}I_{i}=(\alpha_{i})I_{j_{i}}$, there exist an element $t_{j_{i}}\in I_{j_{i}}$ such that

[TABLE]

Let $c_{i}=x/y$, then $x\in I_{i}$ and either $y\in\mathfrak{m}$ or $y\in(\mathcal{O}_{K}/\mathfrak{m})^{\times}$, then we have

[TABLE]

with $t_{j_{i}}x\in I_{j_{i}}$ and either $y_{\mathfrak{q}}y_{i}y\in\mathfrak{m}$ or $y_{\mathfrak{q}}y_{i}y\in(\mathcal{O}_{K}/\mathfrak{m})^{\times}$ then $(\begin{smallmatrix}1&0\\ 0&\alpha_{i}\end{smallmatrix})\cdot c_{i}\in C_{j_{i}}$.

Now, going back to the proof, write $\gamma=\gamma_{j_{i}}(\begin{smallmatrix}1&0\\ 0&\alpha_{i}\end{smallmatrix})\gamma_{i}$ with $\gamma_{j}\in\Gamma_{j}(\mathfrak{n})$, then we have $\gamma_{i}\cdot s_{i}=s_{i}^{\prime}\in C_{i}$ by 1); $(\begin{smallmatrix}1&0\\ 0&\alpha_{i}\end{smallmatrix})\cdot s_{i}^{\prime}=s_{j_{i}}\in C_{j_{i}}$ by 2); and $\gamma_{j_{i}}\cdot s_{j_{i}}=s_{j_{i}}^{\prime}\in C_{j_{i}}$ by 1). Then

[TABLE]

Since $f^{j_{i}}$ is $C_{j_{i}}$-cuspidal and $\gamma\sigma_{s_{i}}\cdot\infty=s_{j_{i}}^{\prime}\in C_{j_{i}}$, we obtain that the constant term of $(f^{j_{i}}|_{\gamma\sigma_{s_{i}}})_{n}$ is trivial for $1\leqslant n\leqslant k+\ell+1$. ∎

2.4. $L$-function of $C$-cuspidal Bianchi modular forms

Henceforth, we will consider $\psi$ a Hecke character over $K$ of conductor $\mathfrak{f}$ with $(\mathfrak{f},I_{i})=1$ for each $i$. For each ideal $\mathfrak{a}=\prod_{\mathfrak{q}|\mathfrak{a}}\mathfrak{q}^{n_{\mathfrak{q}}}$ coprime to $\mathfrak{f}$, we define $\psi(\mathfrak{a})=\prod_{\mathfrak{q}|\mathfrak{a}}\psi_{\mathfrak{q}}(\pi_{\mathfrak{q}})^{n_{\mathfrak{q}}}$ and $\psi(\mathfrak{a})=0$ otherwise. In an abuse of notation, we write $\psi$ for both the idelic Hecke character and the function it determines on ideals.

Definition 2.13**.**

The $L$-function of a Bianchi modular form $\mathcal{F}$ twisted by $\psi$ is the function

[TABLE]

The $L$-function of each Bianchi modular form $f^{i}$ corresponding to $\mathcal{F}$ respect to $I_{i}$ is defined by

[TABLE]

with $w=|\mathcal{O}_{K}^{\times}|$.

Remark 2.14*.*

In [Wei71, Chap. II] it is proved that the $L$-functions of $\mathcal{F}$ and each $f^{i}$ converge absolutely on some right-half plane. Also note that

[TABLE]

The $L$-function of $C$-cuspidal Bianchi modular forms can be written as a finite sum of Mellin transforms similarly as in [Wil17, Thm1.8]. Before state the integral expression of the $L$-function, we recall some facts about Gauss sums and define suitable coefficients to link $L(\mathcal{F},\psi,s)$ with partial modular symbols in section 3.2

Let $\psi$ be a Hecke character of $K$ with conductor $\mathfrak{f}$. Following [Wil17, §1.2.3], we define the Gauss sum of $\psi$ to be

[TABLE]

Remark 2.15*.*

i) Define $\psi_{\mathfrak{f}}=\prod_{\mathfrak{q}|\mathfrak{f}}\psi_{\mathfrak{q}}$, by [Wil17, Prop 1.7], for all $c\in\mathcal{O}_{K}$, we have

[TABLE]

ii) Suppose $\mathfrak{f}$ coprime to $\mathfrak{m}$ and let $i\in\{1,...,h\}$, then we can choose a representative $a$ for each $[a]\in\mathfrak{f}^{-1}/\mathcal{O}_{K}$, such that $a\in C_{i}$: let $I_{k_{i}}$ be the unique ideal such that $\mathfrak{f}I_{k_{i}}=(\alpha_{k_{i}})I_{i}$ for $\alpha_{k_{i}}\in K$, then we put $a=d_{b}/\alpha_{k_{i}}$ where $d_{b}\in I_{k_{i}}$ and $d_{b}\equiv b$ (mod $\mathfrak{f}$). Note that $\alpha_{k_{i}}^{-1}\in\mathfrak{f}^{-1}I_{k_{i}}^{-1}I_{i}\subset\mathfrak{f}^{-1}I_{k_{i}}^{-1}$, so in particular, as $b$ ranges over all classes of $(\mathcal{O}_{K}/\mathfrak{f})^{\times}$ and as $d_{b}\in I_{k_{i}}$, we see that $d_{b}/\alpha_{i}$ ranges over a full set of coset representatives $[a]$ for $\mathfrak{f}^{-1}/\mathcal{O}_{K}$ with $(a)\mathfrak{f}$ coprime to $\mathfrak{f}$. Similarly to (2.7), we have that $\alpha_{k_{i}}=y_{\mathfrak{f}}y_{k_{i}}/t_{i}$ with $y_{\mathfrak{f}}\in(\mathcal{O}_{K}/\mathfrak{m})^{\times}$, $y_{k_{i}}\in(\mathcal{O}_{K}/\mathfrak{m})^{\times}$ and $t_{i}\in I_{i}$. Since $y_{\mathfrak{f}}y_{k_{i}}\in(\mathcal{O}_{K}/\mathfrak{m})^{\times}$, then we obtain $a=d_{b}/\alpha_{k_{i}}=d_{b}t_{i}/y_{\mathfrak{f}}y_{k_{i}}\in C_{i}$.

iii) Let $x_{\mathfrak{f}}$ be the idele associated to the ideal $\mathfrak{f}$ as in [Wil17, §2.6]. Since $\psi(a\mathfrak{f})^{-1}\psi_{\infty}(a)^{-1}=\psi(x_{\mathfrak{f}})^{-1}\psi_{\mathfrak{f}}(ax_{\mathfrak{f}})$ for $a\in\mathfrak{f}^{-1}/\mathcal{O}_{K}$ with $(a\mathfrak{f},\mathfrak{f})=1$, we can rewrite $\tau(\psi)$ as

[TABLE]

Definition 2.16**.**

Let $\mathcal{F}$ be a $C$-cuspidal form of weight $(k,\ell)$, for each $1\leqslant i\leqslant h$, $0\leqslant(s,u)\leqslant(k,\ell)$ and $a\in C_{i}$ we define

[TABLE]

Remark 2.17*.*

The integral in the above definition converges since $1\leqslant\ell+s-u+1\leqslant k+\ell+1$ and $f^{i}$ is $C_{i}$-cuspidal. Also, note that $c_{s,u}^{i}$ is not symmetric in $k$ and $\ell$, this comes from part i) of Remark 2.5.

Proposition 2.18**.**

Let $\mathcal{F}\in\mathcal{M}_{(k,\ell)}(K_{0}(\mathfrak{n}))$ be a $C$-cuspidal Bianchi modular form with $\mathfrak{n}=(p)\mathfrak{m}$ and $\mathfrak{m}$ coprime to $(p)$, then for a Hecke character $\psi$ of $K$ of conductor $\mathfrak{f}$ coprime to $\mathfrak{m}$ and infinity type $0\leqslant(q,r)\leqslant(k,\ell)$ we have

[TABLE]

where

[TABLE]

Proof.

Let $\chi$ be a Hecke character with conductor $\mathfrak{f}$ coprime to $\mathfrak{m}$ and infinity type $(-\frac{\ell+1-n}{2},\frac{\ell+1-n}{2})$ with $n\in\{1,...,k+\ell+1\}$ (note that we did not include $n=0,k+\ell+2$ as was done in [Wil17, Thm. 1.8] for the case of parallel weight).

Let $f^{1},...,f^{h}$ be the $h$ Bianchi modular forms corresponding to $\mathcal{F}$. Note that we can write

[TABLE]

Consider $\tau(\chi)$ as in part ii) of Remark 2.15, using part i) to replace $\chi_{\mathfrak{f}}(\alpha\delta)^{-1}$ in (2.10), noting that $\chi_{\infty}(\alpha)^{-1}=\left(\frac{\alpha}{|\alpha|}\right)^{\ell+1-n}$ and rearranging, we have

[TABLE]

Since

[TABLE]

(see the integral in the proof of [Wil17, Thm 1.8]), after substituting, we obtain

[TABLE]

where

[TABLE]

Now, by setting $n=\ell+q-r+1$ and $s=\frac{q+r+2}{2}$ in (2.11) and using Definition 2.16, we have

[TABLE]

Furthermore, note that $L\left(f^{j},\chi,\frac{q+r+2}{2}\right)=L(f^{j},\chi^{\prime},1)$ where $\chi^{\prime}=\chi|\cdot|_{\mathbb{A}_{K}}^{\frac{q+r}{2}}$ has infinity type $(q,r)$.

