1 Introduction

Let $\mathbf {F}$ be a fixed real quadratic field over $\mathbf {Q}$ , with ring of integers $O = O_{\mathbf {F}}$ and the real imbeddings $\sigma _{1} = 1,\; \sigma _{2}$ . For simplicity, we assume the narrow class number of $\mathbf {F}$ is $1$ , so the totally positive units are squares of units and every ideal has a totally positive generator. Let $SL(2, O)$ be the Hilbert modular group. For any ideal ${\cal C} \subset O$ , the Hecke congruence subgroups $\Gamma _{0}({\cal C}) = \left \{\left(\!\!\!\begin{array}{cc} a & b \\ c & d \end {array} \right ) \in SL(2, O),\;\;\; c \equiv 0\ \ \pmod {{\cal C}} \right \}$ , act discontinuously on the upper half-space $\mathbf {H}^{2}$ in the usual way with finite co-volumes, i.e., for

$$\begin{align*}\gamma = \left(\!\!\!\begin{array}{cc} a & b \\ c & d \end{array} \right) \in \Gamma _{0}({\cal C}) \; \mbox{and} \; z = (z_{1},\; z_{2}) \in \mathbf{H}^{2},\end{align*}$$

we have

$$\begin{align*}\gamma (z) = \left( \frac{\sigma _{1}(a) z_{1} + \sigma _{1}(b)} {\sigma _{1}(c) z_{1} + \sigma _{1}(d)},\; \frac{\sigma _{2}(a) z_{1} + \sigma _{2}(b)} {\sigma _{2}(c) z_{1} + \sigma _{2}(d)} \right) .\end{align*}$$

Denote by $M_{k}(\Gamma _{0}({\cal C}))(k \in 2\mathbf {Z}\; \mbox {and}\;\geq 2),$ the space of Hilbert modular forms of parallel even weight $(k, k)$ , level ${\cal C}$ with trivial character, i.e., the space of holomorphic functions $f(z)$ on $\mathbf {H}^{2}$ such that for $\gamma = \left (\!\!\!\begin {array}{cc} a & b \\ c & d \end {array} \right ) \in \Gamma _{0}({\cal C}), f(\gamma (z)) = N(cz+d)^{k}f(z)$ , where for $ z = (z_{1},\; z_{2}) \in \mathbf {H}^{2} $ ,

$$\begin{align*}N(cz+d)^{k} = (\sigma _{1}(c)z_{1} + \sigma _{1}(d))^{k} \cdot (\sigma _{2}(c)z_{2} + \sigma _{2}(d))^{k} .\end{align*}$$

Any $f(z)$ in $M_{k}(\Gamma _{0}({\cal C}))$ has the following Fourier expansion (we assume that the different of $\mathbf {F}$ is generated by $\delta = \delta _{\mathbf {F}}> 0$ , where and henceforth $\xi> 0 $ for $\xi \in \mathbf {F}$ means that $\xi $ is a totally positive element in $\mathbf {F}$ , and denote $\nu ^{(i)} = \sigma _{i}(\nu )$ , the ith conjugate of $\nu $ for $i = 1, 2$ ):

(1) $$ \begin{align} f(z) = \sum _{\nu \in O,\;\; \nu \geq 0} a(\nu) \exp (2\pi i \mbox{Tr}(\nu z)) , \end{align} $$

where

$$\begin{align*}\mbox{Tr}(\nu z) = \sum _{i=1}^{2} \nu^{(i)} z_{i} \delta ^{(i)-1} .\end{align*}$$

Since any $f(z)$ in $M_{k}(\Gamma _{0}({\cal C}))$ is invariant under $ \left(\begin{array}{cc} \epsilon & 0 \\ 0 & \epsilon ^{-1} \end {array} \right ) $ , where $\epsilon $ is an unit in O, we have $a(\epsilon ^{2} \nu ) = a(\nu ) $ .

$f(z) \in M_{k}(\Gamma _{0}({\cal C}))$ is called a Hilbert modular cusp form if the Fourier expansion of $ f(g(z)) N(cz+d)^{-k} $ (see [Reference LuoLu, p. 130]) has no constant term for all $g = \left(\begin {array}{cc} a & b \\ c & d \end {array} \right ) \in SL(2, \mathbf {F})$ . Space of all such cusp forms is denoted by $S_{k}(\Gamma _{0}({\cal C}))$ .

