Kernels for products of Hilbert L-functions

TL;DR

This paper investigates kernel functions associated with L-functions and their products for Hilbert cusp forms over real quadratic fields, extending previous work on elliptic modular forms.

Contribution

It introduces new kernel functions for Hilbert L-functions, broadening the scope of prior elliptic modular form results.

Findings

Extended kernel function theory to Hilbert cusp forms

Connected kernel functions to products of L-functions over real quadratic fields

Generalized previous elliptic modular form results

Abstract

We study kernel functions of L-functions and products of L-functions of Hilbert cusp forms over real quadratic fields. This extends the results on elliptic modualr forms by Diamantis and C. O'Sullivan. .

Equations

$$j(\gamma ,z{)}^{k1}=N(j(\gamma ,z){)}^k=j(\gamma ,z_1{)}^{k}j({\gamma }^{\prime },z_2{)}^k=(c_{\gamma }{z}_1+d_{\gamma }{)}^k(c_{\gamma }^{\prime }{z}_2+d_{\gamma }^{\prime }{)}^k.$$

$$(f{|}_k\gamma )(z)=N(\det (\gamma ){)}^{\frac{k}{2}}N(j(\gamma ,z){)}^{-k}f(\gamma z).$$

$$f(z)=a_f(0)+\sum _{\alpha \in {\mathfrak{d}}_{+}^{-1}}{a}_f(\alpha )e^{2\pi i\text{tr}(\alpha z)}.$$

$$⟨f,g⟩:={\int }_{\Gamma \backslash {\mathbb{H}}^2}f(z)\overline{g(z)}d\mu ={\int }_{\mathcal{F}}f(z)\overline{g(z)}d\mu ,$$

$$d\mu =(y_1{y}_2{)}^{-2}d{x}_{1}d{x}_{2}d{y}_{1}d{y}_2=N(y{)}^{-2}N(dx)N(dy).$$

$$L(f,s)=\sum _{\alpha \in {\mathfrak{d}}_{+}^{-1}/{\mathcal{O}}_{+}^{\times }}{a}_f(\alpha )N(\alpha \mathfrak{d}{)}^{-s}=\sum _{\mathfrak{a}}{a}_f(\mathfrak{a})N(\mathfrak{a}{)}^{-s},$$

$$\Lambda (f,s):=D^s(2\pi {)}^{-2s}\Gamma (s{)}^{2}L(f,s)=(-1{)}^k\Lambda (f,k-s)$$

$$T_{\mathfrak{n}}(f(z))=N(\mathfrak{n}{)}^{\frac{k}{2}-1}\sum _{\gamma \in Z\Gamma \backslash M_{\mathfrak{n}}}f{|}_k\gamma (z).$$

$${\mathcal{C}}_k^{Hil}(z;s)=\frac{1}{2}{c}_{k,s,D}^{-2}\sum _{\gamma \in A\backslash \Gamma }(\gamma z{)}^{-s1}j(\gamma ,z{)}^{-k1},$$

$$c_{k,s,D}=\frac{D^{\frac{k-1}{2}}{2}^{2-k}\pi \Gamma (k-1)}{e^{\frac{\pi is}{2}}\Gamma (s)\Gamma (k-s)}$$

$$1<\text{Re}(s)<k-1,z\in {\mathbb{H}}^2.$$

$${\mathcal{C}}_k^{Hil}(z;s)=\sum _{f\in {\mathcal{H}}_k}\frac{\Lambda (f,k-s)}{⟨f,f⟩}f(z),$$

$$E_{s,k-s}^{Hil}(z;w)=\sum _{\gamma ,\delta \in {\Gamma }_{\infty }^{+}\backslash \Gamma ,c_{\gamma {\delta }^{-1}}\gg 0}(c_{\gamma {\delta }^{-1}}{)}^{(w-1)1}{\left(\frac{j(\gamma ,z)}{j(\delta ,z)}\right)}^{-s1}j(\delta ,z{)}^{-k1},$$

$$E_{s,k-s}^{\ast ,Hil}(z;w)=2{\alpha }_{k,s,w,D}\cdot E_{s,k-s}^{Hil}(z;w)$$

$${\alpha }_{k,s,w,D}$$

$$2<\text{Re}(s)<k-2,\text{Re}(w)<\min \{\text{Re}(s)-1,k-1-\text{Re}(s)\}.$$

$$E_{s,k-s}^{\ast ,Hil}(\cdot ;w)=\sum _{f\in {\mathcal{H}}_k}\frac{\Lambda (f,s)\Lambda (f,w)}{⟨f,f⟩}f,$$

$$E_{s,k-s}^{\ast ,Hil}(z;w)=E_{w,k-w}^{\ast ,Hil}(z;s),E_{k-s,s}^{\ast ,Hil}(z;w)=E_{s,k-s}^{\ast ,Hil}(z;w).$$

