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- W4386392327 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 $widehatphi$ has compact support. In general, we show the Katz-Sarnak density conjecture holds for test functions $phi$ with $operatorname{supp} widehatphi subset (-frac{1}{2}, frac{1}{2})$. When $N_1 = N_2$, we prove the density conjecture for $operatorname{supp} widehatphi subset (-frac{5}{4}, frac{5}{4})$ when $k_1 ne k_2$, and $operatorname{supp} widehatphi subset (-frac{29}{28}, frac{29}{28})$ when $k_1 = k_2$. A lower order term emerges when the support of $widehatphi$ exceeds $(-1, 1)$, which makes these results particularly interesting. The main idea which allows us to extend the support of $widehatphi$ 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." @default.
- W4386392327 created "2023-09-03" @default.
- W4386392327 creator A5042872987 @default.
- W4386392327 date "2023-08-23" @default.
- W4386392327 modified "2023-09-27" @default.
- W4386392327 title "Low lying zeros of Rankin-Selberg $L$-functions" @default.
- W4386392327 doi "https://doi.org/10.48550/arxiv.2308.16302" @default.
- W4386392327 hasPublicationYear "2023" @default.
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