It is well-known (see [Reference GarrettGa]) that $ \mbox {dim}_{\mathbf {C}}S_{k}(\Gamma _{0}({\cal C}))$ is finite, and (see [Reference ShimizuSh]) $J =:\mbox {dim}_{\mathbf {C}}S_{k}(\Gamma _{0}({\cal C})) \sim \frac {\mbox {vol}(\Gamma _{0}({\cal C}) \backslash \mathbf {H}^{2})}{(4\pi )^{2}} (k-1)^{2} $ as $k \rightarrow \infty $ . Moreover,

$$ \begin{align*} \mbox{vol}(\Gamma _{0}({\cal C}) \backslash \mathbf{H}^{2})) & = [SL(2, O) : \Gamma _{0}({\cal C})]\; \mbox{vol}(SL(2, O) \backslash \mathbf{H}^{2}) \\ & = 2 N({\cal C}) \prod _{{\cal P}|{\cal C}} (1 + N({\cal P})^{-1}) \times \pi ^{-2}\zeta_{\mathbf{F}}(2)D^{3/2}, \end{align*} $$

where $\zeta _{\mathbf {F}}(s) $ is the Dedekind zeta-function of $\mathbf {F}$ and $D = D_{\mathbf {F}}$ is the discriminant. The Petersson inner product on $S_{k}(\Gamma )$ is defined by

$$\begin{align*}<g_{1},\;g_{2}>= \int _{\Gamma \backslash \mathbf{H}^{2}} g_{1}(z) \overline{g_{2}(z)} \prod _{i=1}^{2} y_{i}^{k-2} dx_{i}dy_{i} ,\end{align*}$$

where $z = (z_{1},\;z_{2})$ with $z_{i} = x_{i} + y_{i} \sqrt {-1}, \;i = 1, 2$ .

Now, let f be a cuspidal Hilbert modular form of parallel weight $(k, k)$ for even $k \geq 2$ and with respect to $GL^{+}(2, O) \supset SL(2, O)$ . We assume f is a normalized Hecke eigenform with Fourier coefficients $a_{f}(\nu ) = a_{f}(1) \lambda _{f}(\nu ) N(\nu )^{(k-1)/2}, \;\nu \in O$ , where $\lambda _{f}(\mu ) $ is the eigenvalue of $f(z)$ for the Hecke operator $T_{(\mu )}$ (see, e.g., [Reference GarrettGa]). We have

$$\begin{align*}\lambda _{f}(\mu) \lambda _{f}(\nu) = \sum _{(d),\; d|(\mu, \nu),\; d> 0 } \lambda _{f}\left( \frac{\mu \nu}{d^{2}} \right). \end{align*}$$

The standard L-function associated with f is defined, for $\Re (s)> 1$ , by

$$\begin{align*}L(s,\;f) = \sum _{(\mu),\; \mu> 0} \lambda _{f}(\mu) N(\mu)^{-s} ,\end{align*}$$

which has Euler product

$$\begin{align*}\prod _{(\pi),\; \pi> 0} (1 - \lambda _{f}(\pi) N(\pi)^{-s} + N(\pi)^{-2s})^{-1} ,\end{align*}$$

where $\pi $ stands for prime element of O. It is well-known that $L(s,\;f)$ has analytic continuation to the whole complex plane as an entire function. Let

$$\begin{align*}\Lambda (s, f) = (2\pi)^{-2s} \Gamma ^{2}(s+(k-1)/2) L(s, f) .\end{align*}$$

We then have the functional equation

$$\begin{align*}\Lambda (s, f) = \epsilon _{f} D^{1 - 2s} \Lambda (1-s, f) ,\end{align*}$$

where $\epsilon _{f}$ is the root number of absolute value $1$ .

