Low lying zeros of Rankin-Selberg $L$-functions

TL;DR

This paper investigates the distribution of low-lying zeros of Rankin-Selberg $L$-functions for $GL(2) imes GL(2)$, confirming the Katz-Sarnak density conjecture for certain test functions and levels, and analyzing implications for non-vanishing at the central point.

Contribution

It extends the support of the 1-level density analysis beyond previous limits by analyzing Kloosterman sums, and provides conditional bounds on non-vanishing and central $L$-values.

Findings

Supports Katz-Sarnak conjecture for $ ext{supp} \hat{\phi} extless (-rac{1}{2}, rac{1}{2})$

Extends support to $(-rac{5}{4}, rac{5}{4})$ when levels are equal and weights differ

Identifies lower order terms when support exceeds $(-1, 1)$

Abstract

We study the low lying zeros of $GL(2) \times GL(2)$ Rankin-Selberg $L$-functions. Assuming the generalized Riemann hypothesis, we compute the $1$-level density of the low-lying zeroes of $L(s, f \otimes g)$ averaged over families of Rankin-Selberg convolutions, where $f, g$ are cuspidal newforms with even weights $k_1, k_2$ and prime levels $N_1, N_2$, respectively. The Katz-Sarnak density conjecture predicts that in the limit, the $1$-level density of suitable families of $L$-functions is the same as the distribution of eigenvalues of corresponding families of random matrices. The 1-level density relies on a smooth test function $\phi$ whose Fourier transform $\widehat\phi$ has compact support. In general, we show the Katz-Sarnak density conjecture holds for test functions $\phi$ with $\operatorname{supp} \widehat\phi \subset (-\frac{1}{2}, \frac{1}{2})$. When $N_1 = N_2$, we prove the density conjecture for $\operatorname{supp} \widehat\phi \subset (-\frac{5}{4}, \frac{5}{4})$ when $k_1 \ne k_2$, and $\operatorname{supp} \widehat\phi \subset (-\frac{29}{28}, \frac{29}{28})$ when $k_1 = k_2$. A lower order term emerges when the support of $\widehat\phi$ exceeds $(-1, 1)$, which makes these results particularly interesting. The main idea which allows us to extend the support of $\widehat\phi$ beyond $(-1, 1)$ is an analysis of the products of Kloosterman sums arising from the Petersson formula. We also carefully treat the contributions from poles in the case where $k_1 = k_2$. Our work provides conditional lower bounds for the proportion of Rankin-Selberg $L$-functions which are non-vanishing at the central point and for a related conjecture of Keating and Snaith on central $L$-values.