Low lying zeros of Rankin-Selberg $L$-functions
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arXiv
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| Formato: | Preprint |
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2023
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| _version_ | 1866912841990144000 |
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| author | Shashkov, Alexander |
| author_facet | Shashkov, Alexander |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2308_16302 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Low lying zeros of Rankin-Selberg $L$-functions Shashkov, Alexander Number Theory 11M26, 11M50 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. |
| title | Low lying zeros of Rankin-Selberg $L$-functions |
| topic | Number Theory 11M26, 11M50 |
| url | https://arxiv.org/abs/2308.16302 |