One-level density of quadratic twists of $L$-functions

According to the Langlands program (see [Langlands]), the most general $L$-function is that attached to an automorphic representation of $\text{GL}_{N}$ over a number field, which in turn can be written as products of the $L$-functions attached to cuspidal automorphic representations of $\text{GL}_{M}$ over $\mathbb{Q}$.

Let $L(s,\pi)$ be the $L$-function attached to an automorphic cuspidal representation $\pi$ of $\text{GL}_{M}$ over $\mathbb{Q}$. For $\Re(s)$ large enough, $L(s,\pi)$ can be expressed as an Euler product of the form

The order of vanishing of $L(s,\pi)$ at the central point has rich arithmetic applications as can be seen from the Birch and Swinnerton-Dyer conjecture on the rank of an elliptic curve. One way to address the non-vanishing issue is to apply the density conjecture of N. Katz and P. Sarnak [KS1, K&S], which suggests that the distribution of zeros near the central point of a family of $L$-functions is the same as that of eigenvalues near $1$ of a corresponding classical compact group. The truth of the density conjecture would imply that $L(\tfrac{1}{2},\chi)\neq 0$ for almost all quadratic Dirichlet characters $\chi$.

In this direction, it was estbalished unconditionally by K. Soundararajan [sound1] that $L(\tfrac{1}{2},\chi_{8d})\neq 0$ for at least 87.5% of odd square-free $d\geq 0$. Here and after, $\chi_{d}=\left(\frac{d}{\cdot}\right)$ stands for the Kronecker symbol. Conditionally under the truth of the generalized Riemann hypothesis (GRH), A. E. Özluk and C. Snyder [O&S] computed the one level density for the family of quadratic Dirichlet $L$-functions. Their result implies that, after optimizing the test function involved as done in [B&F, ILS], we have $L(1/2,\chi_{d})\neq 0$ for at least $(19-\cot\frac{1}{4})/16=94.27\ldots\%$ of the fundamental discriminants $|d|\leq X$.

The work of A. E. Özluk and C. Snyder has been extended by M. O. Rubinstein [Ru] to all $n$-level densities for families of quadratic twists of automorphic $L$-functions. In [Gao], the first-named author computed the $n$-level densities for the family of quadratic Dirichlet $L$-functions under GRH. In [G&Zhao2], the authors studied the one-level density for the family for quadratic twists of modular $L$-functions under GRH. In [ER-G&R], A. Entin, E. Roditty-Gershon and Z. Rudnick considered the $n$-level densities for the family for quadratic Dirichlet $L$-functions over function field. For other works on the $n$-level densities of families of quadratic twists of $L$-functions and their relations to the random matrix theory, see [A&B, BFK21-1, B&F, MS, MS1, Miller1, FPS, FPS1, FPS2, HMM, HKS, G&Zhao4, G&Zhao5, G&Zhao6, G&Zhao9].

The aim of this paper to compute the one-level density for the family for quadratic twists of automorphic $L$-functions under GRH. For this, we fix a self-contragredient representation $\pi$ of $\text{GL}_{M}$ over $\mathbb{Q}$ so that $\pi=\tilde{\pi}$. As the cases of $M=1,2$ have been studied in [O&S, G&Zhao2], we assume that $M\geq 3$. We also assume the Ramanujan-Petersson conjecture, which implies that

The Rankin-Selberg symmetric square $L$-function $L(s,\pi\otimes\pi)$ factors as the product of the symmetric and exterior square $L$-functions ( see [BG, p. 139]):

and has a simple pole at $s=1$ which is carried by one of the two factors. Write the order of the pole of $L(s,\wedge^{2})$ as $(\delta(\pi)+1)/2$ so that $\delta(\pi)=\pm 1$. The order of the pole of $L(s,\vee^{2})$ then equals to $(1-\delta(\pi))/2$. In this paper, we restrict our attention to those $\pi$ with $\delta(\pi)=-1$. The case in which $\delta(\pi)=1$ will be discussed later at the end of this section.