By using $\chi=\chi^{\prime}|\cdot|_{\mathbb{A}_{K}}^{\frac{-(q+r)}{2}}$ and part iii) of Remark 2.15, we have

[TABLE]

Also, using the definition of $W(\chi^{\prime})$ and the fact $\tau((\chi^{\prime})^{-1})=|\delta|^{q+r}|x_{\mathfrak{f}}|_{f}^{\frac{-(q+r)}{2}}\tau(\chi^{-1})$ (see [Wil17, §2.6]) we have

[TABLE]

The result follows by substituting (2.13) and (2.14) in 2.12, and by recalling that $L(\mathcal{F},\psi,1)=\sum_{i=1}^{h}L^{i}(\mathcal{F},\psi,1)$. ∎

Definition 2.19**.**

Let $\psi$ be a Hecke character of infinity type $(q,r)$. Define the completed $L$-function of $\mathcal{F}$ by

[TABLE]

Then under the hypothesis of Proposition 2.18, we obtain

[TABLE]

We finish this section with some remarks about algebraicity of $L$-values of $C$-cuspidal Bianchi modular forms.

In the case when we are dealing with cuspidal Bianchi modular forms, the “critical” values of this $L$-function can be controlled. We have the following theorem (see [Hid94, Thm 8.1]):

Theorem 2.20**.**

Let $\mathcal{F}$ be a cuspidal Bianchi eigenform of weight $(k,k)$, there exists a period $\Omega_{\mathcal{F}}\in\mathbb{C}^{\times}$ and a number field $E$ such that, if $\psi$ is a Hecke character of infinity type $0\leqslant(q,r)\leqslant(k,k)$, with $q,r\in\mathbb{Z}$, we have

[TABLE]

where $E(\psi)\subset\overline{\mathbb{Q}}$ is the extension of $E$ generated by the values of $\psi$.

Remark 2.21*.*

In the non-cuspidal case, depending on the Bianchi modular forms we are interested, we can prove algebraicity of critical $L$-values (see Proposition 5.7 for an example). For the construction of the $p$-adic $L$-function, we are interested in view $p$-adically the critical $L$-values. For this, in section 4 we will use an isomorphism between $\mathbb{C}$ and $\overline{\mathbb{Q}}_{p}$.

3. Partial Bianchi modular symbols

In this section, we introduce partial Bianchi modular symbols. These are algebraic analogues of $C$-cuspidal Bianchi modular forms that are easier to study $p$-adically. In section 3.2, we attach partial symbols to $C$-cuspidal forms and link them with $L$-values.

3.1. Partial modular symbols

Let $\Gamma$ be a discrete subgroup of $\mathrm{SL}_{2}(K)$ and let $\mathcal{C}$ be a non-empty $\Gamma$-invariant subset of $\mathbb{P}^{1}(K)$.

We denote by $\Delta_{\mathcal{C}}$ the abelian group of divisors on $\mathcal{C}$, i.e.

[TABLE]

and by $\Delta_{\mathcal{C}}^{0}$ the subgroup of divisors of degree 0 (i.e., such that $\sum_{c\in\mathcal{C}}n_{c}=0$). Note that $\Delta_{\mathcal{C}}^{0}$ has a left action by the group $\Gamma$ (and indeed, of $\mathrm{SL}_{2}(K)$) by fractional linear transformations on $\mathcal{C}$.

Let $V$ be a right $\Gamma$-module, we provide the space $\mathrm{Hom}(\Delta_{\mathcal{C}}^{0},V)$ with a right $\Gamma$-action by setting

[TABLE]

Definition 3.1**.**

We define the space of partial modular symbols on $\mathcal{C}$ for $\Gamma$ with values in $V$, to be the space $\mathrm{Symb}_{\Gamma,\mathcal{C}}(V):=\mathrm{Hom}_{\Gamma}(\Delta_{\mathcal{C}}^{0},V)$ of $\Gamma$-invariant maps from $\Delta_{\mathcal{C}}^{0}$ to $V$.

Remark 3.2*.*

When $\mathcal{C}=\mathbb{P}^{1}(K)$ we drop $\mathcal{C}$ from the notation and call $\mathrm{Symb}_{\Gamma}(V)$ the space of modular symbols for $\Gamma$ with values in $V$ recovering [Wil17, Def 2.3].

Recall from section 2.2 the group $K_{0}(\mathfrak{n})$ and its twist $\Gamma_{i}(\mathfrak{n})$ for each $i=1,...,h$. Setting $\Gamma=\Gamma_{i}(\mathfrak{n})$ in Definition 3.1 and taking suitable modules $V$ to be defined below, we can obtain more concrete partial modular symbols.

For a commutative ring $R$ recall that $V_{k}(R)$ denotes the space of homogeneous polynomials over $R$ in two variables of degree $k$. Furthermore, for integers $k,\ell\geqslant 0$ we define $V_{k,\ell}(R):=V_{k}(R)\otimes_{R}V_{\ell}(R)$.

We identify $V_{k,\ell}(R)$ with the space of polynomials that are homogeneous of degree $k$ in two variables $X,Y$ and homogeneous of degree $\ell$ in two further variables $\overline{X},\overline{Y}$.

Definition 3.3**.**

Let $R$ be a $K$-algebra, we have a left $\Gamma_{i}(\mathfrak{n})$-action on $V_{k}(R)$ defined by $\gamma\cdot P\binom{X}{Y}=P\binom{dX+bY}{cX+aY}$, for $\gamma=(\begin{smallmatrix}a&b\\ c&d\end{smallmatrix})$. We obtain a left $\Gamma_{i}(\mathfrak{n})$-action on $V_{k,\ell}(R)$ given by

[TABLE]

This induces a right $\Gamma_{i}(\mathfrak{n})$-action on the dual space $V_{k,\ell}^{*}(R)$ by setting

[TABLE]

Let $\mathbf{C}=(\mathbf{C}_{1},...,\mathbf{C}_{h})$ with $\mathbf{C}_{i}$ a non-empty $\Gamma_{i}(\mathfrak{n})$-invariant subset of $\mathbb{P}^{1}(K)$.

Definition 3.4**.**

(i) Define the space of partial Bianchi modular symbols on $\mathbf{C}_{i}$ of weight $(k,\ell)$ and level $\Gamma_{i}(\mathfrak{n})$ to be the space $\mathrm{Symb}_{\Gamma_{i}(\mathfrak{n}),\mathbf{C}_{i}}(V_{k,\ell}^{*}(\mathbb{C}))$.

(ii) Define the space of partial Bianchi modular symbols on $\mathbf{C}$ of weight $(k,\ell)$ and level $K_{0}(\mathfrak{n})$ to be the space

[TABLE]

Remark 3.5*.*

When $\mathbf{C}_{i}=\mathbb{P}^{1}(K)$ for all $i$, we drop $\mathbf{C}$ from the notation and recover the space $\mathrm{Symb}_{K_{0}(\mathfrak{n})}(V_{k,\ell}^{*}(\mathbb{C}))$ of Bianchi modular symbols in [Wil17, Def 2.4].

3.2. Partial modular symbols and $C$-cuspidal forms

To relate partial modular symbols with $C$-cuspidal Bianchi modular forms we henceforth take $\mathbf{C}=C=(C_{1},...,C_{h})$ with $C_{i}=\Gamma_{i}(\mathfrak{m})\infty\cup\Gamma_{i}(\mathfrak{m})0$ as in section 2.3 and consider the space of partial Bianchi modular symbols on $C$ of weight $(k,\ell)$ and level $K_{0}(\mathfrak{n})$.

In section 2.3 we defined Hecke operators on Bianchi modular forms. Similarly, we can define Hecke operators on the space of Bianchi modular symbols.

Definition 3.6**.**

Let $\mathfrak{q}$ be a prime ideal, then the Hecke operator $T_{\mathfrak{q}}$ is defined on the space of Bianchi modular symbols $\mathrm{Symb}_{K_{0}(\mathfrak{n})}(V_{k,\ell}^{*}(\mathbb{C}))$ by

[TABLE]

If $\mathfrak{q}|\mathfrak{n}$ we denote the Hecke operator by $U_{\mathfrak{q}}$.

Analogously with $C$-cuspidal Bianchi modular forms, not all Hecke operators act on $\mathrm{Symb}_{K_{0}(\mathfrak{n}),C}(V_{k,\ell}^{*}(\mathbb{C}))$.

Lemma 3.7**.**

The Hecke algebra $\mathcal{H}_{\mathfrak{n},p}$ acts on $\mathrm{Symb}_{K_{0}(\mathfrak{n}),C}(V_{k,\ell}^{*}(\mathbb{C}))$.

Proof.

Let $(\phi_{1},...,\phi_{h})\in\mathrm{Symb}_{K_{0}(\mathfrak{n}),C}(V_{k,\ell}^{*}(\mathbb{C}))$, we have to show that for all prime $\mathfrak{q}\nmid\mathfrak{m}$ with $\mathfrak{q}I_{i}=(\alpha_{i})I_{j_{i}}$ and for each $i=1,...,h$ then

[TABLE]

It suffices to prove $\gamma\cdot s_{i}\in C_{j_{i}}$ for all $\gamma=\gamma_{j_{i}}(\begin{smallmatrix}1&0\\ 0&\alpha_{i}\end{smallmatrix})\gamma_{i}\in\left[\Gamma_{j_{i}}(\mathfrak{n})(\begin{smallmatrix}1&0\\ 0&\alpha_{i}\end{smallmatrix})\Gamma_{i}(\mathfrak{n})\right]$ and $s_{i}\in C_{i}$.