Asai [Reference AsaiAs] defined a new Dirichlet series by restricting the coefficients on rational integers,

$$\begin{align*}L(s, \mbox{As}(f)) = \zeta (2s) \sum _{m=1}^{\infty} \lambda _{f}(m) m^{-s},\;\Re(s)> 1.\end{align*}$$

He showed that the function

$$\begin{align*}\Lambda (s, \mbox{As}(f)) = D^{s/2} (2\pi)^{-2s} \Gamma (s + k -1) \Gamma (s) L(s, \mbox{As}(f)) \end{align*}$$

admits analytic continuation to the whole s-plane with possible simple poles at $s = 0, 1$ , and satisfies the functional equation

$$\begin{align*}\Lambda (s, \mbox{As}(f)) = \Lambda (1-s, \mbox{As}(f)) .\end{align*}$$

Moreover, if

$$ \begin{align*} L(s, f) & = \prod _{(\pi),\; \pi> 0} (1 - \lambda _{f}(\pi) N(\pi)^{-s} + N(\pi)^{-2s})^{-1} \\ & = \prod _{(\pi),\; \pi > 0} [(1 - \alpha _{f}(\pi)N\pi^{-s}) (1 - \beta _{f}(\pi)N\pi^{-s})]^{-1}, \end{align*} $$

then we have

$$\begin{align*}L(s, \mbox{As}(f)) = \prod _{p} L_{p}(s),\end{align*}$$

where

$$\begin{align*}L_{p}^{-1}(s) = \left\{\!\!\begin{array}{lll} (1 - \alpha _{f}(\pi _{1}) \alpha _{f}(\pi _{2}) p^{-s}) (1 - \alpha _{f}(\pi _{1}) \beta _{f}(\pi_{2}) p^{-s}) & & \\ (1 - \beta _{f}(\pi_{1}) \alpha _{f}(\pi _{2}) p^{-s}) (1 - \beta _{f}(\pi_{1}) \beta _{f}(\pi_{2}) p^{-s}), & \;\mbox{if}\; & p = \pi _{1} \pi _{2}, \pi_{1} \neq \pi _{2}; \\ (1 - \alpha _{f}(\pi) p^{-s}) (1 - \beta _{f}(\pi) p^{-s}) (1 - p^{-2s}), & \;\mbox{if}\; & p = \pi; \\ (1 - \alpha ^{2}_{f}(\pi) p^{-s}) (1 - \beta ^{2}_{f}(\pi) p^{-s}) (1 - p^{-s}), & \;\mbox{if}\; & p = \pi^{2}. \end{array} \right. \end{align*}$$

Ramakrishnan [Reference RamakrishnanRa] and Krishnamurthy [Reference KrishnamurthyKr] proved that $\Lambda (s, \mbox {As}(f))$ is in fact the L-function associated with an automorphic form on $GL(4, A_{\mathbf {Q}})$ , the Asai lift $\mbox {As}(f)$ of f. Then, in view of the Splitting Formula in [Reference AsaiAs] and assuming $D = D_{\mathbf {F}}$ is odd, we have

$$\begin{align*}L(s, f\otimes f^{t}) = L(s, \mbox{As}(f))\; L(s, \mbox{As}(f) \otimes \chi _{D}) ,\end{align*}$$

where

$$\begin{align*}\chi _{D}(\cdot) = \left(\frac{D}{\cdot}\right) \end{align*}$$

is the Kronecker symbol, and

$$\begin{align*}f^{t}(z_{1}, z_{2}) = f(z_{2}, z_{1}) .\end{align*}$$

If f is a base change from an Hecke eigenform $h \in S_{k}(SL_{2}(\mathbf {Z}))$ , then f is symmetric, i.e., $f = f^{t}$ , and

$$\begin{align*}L(s, \mbox{As}(f)) = L(s, \mbox{sym}^{2}(h))\;L(s, \chi _{D}),\end{align*}$$

while if f is a base change from an Hecke eigenform $h \in S_{k}(\Gamma _{0}(D), \chi _{D})$ , then also $f = f^{t}$ , and

$$\begin{align*}L(s, \mbox{As}(f)) = L(s, \mbox{sym}^{2}(h))\; \zeta(s) \end{align*}$$

(see [Reference AsaiAs, Section 5]).