$$[f_{1},f_{2}]^{Hil}_{\nu}=\sum_{\begin{array}[]{cc}0\leq\ell_{j}\leq\nu_{j},j=1,2\end{array}}(-1)^{\ell_{1}+\ell_{2}}\left(\begin{smallmatrix}k_{1}\mathbf{1}+\nu-\mathbf{1}\\ \nu-\ell\end{smallmatrix}\right)\left(\begin{smallmatrix}k_{2}\mathbf{1}+\nu-\mathbf{1}\\ \ell\end{smallmatrix}\right)f_{1}^{(\ell)}f_{2}^{(\nu-\ell)}.$$

$${\left(\frac{\Gamma (k_1)\Gamma (\nu +1)}{\Gamma (k_1+\nu )}\right)}^2[E_{k_1},E_{k_2}{]}_{(\nu ,\nu )}^{Hil}=4{\left(\frac{\Gamma (k_2+\nu )}{\Gamma (k_2)}\right)}^2{E}_{k_1+\nu ,k_2+\nu }^{Hil}(z;\nu +1),$$

$$E_k(z):=\sum _{\gamma \in {\Gamma }_{\infty }^{+}\backslash \Gamma }j(\gamma ,z{)}^{-k1}.$$

$$⟨f,R_n⟩=n!(2\pi {)}^{-n-1}L(f,n+1),\text{ for all }f\in S_{2k}({\text{SL}}_2(\mathbb{Z})).$$

$$⟨f,{\mathcal{C}}_k^{ell}(\tau ,s)⟩=\Gamma (s)(2\pi {)}^{s}L(f,s),s\in \mathbb{C}.$$

$$\frac{\Lambda (f,m)}{w_{+}(f)},\frac{\Lambda (f,\ell )}{w_{-}(f)}\in \mathbb{Q}(f),$$

$${\left(\frac{\Lambda (f,m)}{{\omega }_{+}(f)}\right)}^{\sigma }=\frac{\Lambda (f^{\sigma },m)}{{\omega }_{+}(f^{\sigma })},{\left(\frac{\Lambda (f,\ell )}{{\omega }_{-}(f)}\right)}^{\sigma }=\frac{\Lambda (f^{\sigma },\ell )}{{\omega }_{-}(f^{\sigma })}.$$

$$\sum _{x\in \mathcal{O}}(z+x{)}^{-s1}=\frac{(2\pi {)}^{2s}}{e^{\pi is}\Gamma (s{)}^2{D}^{1/2}}\sum _{\xi \in {\mathfrak{d}}_{+}^{-1}}N(\xi {)}^{s-1}\exp (2\pi i\text{Tr}(\xi z)),$$

$$\sum _{x\in \mathcal{O}}(z+x{)}^{-s1}=\sum _{x\in \mathcal{O}}N(z+x{)}^{-s}=\sum _{x\in \mathcal{O}}(z_1+x{)}^{-s}(z_2+x^{\prime }{)}^{-s}.$$

$$f(x)=N(x{)}^{s-1}\exp (2\pi i\text{Tr}(xz))$$

$$\hat{f}(w)=\frac{\Gamma (s{)}^2}{(-2\pi i{)}^{2s}}(z+w{)}^{-s1},w\in {\mathbb{R}}^2.$$

$$|f(x)|+|\hat{f}(x)|\ll (1+||x||{)}^{-2-\delta }$$

Full text

Keynote: Hilbert modular form, special $L$-values, cusp form, double Eisenstein series, Petersson inner product, Rankin-Cohen bracket, kernel function

2010 Mathematics Subject Classification: primary 11F67, 11F41 ; secondary 11F03

1. Introduction

One of the central problems in number theory is to explore the nature of special values of various Dirichlet series such as Riemann zeta function, modular $L$-functions, automorphic $L$-functions, etc. The known main idea to study arithmetic properties of the special values of modular $L$-functions is to compare such values with certain inner product of modular forms.

Such an idea was first introduced by Rankin [13], expressing the product of two critical $L$-values of an elliptic Hecke eigenform in terms of the Petersson scalar product of an elliptic Hecke eigenform with a product of Eisenstein series. Much later Zagier ([16], p 149 ) extended Rankin’s result to express the product of any two critical $L$-values of an elliptic Hecke eigenform in terms of the Petersson scalar product of the Hecke eigenform with the Rankin-Cohen brackets of two Eisenstein series. Shimura [14] and Manin [11] developed theories to study arithmetic properties of modular $L$-values on the critical strip. Kohnen-Zagier and Choie-Park-Zagier [10, 2] further studied the space of modular forms whose $L$-values on the critical strip are rational and showed that such a space can be spanned by Cohen kernel introduced by Cohen [3]. Recently double Eisenstein series has been introduced by Diamantis and O’Sullivan[4, 5] as a kernel yielding products of two $L$-values of elliptic Hecke eigenforms. It turns out that Rankin-Cohen brackets [17] of two Eisenstein series can be realized as a double Eisenstein series [5]. Generalizing Cohen kernel, the arithmetic results of $L$-values by Manin [11] and Shimura [14] could be recovered [4, 5].