Following the proof of Proposition 2.12 , for all $s_{i}\in C_{i}$ we have $\gamma_{i}\cdot s_{i}=s_{i}^{\prime}\in C_{i}$ by fact 1), $(\begin{smallmatrix}1&0\\ 0&\alpha_{i}\end{smallmatrix})\cdot s_{i}^{\prime}=s_{j_{i}}\in C_{j_{i}}$ by fact 2) and $\gamma_{j_{i}}\cdot s_{j_{i}}=s_{j_{i}}^{\prime}\in C_{j_{i}}$ again by fact 1). ∎

Let $\mathcal{F}$ be a $C$-cuspidal Bianchi eigenform of weight $(k,\ell)$ and level $K_{0}(\mathfrak{n})$. For $1\leqslant i\leqslant h$, recall the factor $c_{s,u}^{i}$ from Definition 2.16. Let $\mathcal{X}^{k-q}\mathcal{Y}^{q}\overline{\mathcal{X}}^{\ell-r}\overline{\mathcal{Y}}^{r}$ be the element of the dual basis of $V_{k,\ell}^{*}(\mathbb{C})$ defined by

[TABLE]

Definition 3.8**.**

For each descent $f^{i}$ of $\mathcal{F}$, we define

[TABLE]

by setting

[TABLE]

for each $a\in C_{i}$.

Proposition 3.9**.**

We have $\phi_{\mathcal{F}}:=(\phi_{f^{1}},...,\phi_{f^{h}})\in\mathrm{Symb}_{K_{0}(\mathfrak{n}),C}(V_{k,\ell}^{*}(\mathbb{C}))$. Moreover, the map $\mathcal{F}\rightarrow\phi_{\mathcal{F}}$ is $\mathcal{H}_{\mathfrak{n},p}$-equivariant.

Proof.

First note that for $i=1,...,h$, since $f^{i}$ is $C_{i}$-cuspidal, we can attach to it a vector valued differential 1-form following [Hid94, §2.5] (see also $\S 10$ in op.cit.). After the previous observation, we can proceed as [Wil17, §2.4] and integrate the resulting differential 1-form between cusps in $C_{i}$ and similarly to [Wil17, Prop2.9] and [Gha99, §5.2], obtain $\phi_{f^{i}}\in\mathrm{Symb}_{\Gamma_{i}(\mathfrak{n}),C_{i}}(V_{k,\ell}^{*}(\mathbb{C}))$.

The $\mathcal{H}_{\mathfrak{n},p}$-equivariance is an easy check using the definition of each $\phi_{f^{i}}$ and recalling that for non-principal ideals, the corresponding Hecke operators permutes the $h$ partial symbols. ∎

Remark 3.10*.*

For each $i$ and $0\leqslant(q,r)\leqslant(k,\ell)$ we have

[TABLE]

then we can link $\phi_{\mathcal{F}}$ with the $L$-values of $\mathcal{F}$ using (2.15)

4. $p$-adic $L$-function of $C$-cuspidal Bianchi modular forms

In this section, we define overconvergent partial Bianchi modular symbols to link partial symbols with $p$-adic distributions and prove a control theorem. In section 4.4 we construct $p$-adic $L$-functions of $C$-cuspidal Bianchi modular forms.

Henceforth, we denote $\mathcal{O}_{K,p}:=\mathcal{O}_{K}\otimes_{\mathbb{Z}}\mathbb{Z}_{p}$ to ease notation.

4.1. Locally analytic distributions

Suppose $p\mathcal{O}_{K}=\prod_{\mathfrak{p}|p}\mathfrak{p}^{e_{\mathfrak{p}}}$ and define $f_{\mathfrak{p}}$ to be the residue class degree of $\mathfrak{p}$. Note that $\sum f_{\mathfrak{p}}e_{\mathfrak{p}}=2$. Using the embedding $\iota_{p}:\overline{\mathbb{Q}}\hookrightarrow\overline{\mathbb{Q}}_{p}$ from section 2.1, for each prime $\mathfrak{p}|p$, we have $f_{\mathfrak{p}}e_{\mathfrak{p}}$ embeddings $K_{\mathfrak{p}}\hookrightarrow\overline{\mathbb{Q}}_{p}$, and combining these for each prime, we get an embedding $\sigma:K\otimes\mathbb{Q}_{p}\hookrightarrow\overline{\mathbb{Q}}_{p}\times\overline{\mathbb{Q}}_{p}$ given by $\sigma(a)=(\sigma_{1}(a),\sigma_{2}(a))$.

For $r,s\in\mathbb{R}_{>0}$, define the rigid analytic $(r,s)$-neighborhood of $\mathcal{O}_{K,p}$ in $\mathbb{C}_{p}^{2}$ to be

[TABLE]

Let $L$ be a finite extension of $\mathbb{Q}_{p}$ containing the image of $\sigma$. For $(r,s)$ as above, we write $\mathbb{A}[L,r,s]$ for the $L$-Banach space of rigid analytic functions on $B(\mathcal{O}_{K,p},r,s)$ and $\mathbb{D}[L,r,s]$ for its Banach dual (see [Wil17, §5.1]).

Define the space of $L$-valued locally analytic distributions to be the projective limit

[TABLE]

We endow $\mathbb{A}[L,r,s]$ with a weight $(k,\ell)$-action of the semigroup

[TABLE]

by setting

[TABLE]

These actions are compatible for various $(r,s)$. By duality, they induce actions on $\mathbb{D}[L,r,s]$, and hence on $\mathcal{D}(L)$. We write $\mathbb{D}_{k,\ell}[L,r,s]$ (resp. $\mathcal{D}_{k,\ell}(L)$) for the spaces $\mathbb{D}[L,r,s]$ (resp. $\mathcal{D}(L)$) together with the weight $(k,\ell)$-action of $\Sigma_{0}(p)$.

4.2. Overconvergent partial Bianchi modular symbols

Recall the level $K_{0}(\mathfrak{n})$ with $(p)|\mathfrak{n}$ from section 2.2 and for each $1\leqslant i\leqslant h$, the $\Gamma_{i}(\mathfrak{n})$-invariant subset $C_{i}$ of $\mathbb{P}^{1}(K)$ from section 2.3. Since the lower left entry of a matrix in $\Gamma_{i}(\mathfrak{n})$ is in $\mathfrak{p}$ for all $\mathfrak{p}|p$ (because $(p)|\mathfrak{n}$), we have that $\Gamma_{i}(\mathfrak{n})\subset\Sigma_{0}(p)$. Then we can equip $\mathcal{D}_{k,\ell}(L)$ with an action of $\Gamma_{i}(\mathfrak{n})$ and define partial modular symbols with values in $\mathcal{D}_{k,\ell}(L)$.

Definition 4.1**.**

(i) Define the space of overconvergent partial Bianchi modular symbols on $C_{i}$ of weight $(k,\ell)$ and level $\Gamma_{i}(\mathfrak{n})$ with coefficients in $L$ to be the space $\mathrm{Symb}_{\Gamma_{i}(\mathfrak{n}),C_{i}}(\mathcal{D}_{k,\ell}(L))$.

(ii) Define the space of overconvergent partial Bianchi modular symbols on $C$ of weight $(k,\ell)$ and level $K_{0}(\mathfrak{n})$ with coefficients in $L$ to be the space

[TABLE]

Note that the matrices appearing in Definition 3.6 of Hecke operators can be seen inside $\Sigma_{0}(p)$, then the Hecke algebra $\mathcal{H}_{\mathfrak{n},p}$ acts on $\mathrm{Symb}_{K_{0}(\mathfrak{n}),C}(\mathcal{D}_{k,\ell}(L))$.

There is a natural map $\mathcal{D}_{k,\ell}(L)\rightarrow V_{k,\ell}^{*}(L)$ given by dualizing the inclusion of $V_{k,\ell}(L)$ into $\mathcal{A}_{k,\ell}(L)$. This induces a $\mathcal{H}_{\mathfrak{n},p}$-equivariant specialization map

[TABLE]

Before proving the partial Bianchi control theorem, we record the following lemma (analog to [Wil17, Lem 3.8]) that allows us to work with integral distributions by imitating [Wil17, Prop 3.9].

Lemma 4.2**.**

$\Delta_{C_{i}}^{0}$* is finitely generated as a $\mathbb{Z}[\Gamma_{i}(\mathfrak{n})]$-module for $i=1,...,h$.*

Proof.

This follows from the fact that $\Gamma_{i}(\mathfrak{n})$ is a finitely generated group and the set of orbits of the action of $\Gamma_{i}(\mathfrak{n})$ in $C_{i}$ is finite. ∎

Proposition 4.3**.**

(Williams’ Bianchi control theorem). For each prime $\mathfrak{p}$ above $p$, let $\lambda_{\mathfrak{p}}\in L^{\times}$. Suppose that $v_{p}(\lambda_{\mathfrak{p}})<(\mathrm{min}\{{k,\ell\}}+1)/e_{\mathfrak{p}}$ when $p=\mathfrak{p}^{e_{\mathfrak{p}}}$ is inert or ramified, or $v_{p}(\lambda_{\mathfrak{p}})<k+1$ and $v_{p}(\lambda_{\overline{\mathfrak{p}}})<\ell+1$ when $p$ splits as $\mathfrak{p}\overline{\mathfrak{p}}$, then the restriction of the specialization map

[TABLE]

to the simultaneous $\lambda_{\mathfrak{p}}$-eigenspaces of the $U_{\mathfrak{p}}$ operators is an isomorphism.

Proof.

This result is proved like its counterpart for Bianchi modular symbols of parallel weight $(k,k)$ in [Wil17], doing the corresponding adaptations for weight $(k,\ell)$ and cusps. Here we just present a brief idea of how to follow Williams’ proof.