We are interested in the following family of quadratic twists of $L$-functions given by

Note that $\chi_{8d}$ is a primitive quadratic Dirichlet character modulo $8d$ when $d$ is positive, odd and square-free. Any $L(s,\pi\otimes\chi_{8d})\in\mathcal{F}$ has an Euler product given by (see [Ru, Section 3.6])

As $\pi=\tilde{\pi}$, $L(s,\pi\otimes\chi_{8d})$ has a functional equation of the form

where $\mu_{\pi\otimes\chi_{8d}}(j)\in\mathbb{C}$ and satisfies $\Re(\mu_{\pi\otimes\chi_{8d}}(j))\geq 0$ since we are assuming GRH.

Moreover, as $\delta(\pi)=-1$, we have

where $Q_{\pi\otimes\chi_{8d}}$ is the conductor of $L(s,\pi\otimes\chi_{8d})$ and it is known (see [M&T-B, Section 2]) that

As we assume GRH, we may write the zeros of $\Lambda(s,\pi\otimes\chi_{8d})$ as $\tfrac{1}{2}+i\gamma_{\pi\otimes\chi_{8d},j}$ with $\gamma_{\pi\otimes\chi_{8d},j}\in\mathbb{R}$ and we order them as

Let $X$ be a large real number and we normalize the zeros by defining

We define for a fixed even Schwartz class function $\phi$,

We shall regard the function $\phi$ in the above expression as a test function and we also fix a non-zero, non-negative function $w$ compactly supported on $\mathbb{R}^{+}$, which we shall regard as a weight function. We then define the one-level density of the family $\mathcal{F}$ with respect to $w$ by

where we use $\sum^{*}$ to denote a sum over square-free integers throughout the paper and $W(X)$ is the total weight given by

We evaluate the one-level density $D(X;\phi,w,\mathcal{F})$ asymptotically in this paper and our result is as follows.

Note that the kernel $W_{USp}$ in the integral of (1.5) is the same one appearing on the random matrix theory side, when studying the eigenvalues of unitary symplectic matrices. This implies that the family $\mathcal{F}$ of quadratic twists of $L$-functions is a symplectic family and hence verifies the density conjecture of N. Katz and P. Sarnak for this family when the support of the Fourier transform of the test function $\phi$ is contained in the interval $(-2/M,2/M)$. Our result improves upon a result of Rubinstein given in [Ru, Theorem 3.2] by doubling the size of the allowable support of $\hat{\phi}$ in the case of the one-level density.

Theorem 1.1 can be regarded as a generalization of the result of Özluk and Snyder [O&S] on the one-level density of the family of quadratic Dirichlet $L$-functions and Theorem 1.1 of [G&Zhao2] on the one-level density of the family of quadratic twists of modular $L$-functions. A key step in the proofs of all cases is to estimate certain character sums. The treatments in [O&S] and [G&Zhao2] use Poisson summations to convert the character sums into corresponding dual sums (see the work of K. Soundararajan [sound1] for a systematic development of this Poisson summation formula). Our proof of Theorem 1.1 is different as we only makes use of functional equations of the $L$-functions involved, although we shall call this approach Poisson summation as well. In fact, our approach here is motivated by a Poisson summation formula over number fields established by L. Goldmakher and B. Louvel in [G&L, Lemma 3.2]. The equivalence of this version of Poisson summation formula and that developed by Soundararajan [sound1] over primitive Hecke character of the Gaussian filed $\mathbb{Q}(i)$ has already been pointed out in [Gao2, Section 2.6]. More generally, one may evaluate the expression for $D(X;\phi,w,\mathcal{F})$ in (1.4) by first applying the explicit formula given in Lemma 2.2 to convert the sum over zeros of $L$-function into a sum over prime powers. The resulting double sum can then be viewed as a double Dirichlet series and the use of functional equations can be thought of as an effort to understand the analytical behavior of the double Dirichlet series. One may find in [DGH] a build-up of the theory of double Dirichlet series.

We remark here that Theorem 1.1 may be extended to the case in which $\delta(\pi)=1$. This requires generalizing the Possion summation formula in Lemma 2.5 to smoothed character sums over arithmetic progressions. As noted in [Ru, pp. 173–176], the sign of the functional equation of $L(s,\pi\otimes\chi_{d})$ depends on $d$ lying in certain fixed arithmetic progressions.