The purpose of this paper to state above results to the space of Hilbert modular forms by extending kernel functions introduced in [4, 5]. More precisely, a double Hilbert Eisenstein series is a kernel function of two $L$-values of a primitive form in terms of the Petersson scalar product. Also one can recover the arithmetic results [14] of $L$-values of Hilbert cusp forms by studying Cohen kernel over real quadratic fields. Furthermore it turns out that the Rankin-Cohen bracket of two Hilbert Eisenstein series is the special value of a double Hilbert Eisenstein series.

Acknowledgment: The first author was supported by NRF2018R1A4A1023 590 and NRF2017R1A2B2001807. The second author was partially supported by by HIT Youth Talent Start-Up Grant and Grant of Technology Division of Harbin (RC2016XK001001).

We thank the referee for the valuable remarks, which led to improvements in the paper.

2. Notations and Main Theorems

Throughout of this paper, for simplicity, we only consider the space of Hilbert modular forms over real quadratic fields $F$ with narrow class number one on the full Hilbert modular group $\Gamma=\text{SL}_{2}(\mathcal{O}).$

2.1. Notations

Let $F$ be a real quadratic field with narrow class number equal to $1$. Let $D$, $\mathcal{O}$ and $\mathfrak{d}$ be the fundamental discriminant, the ring of integers and the different of $F$ respectively. Let N and Tr be the norm and the trace on $F$， defined by $\textrm{N}(a)=aa^{\prime},\textrm{Tr}(a)=a+a^{\prime}$ with $a^{\prime}$ the algebraic conjugate of $a\in F$. We denote $a\gg 0$ for $a\in F$ if $a$ is totally positive, that is $a>0$ and $a^{\prime}>0$. For $B\subset F$, let $B_{+}$ denote the subset of totally positive elements in $B$. So $\mathcal{O}_{+}$ and $\mathcal{O}^{\times}_{+}$ denote the set of totally positive integers and the set of totally positive units respectively.

For a $2\times 2$ matrix $\mathbf{\gamma}$ in $GL_{2}^{+}(F)$, we usually denote its entries by ${\mathbf{\gamma}}=\begin{smallmatrix}\begin{pmatrix}a_{\mathbf{\gamma}}&b_{\mathbf{\gamma}}\\ c_{\mathbf{\gamma}}&d_{\mathbf{\gamma}}\end{pmatrix}\end{smallmatrix}$ and $\gamma^{\prime}=\begin{smallmatrix}\begin{pmatrix}a_{\mathbf{\gamma}}^{\prime}&b_{\mathbf{\gamma}}^{\prime}\\ c_{\mathbf{\gamma}}^{\prime}&d_{\mathbf{\gamma}}^{\prime}\end{pmatrix}\end{smallmatrix}$. The group $GL_{2}^{+}(F)$ acts on two copies of the complex upper half plane $\mathbb{H}^{2}$ by $\gamma z:=(\gamma z_{1},\gamma^{\prime}z_{2})=(\frac{a_{\gamma}z_{1}+b_{\gamma}}{c_{\gamma}z_{1}+d_{\gamma}},\frac{a_{\gamma}^{\prime}z_{2}+b_{\gamma}^{\prime}}{c_{\gamma}^{\prime}z_{2}+d_{\gamma}^{\prime}})$ as linear fractional transformations for all $\gamma\in GL_{2}^{+}(F)$ and $z=(z_{1},z_{2})\in\mathbb{H}^{2}$.

Let $\Gamma=\textrm{SL}_{2}(\mathcal{O})$ be the modular group of $2\times 2$ matrices with determinant equal to one over $\mathcal{O}$. Denote $\Gamma_{\infty}$ the subgroup of upper-triangular elements and $\Gamma_{\infty}^{+}$ the subgroup of elements with totally positive diagonal entries in $\Gamma_{\infty}$. Let $A$ denote the subgroup of diagonal elements in $\Gamma_{\infty}^{+}$, so $A=\{\text{diag}(\varepsilon,\varepsilon^{-1})\colon\varepsilon\in\mathcal{O}^{\times}_{+}\}$. Throughout the note, we employ the standard multi-index notation. In particular, for $\gamma\in\text{GL}_{2}^{+}(F)$, $z=(z_{1},z_{2})\in\mathbb{H}^{2}$ and $k\in\mathbb{Z}$, we denote $\mathbf{1}=(1,1)$, $(\gamma z)^{k\mathbf{1}}=N(\gamma z)^{k}=(\gamma z_{1})(\gamma^{\prime}z_{2})$, $|z|=(|z_{1}|,|z_{2}|)$, $|z|^{k\mathbf{1}}=|z_{1}|^{k}|z_{2}|^{k}$ and the automorphic factor by