First, the control theorem is proved for the space of integral rigid analytic distributions, $\mathbb{D}_{k,\ell}[\mathcal{O}_{L},1,1]$, with the operator $U_{p}$ using [Wil17, §4] and noting the change of the condition $v_{p}(\lambda)<k+1$ in [Wil17, Lem 3.15] to $v_{p}(\lambda)<\mathrm{min}\{k,\ell\}+1$. Then, we can extend the result for $\mathbb{D}_{k,\ell}[L,1,1]$ and by [Wil17, §5.2], we obtain the result for $\mathcal{D}_{k,\ell}(L)$.

When $p$ is inert, the process described above allows us to prove the theorem. For $p$ ramified as $\mathfrak{p}^{2}$, we note that $U_{\mathfrak{p}}=U_{\mathfrak{p}^{2}}=U_{\mathfrak{p}}^{2}$ and use [Wil17, Lem 6.9] to obtain the result. Finally, when $\mathfrak{p}$ splits, we adapt the methods in [Wil17, §6] to lift simultaneous eigensymbols of $U_{\mathfrak{p}}$ and $U_{\overline{\mathfrak{p}}}$, obtaining the more subtle result on the slope. ∎

Remark 4.4*.*

Proposition 4.3 can be proved using a cohomological interpretation of partial Bianchi modular symbols as [BD15, §2.4] and following [Urb11], [HN17], [BW19]. In this paper we did not use such an interpretation, however, in a forthcoming paper where we use partial Bianchi modular symbols to construct $p$-adic $L$-function in families, cohomology will be key.

Definition 4.5**.**

Let $\mathcal{F}$ be an eigenform with eigenvalues $\lambda_{I}$, we say $\mathcal{F}$ has small slope if $v_{p}(\lambda_{\mathfrak{p}})<(\mathrm{min}\{{k,\ell\}}+1)/e_{\mathfrak{p}}$ when $p=\mathfrak{p}^{e_{\mathfrak{p}}}$ is inert or ramified; or if $v_{p}(\lambda_{\mathfrak{p}})<k+1$ and $v_{p}(\lambda_{\overline{\mathfrak{p}}})<\ell+1$ when $p$ splits as $\mathfrak{p}\overline{\mathfrak{p}}$. We say $\mathcal{F}$ has critical slope if does not have small slope.

4.3. Admissible distributions

For each pair $r$, $s$, the space $\mathcal{D}_{k,\ell}[L,r,s]$ from section 4.1 admits an operator norm $||\cdot||_{r,s}$ via

[TABLE]

where $|\cdot|_{p}$ is the usual $p$-adic absolute value on $L$ and $|\cdot|_{r,s}$ is the sup norm on $\mathcal{A}_{k,\ell}[L,r,s]$. Note that if $r\leqslant r^{\prime}$, $s\leqslant s^{\prime}$, then $||\mu||_{r,s}\geqslant||\mu||_{r^{\prime},s^{\prime}}$ for $\mu\in\mathcal{D}_{k,\ell}[L,r^{\prime},s^{\prime}]$.

These norms give rise to a family of norms on the space of locally analytic functions that allow us to classify locally analytic distributions by growth properties as we vary in this family.

Definition 4.6**.**

Let $\mu\in\mathcal{D}_{k,\ell}(L)$ be a locally analytic distribution.

(i) Suppose $p$ is inert or ramified in $K$, we say $\mu$ is $h$-admissible if $||\mu||_{r,r}=O(r^{-h})$ as $r\rightarrow 0^{+}$.

(ii) Suppose $p$ splits in $K$, we say $\mu$ is $(h_{1},h_{2})$-admissible if $||\mu||_{r,s}=O(r^{-h_{1}})$ uniformly in $s$ as $r\rightarrow 0^{+}$, and $||\mu||_{r,s}=O(r^{-h_{2}})$ uniformly in $r$ as $s\rightarrow 0^{+}$.

Proposition 4.7**.**

Let $\Psi\in\mathrm{Symb}_{\Gamma_{i}(\mathfrak{n}),C_{i}}(\mathcal{D}_{k,\ell}(L))$, and $r,s\leqslant 1$. Defining

[TABLE]

gives a well-defined norm on $\mathrm{Symb}_{\Gamma_{i}(\mathfrak{n}),C_{i}}(\mathcal{D}_{k,\ell}(L))$.

Proof.

By Lemma 4.2, we can write $D\in\Delta_{C_{i}}^{0}$ as $D=\alpha_{1}D_{1}+\cdots+\alpha_{n}D_{n}$ for a finite set of generators $D_{j}$ and $\alpha_{j}\in\mathbb{Z}[\Gamma_{i}(\mathfrak{n})]$.

Now, note that for every $\gamma\in\Gamma_{i}(\mathfrak{n})$ and $f\in\mathcal{A}_{k,\ell}[L,r,s]$ we can prove (in the same way of [Wil17, Lem 5.11]) that exist positive constants $A$ and $A^{\prime}$ such that

[TABLE]

Finally, by the above there exists a constant $B$ such that (without loss of generality)

[TABLE]

In particular, the supremum is finite and hence gives a well-defined norm as required. ∎

The following proposition is the adaptation to overconvergent partial Bianchi modular symbols of [Wil17, Props 5.12, 6.15].

Proposition 4.8**.**

Let $\Psi\in\mathrm{Symb}_{\Gamma_{i}(\mathfrak{n}),C_{i}}(\mathcal{D}_{k,\ell}(L))$,

(i) Suppose $p$ is inert in $K$ and $\Psi$ is a $U_{p}$-eigensymbol with eigenvalue $\lambda$ and slope $h=v_{p}(\lambda)$. Then, for every $D\in\Delta_{C_{i}}^{0}$, the distribution $\Psi(D)$ is $h$-admissible.

(ii) Suppose $p$ splits in $K$ as $\mathfrak{p}\overline{\mathfrak{p}}$ and $\Psi$ is simultaneously a $U_{\mathfrak{p}}^{n}$- and $U_{\overline{\mathfrak{p}}}^{n}$-eigensymbol for some $n$, with non-zero eigenvalues $\lambda_{1}^{n}$ and $\lambda_{2}^{n}$ with slopes $h_{1}=v_{p}(\lambda_{1})$ and $h_{2}=v_{p}(\lambda_{2})$. Then, for every $D\in\Delta_{C_{i}}^{0}$, the distribution $\Psi(D)$ is $(h_{1},h_{2})$-admissible.

Proof.

To prove part i) we follow line by line the proof of [Wil17, Prop 5.13] and conclude that for any $r$ and positive integer $m$, we have

[TABLE]

To prove part ii) we use the same idea of part i) and conclude that for any $r,s$ using $m$-th powers of the operators $U_{\mathfrak{p}}^{n}$ and $U_{\overline{\mathfrak{p}}}^{n}$ we have

[TABLE]

and combining both, we obtain the result. ∎

4.4. Construction of the $p$-adic $L$-function

Throughout this section, we fix an isomorphism $\iota:\mathbb{C}\rightarrow\overline{\mathbb{Q}}_{p}$ compatible with the embeddings of section 2.1. We recall ray class groups and define the Mellin transform of an overconvergent partial Bianchi modular symbol.

Definition 4.9**.**

Define the ray class group of level $p^{\infty}$ to be

[TABLE]

and denote by $\mathfrak{X}(\mathrm{Cl}_{K}(p^{\infty}))$ the two-dimensional rigid space of $p$-adic characters on $\mathrm{Cl}_{K}(p^{\infty})$.

Note that $\mathrm{Cl}_{K}(p^{\infty})$ can be written as $\mathrm{Cl}_{K}(p^{\infty})=\bigcup_{i\in\mathrm{Cl}_{K}}\mathrm{Cl}_{K}^{i}(p^{\infty})$, where $\mathrm{Cl}_{K}^{i}(p^{\infty})$ is the fiber of $i$ under the canonical surjection $\mathrm{Cl}_{K}(p^{\infty})\twoheadrightarrow\mathrm{Cl}_{K}$ to the class group of $K$, also note that, the choice of $t_{i}\in\mathbb{A}_{K}^{f,\times}$ in section 2.2 identifies $\mathrm{Cl}_{K}^{i}(p^{\infty})$ non-canonically with $\mathcal{O}_{K,p}^{\times}/\mathcal{O}_{K}^{\times}$.

Let $\Psi=(\Psi_{1},...,\Psi_{h})\in\mathrm{Symb}_{K_{0}(\mathfrak{n}),C}(\mathcal{D}_{k,\ell}(L))$ be an overconvergent partial Bianchi modular symbol, we define for $i,j\in\mathrm{Cl}_{K}$ a distribution $\mu_{i}(\Psi_{j})\in\mathcal{D}(\mathrm{Cl}_{K}^{i}(p^{\infty}),L)$ as follows.

Since $\{0\}-\{\infty\}\in\Delta_{C_{i}}^{0}$ for all $i$, we have a distribution $\Psi_{j}(\{0\}-\{\infty\})|_{\mathcal{O}_{K,p}^{\times}}$ on $\mathcal{O}_{K,p}^{\times}$. This restricts to a distribution on $\mathcal{O}_{K,p}^{\times}/\mathcal{O}_{K}^{\times}$, which gives the distribution $\mu_{i}(\Psi_{j})$ on $\mathrm{Cl}_{K}^{i}(p^{\infty})$ under the identification above.

Definition 4.10**.**

The Mellin transform of $\Psi\in\mathrm{Symb}_{K_{0}(\mathfrak{n}),C}(\mathcal{D}_{k,\ell}(L))$ is the ($L$-valued) locally analytic distribution on $\mathrm{Cl}_{K}(p^{\infty})$ given by

[TABLE]

Remark 4.11*.*

The distribution $\mathrm{Mel}(\Psi)$ is independent of the choice of class group representatives.