Moreover, Prasad and Ramakrishnan [Reference Prasad and RamakrishnanPR] established the following (special case of) cuspidal criterion for $\mbox {As}(f)$ .

Choosing an orthonormal basis $\{f_{j}(z)\}_{j=1}^{J}$ of $S_{k}(\Gamma _{0}({\cal C}))$ and denote the Fourier coefficients of $f_{j}(z)$ by $a_{j}(\cdot )$ . We normalize the Fourier coefficients $a_{j}(\mu )$ by

$$\begin{align*}\psi _{j}(\mu) = \left( \frac{ N({\cal C}) ((k-1)!)^{2}D^{k+1}} {((4\pi)^{2} N(\mu))^{k-1}} \right) ^{1/2} a_{j}(\mu). \end{align*}$$

We then have the Petersson formula for Hilbert modular forms as proved in [Reference LuoLu],

$$\begin{align*}\sum ^{J}_{j=1} \bar{\psi_{j}}(\nu) \psi _{j}(\mu) = \chi _{\nu}(\mu) D^{3/2} N({\cal C}) (k-1)^{2} + N({\cal C}) (k-1)^{2} D (2\pi)^{2} \sum _{\epsilon \in U} \sum _{c\in {\cal C}^{\times}/U} \end{align*}$$

(2) $$ \begin{align} \frac{1}{|N(c)|} S(\nu,\mu \epsilon ^{2};c) \;NJ_{k-1}(4\pi \sqrt{\mu \nu} |\epsilon|/|c|) , \end{align} $$

where $\chi _{\nu }$ is the characteristic function of the set $\{ \nu \epsilon ^{2}, \;\epsilon \in U \} $ , U is the unit group of $\mathbf {F}$ ,

$$\begin{align*}S(\nu,\mu;c) = \sum _{h\;(\!\!\bmod \;c)}\!^{*} \; e\left( \frac{\nu h + \mu \bar{h}}{c} \right) \end{align*}$$

is the generalized Kloosterman sum, and $e(x) = \exp (2\pi i \mbox {Tr}(x))$ for $x \in \mathbf {F}$ . We will assume that in the above formula, the c’s are chosen among their associates the representatives satisfying $|N(c)|^{1/2} \ll |c ^{(i)}| \ll |N(c)|^{1/2},\; i = 1, 2$ .

If the $L^{2}$ -normalized basis element $f_{j} = \tilde {f}_{j}/|\!| \tilde {f}_{j} |\!|$ is a newform, where $\tilde {f}_{j}$ is the corresponding arithmetically normalized newform with the first Fourier coefficient $1$ , then $\psi _{j}(\mu ) = \psi _{j}(1)\;\lambda _{j}(\mu )$ , where $\lambda _{j}(\cdot )$ denotes the (normalized) Hecke eigenvalues of $f_{j}$ as noted above. For ${\cal C} = (1)$ , from the integral representation for $L(s,\; \tilde {f}_{j}\otimes \overline {\tilde {f}_{j}})$ , and the factorization $L(s, \;\tilde {f}_{j}\otimes \overline {\tilde {f}_{j}}) = \zeta _{\mathbf {F}}(s)\; L(s, \; \mbox {ad}(\tilde {f}_{j}))$ , we have

$$\begin{align*}|a_{j}(1)|^{-2} = |\!| \tilde{f}_{j} |\!|^{2} = 16 D^{1 + k} (4\pi)^{-2k-2} \Gamma^{2}(k)\;L(1,\; \mbox{ad}(\tilde{f}_{j}))/L(1,\;\chi _{D}) .\end{align*}$$

Thus for ${\cal C} = (1)$ ,

$$\begin{align*}\bar{\psi}_{j}(\nu) \; \psi_{j}(\mu) = \frac{(4 \pi)^{4} L(1,\; \mbox{ad}(\tilde{f}_{j}))}{16 L(1,\;\chi _{D})} \lambda _{j}(\nu)\; \lambda _{j}(\mu) .\end{align*}$$