[TABLE]

For any function $f$ on $\mathbb{H}^{2}$ and $\gamma\in\textrm{GL}_{2}^{+}(F)$, define the slash operator by

[TABLE]

A Hilbert modular form of (parallel) weight $k$ for $\Gamma$ is a holomorphic function $f$ on $\mathbb{H}^{2}$ such that $f|_{k}\gamma=f$ for any $\gamma\in\Gamma$. Then $f$ has the following Fourier expansion

[TABLE]

If $a_{f}(0)=0$, we call $f$ a Hilbert cusp form. For a Hilbert cusp form $f$ and a Hilbert modular form $g$ of weight $k$ on $\Gamma$, their Petersson scalar product is defined by

[TABLE]

where $\mathcal{F}$ is a fundamental domain of $\Gamma$ on $\mathbb{H}^{2}$ and

[TABLE]

Here $z=x+iy,$ $Re(z)=x=(x_{1},x_{2})$ and $Im(z)=y=(y_{1},y_{2})$.

Note that this “unnormalized” Petersson inner product is different from Shimura’s [14]. For a Hilbert cusp form $f$ of weight $k$ for $\Gamma$, define the associated L-function by

[TABLE]

where $a_{f}(\mathfrak{a}):=a_{f}(\alpha)$ for $\alpha\mathfrak{d}=\mathfrak{a}$. It is known [7] that the complete $L$-function satisfies

[TABLE]

and has an analytic continuation to the entire $\mathbb{C}$.

Next we recall the theory of Hecke operators on spaces of Hilbert modular forms. For each nonzero integral ideal $\mathfrak{n}$ of $\mathcal{O}$, let $M_{\mathfrak{n}}$ be the set of $2\times 2$ matrices $\gamma$ over $\mathcal{O}$ such that $\det(\gamma)\gg 0$ and $(\det(\gamma))=\mathfrak{n}$. Moreover, let $Z\cong\mathcal{O}^{\times}$ denote the $2\times 2$ scalar matrices with diagonal entries in $\mathcal{O}^{\times}$. The $\mathfrak{n}$-th Hecke operator $T_{\mathfrak{n}}$ on $S_{k}(\Gamma)$, the space of cusp forms for $\Gamma$ of parallel weight-$k$, is defined as

[TABLE]

The operators $T_{\mathfrak{n}}$ are self-adjoint with respect to the Petersson inner product and generate a commutative algebra. It follows that there exists a basis $\mathcal{H}_{k}$, consisting of normalized cuspidal Hecke eigenforms, of $S_{k}(\Gamma)$. We call elements in $\mathcal{H}_{k}$ “primitive forms”. Here $f$ is normalized if the Fourier coefficient $a_{f}(\mathcal{O})=1$ or equivalently if $\mathfrak{d}^{-1}=(\alpha)$ with $\alpha\gg 0$, then $a_{f}(\alpha)=1$. Therefore, for $f\in\mathcal{H}_{k}$, $T_{\mathfrak{n}}f=a_{f}(\mathfrak{n})f$, so $a_{f}(\mathfrak{n})$ is real. For details see Section 1.15 of [7].

2.2. Main Theorems

Fix $k\in\mathbb{Z}.$ We define the Cohen kernel $\mathcal{C}_{k}^{Hil}(z;s)$ on $\mathbb{H}^{2}\times\mathbb{C}$ by

[TABLE]

with

[TABLE]

and $A=\{\textrm{diag}(\varepsilon,\varepsilon^{-1})\colon\varepsilon\in\mathcal{O}_{+}^{\times}\}.$ Note that if $k$ is odd, this definition gives zero function.

Theorem 2.1**.**

(Cohen kernel) Let $k\geq 4$ be even. Then the following hold:

1. (1)

$\mathcal{C}_{k}^{Hil}(z;s)$* converges absolutely and uniformly on all compact subsets in the region given by*

[TABLE] 2. (2)

For each $s\in\mathbb{C}$,

[TABLE]

where $\mathcal{H}_{k}$ is the set of primitive forms s of weight $k$ on $\Gamma.$ 3. (3)

$\mathcal{C}_{k}^{Hil}(z;s)$* can be analytically continued to the whole $s$-plane and for each $s\in\mathbb{C}$, $\mathcal{C}_{k}^{Hil}(z;s)$ is a cusp form for $\Gamma$ of weight $k$ in $z$.*

Next we define the double Eisenstein series as follows: for $s,w\in\mathbb{C},z\in\mathbb{H}^{2}$ and even integer $k\geq 6$,

[TABLE]

and a completed double Eisenstein series by

[TABLE]

with

[TABLE]

Then we have the following:

Theorem 2.2**.**

(double Eisenstein series) Let $k\geq 6$ be even.