The theory of partial Bianchi modular symbols developed in sections 3 and 4, allows us to construct the $p$-adic $L$-function of small slope $C$-cuspidal Bianchi modular forms.

Theorem 4.12**.**

Let $\mathcal{F}$ be a $C$-cuspidal Bianchi eigenform of weight $(k,\ell)$ and level $K_{0}(\mathfrak{n})$, with $U_{\mathfrak{p}}$-eigenvalues $\lambda_{\mathfrak{p}}$, where $v_{p}(\lambda_{\mathfrak{p}})<(\mathrm{min}\{{k,\ell\}}+1)/e_{\mathfrak{p}}$ when $p=\mathfrak{p}^{e_{\mathfrak{p}}}$ is inert or ramified, or $v_{p}(\lambda_{\mathfrak{p}})<k+1$ and $v_{p}(\lambda_{\overline{\mathfrak{p}}})<\ell+1$ when $p$ splits as $\mathfrak{p}\overline{\mathfrak{p}}$. Let $\iota$ be a fixed isomorphism between $\mathbb{C}$ and $\overline{\mathbb{Q}}_{p}$. Then there exists a locally analytic distribution $L_{p}^{\iota}(\mathcal{F},-)$ on $\mathrm{Cl}_{K}(p^{\infty})$ such that for any Hecke character of $K$ of conductor $\mathfrak{f}=\prod_{\mathfrak{p}|p}\mathfrak{p}^{r_{\mathfrak{p}}}|(p^{\infty})$ and infinity type $0\leqslant(q,r)\leqslant(k,\ell)$, we have

[TABLE]

where $\psi_{p-\mathrm{fin}}$ is the $p$-adic avatar of $\psi$ as in [Wil17, §7.3], $U_{\mathfrak{f}}\mathcal{F}=\lambda_{\mathfrak{f}}\mathcal{F}$ with $U_{\mathfrak{f}}=\prod_{\mathfrak{p}|p}U_{\mathfrak{p}}^{r_{\mathfrak{p}}}$ and

[TABLE]

The distribution $L_{p}^{\iota}(\mathcal{F},-)$ is $(h_{\mathfrak{p}})_{\mathfrak{p}|p}$-admissible, where $h_{\mathfrak{p}}=v_{p}(\lambda_{\mathfrak{p}})$, and hence is unique.

We call $L_{p}^{\iota}(\mathcal{F},-)$ the $p$-adic $L$-function of $\mathcal{F}$.

Proof.

The small slope $C$-cuspidal Bianchi modular form $\mathcal{F}$ corresponds to a collection of $C_{i}$-cuspidal forms $f^{i},...,f^{h}$ on $\mathcal{H}_{3}$.

We first attach to $\mathcal{F}$ a complex-valued partial Bianchi modular eigensymbol $\phi_{\mathcal{F}}=(\phi_{f^{1}},...,\phi_{f^{h}})$ using Proposition 3.9.

By applying $\iota$, we obtain from $\phi_{\mathcal{F}}$ a symbol $\phi^{\iota}_{\mathcal{F}}=(\phi^{\iota}_{f^{1}},...,\phi^{\iota}_{f^{h}})$ with values in $V_{k,\ell}^{*}(\overline{\mathbb{Q}}_{p})$. Since by Lemma 4.2, $\Delta_{C_{i}}^{0}$ is finitely generated as a $\mathbb{Z}[\Gamma_{i}(\mathfrak{n})]$-module for each $i$, it follows that for all $i$, $\phi^{\iota}_{f^{i}}$ has values in $V_{k,\ell}^{*}(L)$ for a sufficiently large $L/\mathbb{Q}_{p}$ finite (containing the eigenvalues of $\mathcal{F}$), thus $\phi^{\iota}_{\mathcal{F}}\in\mathrm{Symb}_{K_{0}(\mathfrak{n}),C}(V_{k,\ell}^{*}(L))$.

Since $\mathcal{F}$ has small slope, we can lift $\phi^{\iota}_{\mathcal{F}}$ to a unique $\Psi^{\iota}_{\mathcal{F}}\in\mathrm{Symb}_{K_{0}(\mathfrak{n}),C}(\mathcal{D}_{k,\ell}(L))$ using Proposition 4.3.

Finally, we define the $p$-adic $L$-function of $\mathcal{F}$ as the Mellin transform of $\Psi^{\iota}_{\mathcal{F}}$

[TABLE]

The interpolation property in (4.1) comes from the link between partial Bianchi modular symbols and $\Lambda(\mathcal{F},-)$ in Remark 3.10. Additionally, by Proposition 4.8 the distribution $L_{p}^{\iota}(\mathcal{F},-)$ is $(h_{\mathfrak{p}})_{\mathfrak{p}|p}$-admissible, where $h_{\mathfrak{p}}=v_{p}(\lambda_{\mathfrak{p}})$. Both interpolation and growth properties give the uniqueness of $L_{p}^{\iota}(\mathcal{F},-)$. ∎

Remark 4.13*.*

In general, non-cuspidal Bianchi modular forms do not need to be $C$-cuspidal. Whilst it is not computed in this article, the author believes that we can consider a linear combination as in [BD15, §6.1], to turn a non-cuspidal Bianchi modular form into $C$-cuspidal and then construct its $p$-adic $L$-function. This relies on the fact that, for example, in the parallel weight case, in the Fourier expansion of a Bianchi modular form $\mathcal{F}=\sum_{n=0}^{2k+2}\mathcal{F}_{n}X^{2k+2-n}Y^{n}$, only the constant term of $\mathcal{F}_{0}$, $\mathcal{F}_{k+1}$ and $\mathcal{F}_{2k+2}$ can be non-trivial (see part ii in Remark 2.5) and to achieve $C$-cuspidality we just need to control the constant term of $\mathcal{F}_{k+1}$ at the cusps [math] and $\infty$. Note that in the case of non-parallel weight $(k,\ell)$, we need to control the constant terms of $\mathcal{F}_{k+1}$ and $\mathcal{F}_{\ell+1}$ both at the cusps 0 and $\infty$.

5. $p$-adic $L$-function of non-cuspidal base change Bianchi modular forms

Throughout this section, suppose $p$ splits in $K$ as $\mathfrak{p}\overline{\mathfrak{p}}$ with $\mathfrak{p}$ being the prime corresponding to the embedding $\iota_{p}$ from section 2.1.

For non-cuspidal base change Bianchi modular forms, we can deepen the construction of $p$-adic $L$-functions. More precisely, let $\mathcal{F}$ be such a form, we can prove algebraicity of the critical $L$-values of $\mathcal{F}$ through the existence of a complex period, likewise, we can attach a complex partial Bianchi modular symbol to $\mathcal{F}$ and view it $p$-adically without using an isomorphism between $\mathbb{C}$ and $\overline{\mathbb{Q}}_{p}$. Additionally, we factor the $p$-adic $L$-function of $\mathcal{F}$ as a product of two Katz $p$-adic $L$-functions.

5.1. Base change and $C$-cuspidality

Let $\varphi$ be a Hecke character of $K$ of conductor $\mathfrak{M}$ coprime to $p$ and infinity type $(-k-1,0)$ with $k\geqslant 0$. Let $f_{\varphi}$ be the theta series associated to $\varphi$, which is a newform of weight $k+2$, level $\Gamma_{0}(DN(\mathfrak{M}))$ and nebentypus $\epsilon_{f_{\varphi}}=\chi_{K}\varphi_{\mathbb{Z}}$ where $\chi_{K}$ is the Kroneker character of $K$ and $\varphi_{\mathbb{Z}}$ is the Dirichlet character modulo $M=N(\mathfrak{M})$ given by $\varphi_{\mathbb{Z}}(a)=\varphi(a\mathcal{O}_{K})a^{-k-1}$, for an integer $a$ coprime to $M$.

Let $\pi$ be the automorphic representation of $\mathrm{GL}_{2}(\mathbb{A}_{\mathbb{Q}})$ generated by $f_{\varphi}$ and let BC($\pi$) be the base change of $\pi$ to $\mathrm{GL}_{2}(\mathbb{A}_{K})$ (see [Lan80]). The base change of $f_{\varphi}$ to $K$ is the normalized new vector $f_{\varphi/K}$ in BC($\pi$) which is a non-cuspidal Bianchi modular form of weight $(k,k)$, level $K_{0}(\mathfrak{m})$ with $M\mathcal{O}_{K}|\mathfrak{m}|DM\mathcal{O}_{K}$ (see [Fri83, §2.3]) and nebentypus $\epsilon_{f_{\varphi/K}}=\epsilon_{f_{\varphi}}\circ N$.

Remark 5.1*.*

The Hecke eigenvalues of $f_{\varphi/K}$ are determined from those of $f_{\varphi}$. For every prime $q$ not dividing the level of $f_{\varphi}$, if the eigenvalue of $T_{q}$ on $f$ is $a_{q}$ then the eigenvalues $a_{\mathfrak{q}}$ for ${\mathfrak{q}}\mid q$ of $f_{\varphi/K}$ are given by: $a_{\mathfrak{q}}=a_{\overline{\mathfrak{q}}}=a_{q}=\varphi(\mathfrak{q})+\varphi(\overline{\mathfrak{q}})$ if $q$ splits as $\mathfrak{q}\overline{\mathfrak{q}}$, $a_{\mathfrak{q}}=a_{q}=\varphi(\mathfrak{q})$ if $q$ ramifies as $\mathfrak{q}^{2}$ and $a_{\mathfrak{q}}=a_{q}^{2}-2\chi_{K}(q)\varphi_{\mathbb{Z}}(q)q^{k+1}=2\varphi(\mathfrak{q})$ if $q$ is inert as $q=\mathfrak{q}$.

