Low lying zeros of Rankin-Selberg $L$-functions
Alexander Shashkov
TL;DR
This work studies the low-lying zeros of GL(2)×GL(2) Rankin–Selberg L-functions by averaging the 1-level density over Rankin–Selberg convolution families under GRH. The authors prove Katz–Sarnak-type density results for test functions with compactly supported Fourier transform, first in general up to $\operatorname{supp}\widehat{\phi} \subset (-\tfrac{1}{2},\tfrac{1}{2})$, and then extend the support to $(-\tfrac{5}{4}, \tfrac{5}{4})$ when the levels coincide with $k_1\neq k_2$, and to $(-\tfrac{29}{28}, \tfrac{29}{28})$ when $k_1=k_2$. A key technical advance is the detailed analysis of products of Kloosterman sums arising from the Petersson trace formula, enabling the extension of the admissible Fourier support beyond the unit interval and revealing a lower-order term when the support crosses $(-1,1)$. The method integrates the explicit formula with Petersson averaging and a careful treatment of off-diagonal contributions via GRH for Dirichlet $L$-functions, Gauss sums, and Ramanujan sums. The results yield conditional lower bounds on the proportion of non-vanishing central values and connect to Keating–Snaith’s central-value distribution, illustrating symplectic symmetry in this Rankin–Selberg setting and informing central-value statistics in families of automorphic $L$-functions.
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 $φ$ whose Fourier transform $\widehatφ$ has compact support. In general, we show the Katz-Sarnak density conjecture holds for test functions $φ$ with $\operatorname{supp} \widehatφ\subset (-\frac{1}{2}, \frac{1}{2})$. When $N_1 = N_2$, we prove the density conjecture for $\operatorname{supp} \widehatφ\subset (-\frac{5}{4}, \frac{5}{4})$ when $k_1 \ne k_2$, and $\operatorname{supp} \widehatφ\subset (-\frac{29}{28}, \frac{29}{28})$ when $k_1 = k_2$. A lower order term emerges when the support of $\widehatφ$ exceeds $(-1, 1)$, which makes these results particularly interesting. The main idea which allows us to extend the support of $\widehatφ$ 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.