1. (1)

$E_{s,k-s}^{Hil}(z;w)$* converges absolutely and uniformly on compact subsets in the region $\mathcal{R}$ of points $(z,(s,w))$ in $\mathbb{H}^{2}\times\mathbb{C}^{2}$ subject to*

[TABLE] 2. (2)

$E^{*,Hil}_{s,k-s}(z;w)$* has an analytic continuation to all $s,w\in\mathbb{C}$ and is a Hilbert cusp form of weight $k$ on $\Gamma$ as a function in $z.$* 3. (3)

[TABLE]

where $\mathcal{H}_{k}$ is the set of primitive forms of weight $k.$ 4. (4)

For $f\in\mathcal{H}_{k}$, $\langle E^{*,Hil}_{s,k-s}(\cdot;w),\,f\rangle=\Lambda(f,s)\Lambda(f,w),\text{ for all }s,w\in\mathbb{C}.$ 5. (5)

$E^{*,Hil}_{s,k-s}(z;w)$* satisfies functional equations:*

[TABLE]

The following gives a relation between Rankin-Cohen brackets and a double Eisenstein series. Rankin-Cohen brackets on spaces of Hilbert modular forms have been studied in [1]. Let us recall the definition of Rankin-Cohen brackets: for each $j=1,2,$ let $f_{j}:\mathbb{H}^{2}\rightarrow\mathbb{C}$ be holomorphic, $k_{j}\in\mathbb{N}$ and $\ell=(\ell_{1},\ell_{2}),\nu=(\nu_{1},\nu_{2})\in\mathbb{Z}_{\geq 0}^{2}.$ Define the $\mathbb{\nu}$-th Rankin-Cohen bracket

[TABLE]

Here $f^{(\ell)}(z)=(\frac{\partial^{\ell_{1}+\ell_{2}}}{{\partial z_{1}}^{\ell_{1}}{\partial z_{2}}^{\ell_{2}}}f)(z)$ and $\left(\begin{smallmatrix}k\mathbf{1}+\nu-\mathbf{1}\\ \nu-\ell\end{smallmatrix}\right)=\left(\begin{smallmatrix}{k}+\nu_{1}-1\\ \nu_{1}-\ell\end{smallmatrix}\right)\left(\begin{smallmatrix}{k}+\nu_{2}-1\\ \nu_{2}-\ell\end{smallmatrix}\right)$.

In the following, we only need parallel $\nu$, that is $\nu_{1}=\nu_{2}$.

Theorem 2.3**.**

(Rankin-Cohen brackets and a double Eisenstein series) For $\nu\in\mathbb{Z}_{\geq 0}$ and $k_{j}\in 2\mathbb{N},j=1,2$, we have

[TABLE]

where $E_{{k}}(z)$ is the usual Hilbert Eisenstein series of weight $k$ on $\Gamma$ defined by

[TABLE]

Remark 2.4**.**

1. (1)

Cohen kernel (see [3] and [10]) is an elliptic cusp form $R_{n}$ of weight $2k$ on $\text{SL}_{2}(\mathbb{Z})$ characterized by, for each $0\leq n\leq 2k-2,$

[TABLE]

Diamantis and O’Sullivan in [4] generalized Cohen kernel $\mathcal{C}^{ell}_{k}(\tau,s)$ to get

[TABLE] 2. (2)

Double Eisenstein series was introduced and studied in [4, 5] as a kernel yielding products of the periods of an elliptic Hecke eigenform at critical values as well as producing products of $L$-functions for Maass cusp forms.

In the following theorem, we recover Shimura’s result on the algebraicity of critical values of $L(f,s)$ (Theorem 4.3 of [14]). For a primitive form $f$ of even weight $k$, let $\mathbb{Q}(f)$ denote the number field generated by the Fourier coefficients of $f$ over $\mathbb{Q}$.