Note that the Bianchi modular form $f_{\varphi/K}$ has level $\mathfrak{m}$ coprime to $p$, however, to construct its $p$-adic $L$-function we need $p$ in the level, then, we consider its $p$-stabilizations. We are interested in the $p$-stabilization satisfying the small slope condition of Definition 4.5, i.e., the $p$-adic valuation of the eigenvalues of $U_{\mathfrak{p}}$ and $U_{\overline{\mathfrak{p}}}$ are both less than $k+1$.

The $p$-stabilizations of $f_{\varphi/K}$ can be explicitly described from the $p$-stabilizations of $f_{\varphi}$ by considering its Hecke polynomial at $p$ given by $x^{2}-a_{p}x+\epsilon_{f_{\varphi}}(p)p^{k+1}$. Since $p=\mathfrak{p}\overline{\mathfrak{p}}$, we have that $a_{p}=\varphi(\mathfrak{p})+\varphi(\overline{\mathfrak{p}})$ and $\epsilon_{f_{\varphi}}(p)=\chi_{K}(p)\varphi_{\mathbb{Z}}(p)=\varphi(p\mathcal{O}_{K})/p^{k+1}=\varphi(\mathfrak{p}\overline{\mathfrak{p}})/p^{k+1}$, thus the roots of the Hecke polynomial of $f_{\varphi}$ at $p$ are $\alpha_{p}=\varphi(\mathfrak{p})$ and $\beta_{p}=\varphi(\overline{\mathfrak{p}})$.

Consider the Hecke polynomial of $f_{\varphi/K}$ at $\mathfrak{q}|p$ given by $x^{2}-a_{\mathfrak{q}}x+\epsilon_{f_{\varphi/K}}(\mathfrak{q})N(\mathfrak{q})^{k+1}$. Since $a_{\mathfrak{q}}=a_{p}$ by Remark 5.1, $N(\mathfrak{q})=p$ and $\epsilon_{f_{\varphi/K}}(\mathfrak{q})=\chi_{K}(N(\mathfrak{q}))\varphi_{\mathbb{Z}}(N(\mathfrak{q}))=\chi_{K}(p)\varphi_{\mathbb{Z}}(p)=\epsilon_{f_{\varphi}}(p)$; then the Hecke polynomials of $f_{\varphi/K}$ at $\mathfrak{p}$ and $\overline{\mathfrak{p}}$ and the Hecke polynomial of $f_{\varphi}$ at $p$ are equal.

Therefore, we can take the roots of the Hecke polynomial of $f_{\varphi/K}$ at $\mathfrak{p}$ to be $\alpha_{\mathfrak{p}}=\alpha_{p}=\varphi(\mathfrak{p})$ and $\beta_{\mathfrak{p}}=\beta_{p}=\varphi(\overline{\mathfrak{p}})$ and the roots of the Hecke polynomial of $f_{\varphi/K}$ at $\overline{\mathfrak{p}}$ to be $\alpha_{\overline{\mathfrak{p}}}=\varphi(\mathfrak{p})$ and $\beta_{\overline{\mathfrak{p}}}=\varphi(\overline{\mathfrak{p}})$.

If $f_{\varphi}^{\alpha}$ (resp. $f_{\varphi}^{\beta}$) is the $p$-stabilization of $f_{\varphi}$ corresponding to $\alpha_{p}$ (resp. $\beta_{p}$), we define its base change to $K$ to be the $p$-stabilization $f_{\varphi/K}^{\alpha\alpha}$ (resp. $f_{\varphi/K}^{\beta\beta}$) of $f_{\varphi/K}$ corresponding to $\alpha_{\mathfrak{p}}$, $\alpha_{\overline{\mathfrak{p}}}$ (resp. $\beta_{\mathfrak{p}}$, $\beta_{\overline{\mathfrak{p}}}$).

Remark 5.2*.*

Note that $v_{p}(\alpha_{\mathfrak{p}})=v_{p}(\alpha_{\overline{\mathfrak{p}}})=v_{p}(\varphi(\mathfrak{p}))=k+1$ and $v_{p}(\beta_{\mathfrak{p}})=v_{p}(\beta_{\overline{\mathfrak{p}}})=v_{p}(\varphi(\overline{\mathfrak{p}}))=0$. Since we are interested in the small slope $p$-stabilization of $f_{\varphi/K}$, henceforth we will work with $f_{\varphi/K}^{p}:=f_{\varphi/K}^{\beta\beta}$.

Recall the level of $f_{\varphi/K}^{p}$ given by $K_{0}(\mathfrak{n})$ with $\mathfrak{n}=(p)\mathfrak{m}$ and consider $C=(C_{1},...,C_{h})$ with $C_{i}=\Gamma_{i}(\mathfrak{m})\infty\cup\Gamma_{i}(\mathfrak{m})0$ as in section 2.3.

The following proposition is the analog of [BD15, Def 6.2], however, in our case we do not need to produce a linear combination, since we can take advantage of the nice vanishing properties of $f_{\varphi/K}$.

Proposition 5.3**.**

The Bianchi modular form $f_{\varphi/K}^{p}$ is $C$-cuspidal.

Proof.

Suppose that $K$ has class number $1$. By part ii) of Remark 2.5, the constant term of the Fourier expansion at $\infty$ of $f_{\varphi/K}=(f_{0},...,f_{n},...,f_{2k+2})$ is trivial for $n\notin\{0,k+1,2k+2\}$ and by [Fri83, Thm 3.1], we also have that the constant Fourier coefficient of $f_{k+1}$ is trivial and therefore $f_{\varphi/K}$ quasi-vanishes at $\infty$. Likewise, since $f_{\varphi/K}$ is an eigenform for the Hecke operators, then it is also an eigenform for the Fricke involution $W_{\mathfrak{m}}$, with eigenvalue $\pm 1$ (see [CW94, §2], [Pal23, §2.4]). Note that $W_{\mathfrak{m}}$ sends the cusp $\infty$ to the cusp [math], then $f_{\varphi/K}$ quasi-vanishes at [math], therefore $f_{\varphi/K}$ quasi-vanishes at $C$. To conclude that $f_{\varphi/K}^{p}$ is $C$-cuspidal, we notice that $C$ is stable under multiplication by the matrix $(\begin{smallmatrix}\pi_{\mathfrak{p}}&0\\ 0&1\end{smallmatrix})$ (resp. $(\begin{smallmatrix}\overline{\pi}_{\mathfrak{p}}&0\\ 0&1\end{smallmatrix})$) used in the $\mathfrak{p}$-stabilization (resp. $\overline{\mathfrak{p}}$-stabilization) of $f_{\varphi/K}$.

When $K$ has a higher class number, we proceed similarly as above and obtain that $f_{\varphi/K}^{i}$ quasi-vanishes at $C_{i}$ for each $i$. Then we note that the set of cusps $C_{i}$ is stable under multiplication by the matrix ${(\begin{smallmatrix}\alpha_{i}&0\\ 0&1\end{smallmatrix})}$ with $\mathfrak{p}I_{i}=(\alpha_{i})I_{j_{i}}$ (resp. ${(\begin{smallmatrix}\beta_{i}&0\\ 0&1\end{smallmatrix})}$ with $\overline{\mathfrak{p}}I_{i}=(\beta_{i})I_{j_{i}}$) appearing in the $i$-component of the $\mathfrak{p}$-stabilization (resp. $\overline{\mathfrak{p}}$-stabilization) of $f_{\varphi/K}$. ∎

5.2. $L$-function

The fact that $f_{\varphi/K}$ is constructed from a Hecke character, allows us to relate its complex $L$-function with Hecke $L$-functions and prove algebraicity of critical $L$-values of $f_{\varphi/K}^{p}$.

Lemma 5.4**.**

Let $\psi$ be a Hecke character of $K$ and let $\psi^{c}(\mathfrak{q}):=\psi(\overline{\mathfrak{q}})$ where $\mathfrak{q}$ is an ideal of $K$ and $\overline{\mathfrak{q}}$ is its conjugate ideal. Then

[TABLE]

where $\lambda_{K}=\chi_{K}\circ N$.

Proof.

This follows by comparing the Euler factors on both sides, using Remark 5.1. ∎

Remark 5.5*.*

There are 6 more ways to factor $L(f_{\varphi/K},\psi,s)$ as product of two Hecke $L$-functions, this comes from the fact that for a Hecke character $\nu$ we have $L(\nu,s)=L(\nu^{c},s)$. Similar factorizations in the $p$-adic setting do not necessarily hold, because if $\nu$ has infinity type $(q,r)$ the involution $\nu\rightarrow\nu^{c}$ corresponds to the map $(q,r)\rightarrow(r,q)$ on weight space and therefore does not preserve the lower right quadrant of weights of Hecke characters that lie in the range of classical interpolation of the Katz $p$-adic $L$-functions (see [BDP12, Figure 1] and Theorem 5.13).

We now prove algebraicity of the critical $L$-values of $f_{\varphi/K}$. For this, we first follow [Ber08, §2.3] to recall algebraicity of the critical $L$-value of $L(\chi,0)$ for a Hecke character $\chi$ of infinity type $(a,b)$ with $a,b\in\mathbb{Z}$ satisfying $a>0\geqslant b$.