For each $j,\;1\leq j \leq J$ and any $\epsilon> 0$ , we have (see [Reference TaylorTa])

$$\begin{align*}\lambda _{j}(\mu) \ll N(\mu)^{\epsilon} ,\end{align*}$$

and by a straightforward extension of results of [Reference IwaniecIw] and [Reference Hoffstein and LockhartHL] that

$$\begin{align*}k^{-\epsilon} \ll L(1,\; \mbox{ad}(\tilde{f}_{j})) \ll k^{\epsilon} .\end{align*}$$

In [Reference LuoLu], we proved an asymptotic formula for the mean value of the linear form in $\psi _{j}(\cdot )$ in the level aspect. In this paper, we establish an analogous result for the weight aspect as well in the context of the quadratic field $\mathbf {F}$ , with an application to the second moment of $L(1/2, \mbox {As}(f))$ . The generalization of Theorem 1.2 to the general totally real fields is straightforward.

Theorem 1.2 Let $b(\cdot )$ be an arbitrary complex numbers such that $b(\epsilon ^{2} \mu ) = b(\mu ) $ for $\epsilon \in U$ , and $\eta> 0$ . Then for $S_{k}(\Gamma _{0}({\cal C})),$ we have as $k \rightarrow \infty $ ,

$$\begin{align*}\sum _{j=1}^{J} {\left|\sum _{\mu} b(\mu) \psi_{j}(\mu)\right|}^{2} \ll (N({\cal C})k^{2} + X) (kXN({\cal C}))^{\eta} \sum _{\mu} |b(\mu) |^{2} , \end{align*}$$

where the summation over $\mu $ ’s is restricted to $\mu \in O^{\times }/U^{2},\; \mu> 0,\; N(\mu )\leq X $ , and the implicit constant only depends on the quadratic field $\mathbf {F}$ and $\eta $ .

Assume $\mbox {As}(f)$ is cuspidal. From [Reference Iwaniec and KowalskiIK, p. 98], we have a series representation for the central L-value of $ L(s, \mbox {As}(f))$ ,

(3) $$ \begin{align} L(1/2,\; \mbox{As}(f)) = 2 \sum _{n=1}^{\infty} \frac{\lambda_{f}(n)}{n^{1/2}} V_{1/2}\left( \frac{n}{\sqrt{D}} \right), \end{align} $$

where

$$\begin{align*}V_{1/2}(y) = \frac{1}{2\pi i} \int _{(2)} (4\pi ^{2}y)^{-u} \;\zeta(1 + 2u) \frac{\Gamma (1/2 + u)\; \Gamma (k + u - 1/2)}{\Gamma (1/2)\; \Gamma (k - 1/2)} \frac{du}{u} .\end{align*}$$

Since

$$\begin{align*}\frac{\Gamma (k + u - 1/2)}{\Gamma (k - 1/2)} \ll k^{\Re(u)} \end{align*}$$

by Stirling’s formula, we see that $V_{1/2}(y) \ll k^{-A}$ for any $A \geq 1,$ if $y> k^{1 + \eta }$ for any $\eta> 0$ . Thus, we have

$$\begin{align*}L(1/2, \;\mbox{As}(f)) = 2 \sum _{n \leq k^{1 + \eta}} \frac{\lambda_{f}(n)}{n^{1/2}} V_{1/2}\left( \frac{n}{\sqrt{D}} \right) + O(1) .\end{align*}$$

From Theorem 1.2 and the above formula for $L(1/2,\; \mbox {As}(f))$ , and by extending the orthonormal Hecke basis of $S_{k}(GL^{+}_{2}(O))$ to an orthonormal (Hecke) basis of $S_{k}(SL(2, O))$ and the positivity, we obtain the following theorem.

Theorem 1.3 For the orthonormal Hecke basis $\{f_{j}\}$ of $S_{k}(GL^{+}_{2}(O))$ and any $\eta> 0$ , we have

$$\begin{align*}\sum _{1\leq j \leq J}\!^{*} \; |L(1/2,\; \mbox{As}(f_{j}))|^{2} \ll k^{2 + \eta} ,\end{align*}$$

where the * means that the summation is restricted to cuspidal Asai lifts $\mbox {As}(f_{j})$ , and the constant implicit only depends on the quadratic field $\mathbf {F}$ and $\eta $ .

It remains to prove Theorem 1.2, which is the goal of the next section.