Theorem 2.5**.**

(rationality) Let $f$ be a primitive form of even weight $k\geq 6$ for $\Gamma$. Then there exist complex numbers $\omega_{\pm}(f)$ with $\langle f,\,f\rangle=\omega_{+}(f)\omega_{-}(f)$ such that for even $m$ and odd $\ell$ with $1\leq m,\ell\leq k-1$,

1. (1)

[TABLE] 2. (2)

for each $\sigma\in\text{Gal}(\bar{\mathbb{Q}}/\mathbb{Q})$,

[TABLE]

Remark 2.6**.**

1. (1)

The above theorem is an analogous result of that for elliptic modular forms proved in [10] (Theorem in page 202). We can also extend the rationality easily to arbitrary L-values as did in Theorem 8.3 of [5]. 2. (2)

The above theorem is a special case of Shimura theorem (Theorem 4.3 in [14]) by taking $n=2$, $\psi=1$, and $k_{1}=k_{2}=k.$

3. Proofs

We need the following multi variable Lipschitz summation formula.

Lemma 3.1**.**

(multi-variable Lipschitz summation formula) Assume that $\textrm{Im}(s)>2$. For $z\in\mathbb{H}^{2}$,

[TABLE]

Proof.

By the multi-index notation,

[TABLE]

Following [9], define

[TABLE]

for $x=(x_{1},x_{2})\gg 0$ and [math] otherwise, so for $\textrm{Im}(s)>2$ and $z\in\mathbb{H}^{2}$, $f$ is clearly $L^{1}$ on the quadratic space $V=\mathbb{R}^{2}$ with the trace form. The computation of [9, Theorem 1] shows that the Fourier transform $\hat{f}(w)$ is given by

[TABLE]

It is clear that for $x\in\mathbb{R}^{2}$,

[TABLE]

for any positive $\delta$, where $||\cdot||$ is the Euclidean norm. Therefore, we may apply the Poisson summation formula (see page 252 of [15]), and for a general lattice $M$ in $V$ with integral dual lattice $M^{\vee}$, the Poisson summation formula reads

[TABLE]

Now set $M=\mathfrak{d}^{-1}$, then $M^{\vee}=\mathcal{O}$, $|M/M^{\vee}|=D$ and the Lipschitz summation formula follows easily. ∎

Now we prove Theorem 2.1 about Cohen kernel.

*Proof of Theorem 2.1 *: To show the convergence, we follow the treatment of Section 1.15 in [7]. Firstly, we prove the uniform absolute convergence on compact subsets, using the fact that $L^{1}$-convergence implies uniform convergence on compact subset for series of holomorphic functions (See Lemma on Page 52 of [7]). It suffices to treat the case for $z$ in a small neighborhood $U$ such that $\overline{U}$ is compact, $N(\textrm{Im}z)>X^{-1}$ and $N(\textrm{Im}\gamma z)<X$ for any $\gamma\in\Gamma$ and $z\in U$ for fixed big $X>0$. Note that this essentially picks a Siegel set where $\overline{U}$ lives. In this case, we only have to prove that

[TABLE]

where $\sigma=\textrm{Re}(s)$ and $\mathbb{H}^{2}_{X}$ is the subset of $z$ with $N(\textrm{Im}z)<X$ in $\mathbb{H}^{2}$. Here we denote $|z|=(|z_{1}|,|z_{2}|)$ and employ the multi-index notation. The left-hand side is bounded by

[TABLE]

The space $A\backslash\mathbb{H}^{2}_{X}$ can be viewed as a subspace of

[TABLE]

for some positive $Y$ ($Y$ can be chosen as the smallest totally positive unit bigger than $1$). Moreover, that $N(\textrm{Im}(-1/z))<X$ implies $N(|z|^{2})>X^{-1}N(\textrm{Im}z)$. For $1<r<\sigma<k-1$, the last quantity is equal to

[TABLE]

where in the third line we applied Equation (5.8) of [4] for the integration on $x$. This is part (1).

For part (2), first note that the absolutely uniformly convergence implies that ${\mathcal{C}}^{Hil}_{k}(z;s)$ converges to a Hilbert modular form in the strip $2<\sigma<k-1$ since ${\mathcal{C}}^{Hil}_{k}(z;s)$ is $\Gamma$-invariant with a proper automorphic factor. Secondly, we write

[TABLE]

Applying the Lipschitz summation formula in Lemma 3.1 with $2<\sigma<k-1$, we have

[TABLE]

where $U$ is the subgroup of elements of the form $\begin{pmatrix}1&x\\ 0&1\end{pmatrix}$ in $\Gamma$. On the other hand, recall the $\xi$-th Poincaré series [7]

[TABLE]

and that it is a cusp form with

[TABLE]

We see that up to a constant factor (depending on $s$) ${\mathcal{C}}^{Hil}_{k}(z;s)$ is equal to

[TABLE]

where we used the fact that $a_{f}(\xi)$ is real. Putting everything together, we see that

[TABLE]

It follows that ${\mathcal{C}}^{Hil}_{k}(z;s)$ is cuspidal on the region $2<\sigma<k-1$, and that

[TABLE]