Let $E$ be an elliptic curve with complex multiplication by $\mathcal{O}_{K}$ and $\omega$ its Néron differential. Suppose also that $E$ has good reduction at the place above $p$ and $\overline{\omega}$ is a non-vanishing invariant differential on the reduced curve $\overline{E}$, then Damerell showed that $\pi^{-b}\Omega_{\infty}^{-a+b}L(\chi,0)\in\overline{\mathbb{Q}}$, where $\Omega_{\infty}\in\mathbb{C}$ is the complex period of $\omega$.

Definition 5.6**.**

Define the complex period $\Omega_{f_{\varphi/K}}\in\mathbb{C}^{\times}$ attached to $f_{\varphi/K}$ to be

[TABLE]

Proposition 5.7**.**

For all Hecke character $\psi$ of $K$ with infinity type $0\leqslant(q,r)\leqslant(k,k)$, we have $\Lambda(f_{\varphi/K},\psi)/\Omega_{f_{\varphi/K}}\in\overline{\mathbb{Q}}$.

Proof.

By Lemma 5.4 we have

[TABLE]

Note that $\varphi^{c}\psi|\cdot|_{\mathbb{A}_{K}}$ has infinity type $(q+1,r-k)$ with $q+1>0\geqslant r-k$ and $\varphi^{c}\psi^{c}\lambda_{K}|\cdot|_{\mathbb{A}_{K}}$ has infinity type $(r+1,q-k)$ with $r+1>0\geqslant q-k$, therefore

[TABLE]

The result follows by multiplying the two numbers above, since

[TABLE]

∎

Suppose $f_{\varphi/K}$ does not have level at $p$ and consider its $p$-stabilization $f_{\varphi/K}^{p}$ as in Remark 5.2.

Corollary 5.8**.**

For all Hecke character $\psi$ of $K$ with infinity type $0\leqslant(q,r)\leqslant(k,k)$, we have $\Lambda(f_{\varphi/K}^{p},\psi)/\Omega_{f_{\varphi/K}}\in\overline{\mathbb{Q}}$.

Proof.

The $L$-functions of $f_{\varphi/K}^{p}$ and $f_{\varphi/K}$ are related (see [Pal23, §3.3] for Bianchi modular forms with trivial nebentypus and $K$ with class number 1) by

[TABLE]

By Proposition 5.7 we obtain the result. ∎

5.3. $p$-adic $L$-function

Let $f^{i}$ for $i=1,..,h$ be the collection of $C_{i}$-cuspidal descents to $\mathcal{H}_{3}$ of $f_{\varphi/K}^{p}$. By Proposition 3.9 we can attach to $f_{\varphi/K}^{p}$ a partial Bianchi modular symbol $\phi_{f_{\varphi/K}^{p}}=(\phi_{f^{1}},...,\phi_{f^{h}})$ where

[TABLE]

with $c_{s,u}^{i}(a)$ as in Definition 2.16 for $a\in C_{i}$.

Remark 5.9*.*

Recall that for each $i$, the integrals inside $c_{s,u}^{i}(a)$ are convergent because $f^{i}$ quasi-vanishes at the cusps of $C_{i}$, since $f_{\varphi/K}^{p}$ is $C$-cuspidal.

Proposition 5.10**.**

Let $\Omega_{f_{\varphi/K}}$ be the period in Proposition 5.7, then the partial Bianchi modular symbol $\phi^{\prime}_{f_{\varphi/K}^{p}}:=\phi_{f_{\varphi/K}^{p}}/\Omega_{f_{\varphi/K}}$ takes values in $V_{k,k}^{*}(E)$ for some number field $E$.

Proof.

Let $\psi$ be a Hecke character of $K$ of infinity type $0\leqslant(q,r)\leqslant(k,k)$ then by (2.15) we have

[TABLE]

Dividing both sides by $\Omega_{f_{\varphi/K}}$ we obtain

[TABLE]

By Corollary 5.8 we have $\Lambda(f_{\varphi/K}^{p},\psi)/\Omega_{f_{\varphi/K}}\in\overline{\mathbb{Q}}$ for all $\psi$ with infinity type $0\leqslant(q,r)\leqslant(k,k)$, then the left-hand side of (5.1) is an algebraic number and consequently by linear independence of characters we have that $c_{q,r}^{i}(a)/\Omega^{\prime}_{f_{\varphi/K}}\in\overline{\mathbb{Q}}$ for each $i$.

Consider the polynomial $(X+aY)^{q}Y^{k-q}(\overline{X}+\overline{a}\overline{Y})^{r}\overline{Y}^{k-r}$, we have that

[TABLE]

Then, defining $\phi^{\prime}_{f^{i}}:=\phi_{f^{i}}/\Omega_{f_{\varphi/K}}$ we have

[TABLE]

Now, using (5.2), we will show $\phi^{\prime}_{f^{i}}\in\mathrm{Symb}_{\Gamma_{i}(\mathfrak{n}),C_{i}}(V_{k,k}^{*}(\overline{\mathbb{Q}}))$. For this, consider some divisor $D\in\Delta_{C_{i}}^{0}$ and a polynomial $P\left[\binom{X}{Y}\binom{\overline{X}}{\overline{Y}}\right]\in V_{k,k}(\overline{\mathbb{Q}})$, we want to show $[\phi^{\prime}_{f^{i}}(D)](P)\in\overline{\mathbb{Q}}$.

By Lemma 4.2, $\Delta_{C_{i}}^{0}$ is finitely generated as a $\mathbb{Z}[\Gamma_{i}(\mathfrak{n})]$-module by divisors $\{a\}-\{\infty\}$, with $a\in C_{i}$, then, it is enough to prove $[\phi^{\prime}_{f^{i}}(\{a\}-\{\infty\})](P)\in\overline{\mathbb{Q}}$.

Let $P\left[\binom{X}{Y}\binom{\overline{X}}{\overline{Y}}\right]=\sum_{b,d=0}^{k}t_{b,d}X^{b}Y^{k-b}\overline{X}^{d}\overline{Y}^{k-d}$ with $t_{b,d}\in\overline{\mathbb{Q}}$. For $0\leqslant b,d\leqslant k$, we can write each $X^{b}Y^{k-b}\overline{X}^{d}\overline{Y}^{k-d}$ as a homogeneous polynomial $Q_{b,d}\left[\binom{X+aY}{Y}\binom{\overline{X}+\overline{a}\overline{Y}}{\overline{Y}}\right]$ after replacing $X$ by $(X+aY)-aY$ and $\overline{X}$ by $(\overline{X}+\overline{a}\overline{Y})-\overline{a}\overline{Y}$. Applying (5.2) on each $Q_{b,d}$, we obtain $[\phi^{\prime}_{f^{i}}(\{a\}-\{\infty\})](Q_{b,d})\in\overline{\mathbb{Q}}$ and then $[\phi^{\prime}_{f^{i}}(\{a\}-\{\infty\})](P)\in\overline{\mathbb{Q}}$.

Finally, using again that $\Delta_{C_{i}}^{0}$ is finitely generated as a $\mathbb{Z}[\Gamma_{i}(\mathfrak{n})]$-module, we have for all $i$ that $\phi^{\prime}_{f^{i}}\in\mathrm{Symb}_{\Gamma_{i}(\mathfrak{n}),C_{i}}(V_{k,k}^{*}(E))$ for some sufficiently large number field $E$. Therefore, $\phi^{\prime}_{f_{\varphi/K}^{p}}\in\mathrm{Symb}_{K_{0}(\mathfrak{n}),C}(V_{k,k}^{*}(E))$. ∎

The previous Proposition and the work done in section 4 allow us to obtain the $p$-adic $L$-function of $f_{\varphi/K}^{p}$ without using an isomorphism between $\mathbb{C}$ and $\overline{\mathbb{Q}}_{p}$.

Theorem 5.11**.**

Let $\varphi$ be a Hecke character of $K$ with conductor coprime to $p$ and infinity type $(-k-1,0)$ for $k\geqslant 0$, and denote by $f_{\varphi}^{p}$ the ordinary $p$-stabilization of the CM form $f_{\varphi}$ induced by $\varphi$. Let $f_{\varphi/K}^{p}$ be the base change to $K$ of $f_{\varphi}^{p}$ and $\Omega_{f_{\varphi/K}}$ be the complex period of Definition 5.6. Then there exists a unique measure $L_{p}(f_{\varphi/K}^{p},-)$ on $\mathrm{Cl}_{K}(p^{\infty})$ such that for any Hecke character $\psi$ of $K$ of conductor $\mathfrak{f}=\mathfrak{p}^{t}\overline{\mathfrak{p}}^{s}$ and infinity type $0\leqslant(q,r)\leqslant(k,k)$, we have

[TABLE]

Proof.

By Proposition 3.9 we can attach to $f_{\varphi/K}^{p}$ a complex-valued partial Bianchi modular symbol $\phi_{f_{\varphi/K}^{p}}$. By Proposition 5.10, the partial Bianchi modular symbol $\phi^{\prime}_{f_{\varphi/K}^{p}}=\phi_{f_{\varphi/K}^{p}}/\Omega_{f_{\varphi/K}}$ has values in $V_{k,k}^{*}(L)$ for a sufficiently large $p$-adic field $L$. Since $f_{\varphi/K}^{p}$ has small slope, we can lift $\phi^{\prime}_{f_{\varphi/K}^{p}}$ to its corresponding unique overconvergent partial Bianchi eigensymbol $\Psi_{f_{\varphi/K}^{p}}$ using Proposition 4.3. Taking the Mellin transform of $\Psi_{f_{\varphi/K}^{p}}$ we obtain a locally analytic distribution $L_{p}(f_{\varphi/K}^{p},-)=\mathrm{Mel}(\Psi_{f_{\varphi/K}^{p}})$ on $\mathrm{Cl}_{K}(p^{\infty})$ that is $(h_{\mathfrak{p}},h_{\overline{\mathfrak{p}}})$-admissible, where $h_{\mathfrak{p}}=v_{p}(\lambda_{\mathfrak{p}})=0$ and $h_{\overline{\mathfrak{p}}}=v_{p}(\lambda_{\overline{\mathfrak{p}}})=0$, thus is bounded, i.e., it is a measure.