For part (3): The expression of ${\mathcal{C}}^{Hil}_{k}(z;s)$ in part (2) gives the analytic continuation to $s\in\mathbb{C}$ and that for each $s\in\mathbb{C}$, ${\mathcal{C}}^{Hil}_{k}(z;s)$ is a cusp form. This completes the proof. $\Box$

Next, to prove Theorem 2.2 we first need to show a connection between Cohen kernel and double Eisenstein series, which is obtained in the following lemma:

Lemma 3.2**.**

On the region $\mathcal{R}$, we have

[TABLE]

with $T_{\mathfrak{n}}$ the $\mathfrak{n}$-th Hecke operator and $\zeta_{F}(s)$ the Dedekind zeta function for $F$ defined as

[TABLE]

where $\mathfrak{a}$ runs through all integral nonzero ideals.

Proof.

On $\mathcal{R}$, the series expansions of the two $\zeta_{F}$-factors converge absolutely. Therefore, on $\mathcal{R}$, by sending $\gamma$ to $(c_{\gamma},d_{\gamma})$, the left-hand side is equal to

[TABLE]

where $u,\tilde{u}\in\mathcal{O}_{+}^{\times}\backslash\mathcal{O}_{+}$ and $(c,d),(\tilde{c},\tilde{d})\in\mathcal{O}_{+}^{\times}\backslash\mathcal{O}^{2}$ such that $\mathcal{O}c+\mathcal{O}d=\mathcal{O}\tilde{c}+\mathcal{O}\tilde{d}=\mathcal{O}$ and $c\tilde{d}-d\tilde{c}\gg 0$. Combining the two summations, we have

[TABLE]

where this time $\mathfrak{a},\tilde{\mathfrak{a}}$ are over all nonzero integral ideals and the inner summation is over $(c,d),(\tilde{c},\tilde{d})\in\mathcal{O}_{+}^{\times}\backslash\mathcal{O}^{2}$ such that $\mathcal{O}c+\mathcal{O}d=\mathfrak{a}$, $\mathcal{O}\tilde{c}+\mathcal{O}\tilde{d}=\tilde{\mathfrak{a}}$ and $c\tilde{d}-d\tilde{c}\gg 0$. Then we can remove the summation over $\mathfrak{a},\tilde{\mathfrak{a}}$ and it equals to

[TABLE]

where $\mathfrak{n}$ is over all nonzero integral ideals and the inner summation is over over $(c,d),(\tilde{c},\tilde{d})\in\mathcal{O}_{+}^{\times}\backslash\mathcal{O}^{2}$ such that $c\tilde{d}-d\tilde{c}\gg 0$ and $(c\tilde{d}-d\tilde{c})=\mathfrak{n}$. Note that the two summations over $(c,d),(\tilde{c},\tilde{d})$ in the preceding equation have different ranges.

Let $\tilde{A}$ denote the group of diagonal $2\times 2$ matrices with entries in $\mathcal{O}_{+}^{\times}$, so clearly $\tilde{A}\subset Z\Gamma$ and $\tilde{A}\backslash Z\Gamma\cong A\backslash\Gamma$. Note that the inner summation set is mapped bijectively to $\tilde{A}\backslash M_{\mathfrak{n}}$ via

[TABLE]

Therefore, above expression is equal to

[TABLE]

which is the right-hand side. ∎

Using the preceding lemma we prove the following main theorem:

Proof of Theorem 2.2 : For part (1), apply the proof of Lemma 4.1 in [5] for each component and we have

[TABLE]

for any $\gamma,\delta\in\Gamma$ with $c_{\gamma\delta^{-1}}\gg 0$. Let $r=\max\{\textrm{Re}(w),1\}$. Since $[\Gamma_{\infty}:\Gamma_{\infty}^{+}]$ is finite, $E_{s,k-s}(z;w)$ is absolutely bounded up to a constant by

[TABLE]

which is the product of two Eisenstein series whose absolute convergence is well-known (see, for example, 5.7 Lemma of Chapter I in [6]). So absolute convergence follows if we have

[TABLE]

One sees easily that $E_{s,k-s}(z;w)$ transforms correctly under $\Gamma.$ In the above estimate

[TABLE]

where $E(z,s):=\sum_{\gamma\in\Gamma_{\infty}\backslash\Gamma}N(Im(\gamma z))^{-s}=N(y)^{s}+A(s)N(y)^{1-s}+o(1)$. By removing the highest terms $N(y)^{\frac{k}{2}-r+1}$ from the difference, the rest are all $o(N(y)^{\frac{k}{2}}).$ This shows that $E_{s,k-s}(z;w)\rightarrow 0$ as $N(y)\rightarrow\infty$, and hence proves part (2) that $E_{s,k-s}(z;w)$ is a cuspform since only one cusp exists.

For part (3), by Theorem 2.1 the Cohen kernel are cuspforms. By Lemma 3.2,

[TABLE]

since $\overline{a_{f}(\mathfrak{n})}=a_{f}(\mathfrak{n})$. We have shown in Theorem 2.1 that

[TABLE]

By defining

[TABLE]

with $\alpha_{k,s,w,D}$ in (2.2) and using the result of Theorem 2.1, we obtain

[TABLE]

Part (4) follows easily from part (3) since $f$ is a primitive form. Finally, by part (3), $E^{*,Hil}_{s,k-s}(z;w)$ has meromorphic continuation to all of $s,w\in\mathbb{C}$, and reflected from properties of $\Lambda(f,s)$, it satisfies functional equations

[TABLE]

proving part (5) and hence the whole theorem. $\Box$

Using the result about Rankin-Cohen brackets studied in [1], we prove Theorem 2.3:

Proof of Theorem 2.3: One checks (from Proposition 1 in [1])

[TABLE]

Since

[TABLE]

by Lemma 1 in [1], this in turn is equal to

[TABLE]

In such a particular situation, we see easily that the summand is actually well-defined on $\Gamma_{\infty}\backslash\Gamma$. Denote $S$ the subset of $(\delta,\gamma)\in(\Gamma_{\infty}^{+}\backslash\Gamma)^{2}$ with $c_{\gamma\delta^{-1}}\neq 0$ and $S_{\pm,\pm}\subset S$ consists of elements whose $c_{\gamma\delta^{-1}}$ has the prescribed sign vector. In particular, $S_{+,+}$ consists of elements with $c_{\gamma\delta^{-1}}\gg 0$. It is obvious that the sums over these four subsets are all equal, since we may multiply on left by $\pm I$ and $\pm\text{diag}(\varepsilon_{0},\varepsilon_{0}^{-1})\in\Gamma_{\infty}$ to adjust the signs; here $\varepsilon_{0}$ is the fundamental unit. That said, we have

[TABLE]

and it finishes the proof. $\Box$

Proof of Theorem 2.5: We follow the lines in Section 8A of [5] and first prove that for even $m$ and odd $\ell$ with $1\leq m,\ell\leq k-1$, both of $E^{*,Hil}_{m,k-m}(z;k-1)$ and $E^{*,Hil}_{k-2,2}(z;\ell)$ have rational Fourier coefficients. By the functional equations in Theorem 2.2,

[TABLE]

and it suffices to prove that the Fourier coefficients of $E^{*,Hil}_{m,k-m}(z;\ell)$ are rational for even $m$ and odd $\ell$ with $1\leq\ell<m\leq k/2$. By Theorem 2.3, $E^{*,Hil}_{m,k-m}(z;\ell)=C[E_{m+1-\ell},E_{k+1-m-\ell}]^{Hil}_{\ell-1},$ where $C$ is a rational multiple of $\pi^{2-2\ell}$ by Theorem 9.8 on page 515 of [12]. It follows that the Fourier coefficients of $E^{*,Hil}_{m,k-m}(z;\ell)$ belong to $\mathbb{Q}$.

Next, for primitive $f\in\mathcal{H}_{k}$, by Proposition 4.15 of [14] and Theorem 2.2, we have $\langle f,E^{*,Hil}_{k-1,2}(z;k-1)\rangle=\alpha_{f}\langle f,\,f\rangle=\Lambda(f,k-1)\Lambda(f,k-2),$ for certain $\alpha_{f}\in\mathbb{Q}(f)$. Again by Proposition 4.15, since $E^{*,Hil}_{k-1,2}(z;k-1)$ has rational Fourier coefficients, $\alpha_{f}^{\sigma}=\alpha_{f^{\sigma}}$ for each $\sigma\in\text{Gal}(\bar{\mathbb{Q}}/\mathbb{Q})$. Also note $\alpha_{f}\neq 0$ because of the convergence of the Euler product of $\Lambda(f,s)$ for $\textrm{Re}(s)\geq k/2+1$ (see Kim-Sarnak’s bound in [8]). Define

[TABLE]

Then for even $m$, odd $\ell$ with $1\leq m,\ell\leq k-1$,

[TABLE]

again by Proposition 4.15 of [14] and similarly $\frac{\Lambda(f,\ell)}{\omega_{-}(f)}\in\mathbb{Q}(f)$. It is clear that $\omega_{+}(f)\omega_{-}(f)=\langle f,f\rangle$. Finally, the assertion (4.16) of [14] and that $\alpha_{f}^{\sigma}=\alpha_{f^{\sigma}}$ for each $\sigma\in\text{Gal}(\bar{\mathbb{Q}}/\mathbb{Q})$ implies that

[TABLE]

finishing the proof. $\Box$