Using the connection between Bianchi modular symbols and $L$-values, for any Hecke character $\psi$ of $K$ of conductor $\mathfrak{f}=\mathfrak{p}^{t}\overline{\mathfrak{p}}^{s}$ and infinity type $0\leqslant(q,r)\leqslant(k,k)$ we obtain a similar interpolation to Theorem 4.12 given by

[TABLE]

Since $\lambda_{\mathfrak{p}}=\lambda_{\overline{\mathfrak{p}}}=\varphi(\overline{\mathfrak{p}})$ and $\lambda_{\mathfrak{f}}=\lambda_{\mathfrak{p}}^{t}\lambda_{\overline{\mathfrak{p}}}^{s}=\varphi(\overline{{\mathfrak{p}}})^{t+s}$, we obtain the interpolation desired, which gives us the uniqueness of $L_{p}(f_{\varphi/K}^{p},-)$. ∎

The function $L_{p}(f_{\varphi/K}^{p},-)$ can be related with $L_{p}^{\iota}(f_{\varphi/K}^{p},-)$ from Theorem 4.12 using the factor $\iota(\Omega_{f_{\varphi/K}})$.

Proposition 5.12**.**

We have the following equality of measures on $\mathrm{Cl}_{K}(p^{\infty})$

[TABLE]

Proof.

Comparing the interpolation properties satisfied by both measure, we obtain that

[TABLE]

for all $p$-adic avatar $\psi_{p-\mathrm{fin}}$ of Hecke characters $\psi$ of $K$ of conductor $\mathfrak{f}|(p^{\infty})$ and infinity type $0\leqslant(q,r)\leqslant(k,k)$.

Since there are infinitely many characters $\psi$ as above and both sides of the equation are bounded functions. Then the result follows because a non-zero bounded analytic function on an open ball has at most finitely many zeros. ∎

5.4. Relation with Katz $p$-adic $L$-functions

To relate the $p$-adic $L$-function of $f_{\varphi/K}^{p}$ with Katz $p$-adic $L$-functions, we first introduce some notation and state the interpolation property of Katz $p$-adic $L$-functions, for more details the reader can see [BDP12] and [HT93].

Let $\psi$ be a Hecke character of $K$ of suitable infinity type and conductor $\mathfrak{g}\mathfrak{f}$ with $\mathfrak{g}$ coprime to $p$ and $\mathfrak{f}|p^{\infty}$, then Katz constructed in [Kat78] the $p$-adic $L$-function of $\psi$ when $\mathfrak{g}$ is trivial. Later, Hida and Tilouine in [HT93] extended Katz’ construction for non-trivial $\mathfrak{g}$.

Recall the Gauss sum of $\psi$ from 2.9, we now define the local Gauss sum of $\psi$ at prime ideals $\mathfrak{q}$ dividing the conductor of $\psi$ by

[TABLE]

where $e_{K}$ is the character in section 2.3, $\pi_{\mathfrak{q}}$ is a prime element in $\mathcal{O}_{\mathfrak{q}}$, $t=t(\mathfrak{q})$ is the exponent of $\mathfrak{q}$ in the conductor of $\psi$ and $d_{\mathfrak{q}}$ is the $\mathfrak{q}$ component of the idele $d$ associated to the different ideal of $K$. Outside the conductor of $\psi$, we simply put $\tau_{\mathfrak{q}}(\psi)=1$.

Recall the elliptic curve $E$ with complex multiplication by $\mathcal{O}_{K}$ of section 5.2. In Proposition 5.7, we obtained algebraicity of critical $L$-values of $\psi$ using the complex period $\Omega_{\infty}$. Analogously, there exists a $p$-adic period $\Omega_{p}$, that gives us algebraicity of the $p$-adic $L$-function of $\psi$.

Theorem 5.13**.**

(Katz, Hida-Tilouine) Let $\mathfrak{g}$ be an ideal of $K$ coprime to $p$. There exists a unique measure $L_{p}(-)$ on the ray class group $\mathrm{Cl}_{K}(\mathfrak{g}p^{\infty})$ whose value on the $p$-adic avatar $\psi_{p-\text{fin}}$ of a Hecke character $\psi$ of $K$ of infinity type $(a,b)$ with $a>0\geqslant b$ and conductor $\mathfrak{g}\mathfrak{f}$ with $\mathfrak{f}=\mathfrak{p}^{t}\overline{\mathfrak{p}}^{s}$ is given by:

[TABLE]

where $W_{\mathfrak{p}}(\psi)=N(\mathfrak{p})^{-t}\tau_{\mathfrak{p}}(\psi)$.

In order to link our $p$-adic $L$-function $L_{p}(f_{\varphi/K}^{p},-)$ with Katz $p$-adic $L$-functions, we use the factorization of $L(f_{\varphi/K},\psi,s)$ as the product of two Hecke $L$-functions in section 5.2. For this, we first rewrite the interpolation property of $L_{p}(f_{\varphi/K}^{p},-)$ by considering the relation between $\Lambda(f_{\varphi/K}^{p},-)$ and $\Lambda(f_{\varphi/K},-)$ in the proof of Corollary 5.8. Then, by Theorem 5.11, for any Hecke character $\psi$ of $K$ of conductor $\mathfrak{f}=\mathfrak{p}^{t}\overline{\mathfrak{p}}^{s}$ and infinity type $0\leqslant(q,r)\leqslant(k,k)$, we have

[TABLE]

where

[TABLE]

Theorem 5.14**.**

Let $\varphi$ be a Hecke character of $K$ with conductor $\mathfrak{M}$ coprime to $p$ and infinity type $(-k-1,0)$ for $k\geqslant 0$, and denote by $f_{\varphi}^{p}$ the ordinary $p$-stabilization of the CM form $f_{\varphi}$ induced by $\varphi$. Let $f_{\varphi/K}^{p}$ be the base change to $K$ of $f_{\varphi}^{p}$ and $L_{p}(f_{\varphi/K}^{p},-)$ its $p$-adic $L$-function, then for all $\kappa\in\mathfrak{X}(\mathrm{Cl}_{K}(p^{\infty}))$ we have

[TABLE]

where $\chi_{p}$ is the $p$-adic avatar of the adelic norm character and $\Omega_{p}$ is the $p$-adic period in Theorem 5.13.

Remark 5.15*.*

Note that $L_{p}(f_{\varphi/K}^{p},-)$ is a function on $\mathfrak{X}(\mathrm{Cl}_{K}(p^{\infty}))$ while the right-hand side in the above Theorem is a function on $\mathfrak{X}(\mathrm{Cl}_{K}(\overline{\mathfrak{M}}p^{\infty})$. To relate them, we see the latter as a function on $\mathfrak{X}(\mathrm{Cl}_{K}(p^{\infty}))$ via the map $\mathrm{Cl}_{K}(\overline{\mathfrak{M}}p^{\infty})\rightarrow\mathrm{Cl}_{K}(p^{\infty})$.

Proof.

Since $L_{p}(f_{\varphi/K}^{p},-)$ and $L_{p}(-)$ are measures, to obtain the equality in the Theorem, it suffices to prove it on $p$-adic characters $\psi_{p-\text{fin}}$ coming from finite order Hecke characters $\psi$ of conductor $\mathfrak{f}=\mathfrak{p}^{t}\overline{\mathfrak{p}}^{s}$.

For such characters, from (5.4) and Definition 2.19 we have

[TABLE]

Recall from Lemma 5.4 that

[TABLE]

and denote the Hecke characters above by $\eta=\varphi^{c}\psi|\cdot|_{\mathbb{A}_{K}}$ and by $\eta^{\prime}=\varphi^{c}\psi^{c}\lambda_{K}|\cdot|_{\mathbb{A}_{K}}$. By Theorem 5.13 we obtain the following interpolations

[TABLE]

and

[TABLE]

Note that the Euler factors cancel out and the product of the Gauss sums

[TABLE]

[TABLE]

is related to $W(\psi)$ (see for example ii) in [Nar74, Prop 2.14]) by

[TABLE]

where $x_{\mathfrak{f}}$ is the idele corresponding to $\mathfrak{f}$ satisfying $(x_{\mathfrak{f}})_{\mathfrak{q}}=1$ for $\mathfrak{q}$ coprime to $(p)$, $(x_{\mathfrak{f}})_{\mathfrak{p}}=\pi_{\mathfrak{p}}^{t}$ and $(x_{\mathfrak{f}})_{\overline{\mathfrak{p}}}=\pi_{\overline{\mathfrak{p}}}^{s}$.

Recalling that $\Omega_{f_{\varphi/K}}=\left(\frac{\Omega_{\infty}}{\pi}\right)^{2k+2}$ and putting the interpolations of $L_{p}(f_{\varphi/K}^{p},\psi_{p-\mathrm{fin}})$, $L_{p}(\eta_{p-\text{fin}})$ and $L_{p}(\eta^{\prime}_{p-\text{fin}})$ together, we obtain

[TABLE]

Finally, the result follows by normalizing the period $\Omega_{f_{\varphi/K}}$ in Definition 5.6 to be $\frac{2}{w}\left(\frac{\sqrt{D}\Omega_{\infty}}{2\pi i}\right)^{2k+2}$ and using the same argument in the proof of Proposition 5.12. ∎