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| Main Authors: | , , , , |
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| Format: | Preprint |
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2020
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| Online Access: | https://arxiv.org/abs/2011.10140 |
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| _version_ | 1866917770426318848 |
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| author | Bołdyriew, Elżbieta Chen, Fangu VI, Charles Devlin Miller, Steven J. Zhao, Jason |
| author_facet | Bołdyriew, Elżbieta Chen, Fangu VI, Charles Devlin Miller, Steven J. Zhao, Jason |
| contents | Katz and Sarnak conjectured a correspondence between the $n$-level density statistics of zeros from families of $L$-functions with eigenvalues from random matrix ensembles. In many cases the sums of smooth test functions, whose Fourier transforms are finitely supported, over scaled zeros in a family converge to an integral of the test function against a density $W_{n, G}$ depending on the symmetry $G$ of the family (unitary, symplectic or orthogonal). This integral bounds the average order of vanishing at the central point of the corresponding family of $L$-functions. We can obtain better estimates on this vanishing by finding better test functions to minimize the integral. We pursue this problem when $n=2$, minimizing \[ \frac{1}{Φ(0, 0)} \int_{{\mathbb R}^2} W_{2,G} (x, y) Φ(x, y) dx dy \] over test functions $Φ\colon {\mathbb R}^2 \to [0, \infty)$ with compactly supported Fourier transform. We study a restricted version of this optimization problem, imposing that our test functions take the form $ϕ(x) ψ(y)$ for some fixed admissible $ψ(y)$ and $\mathrm{supp}({\hat ϕ}) \subseteq [-1, 1]$. Extending results from the $1$-level case, namely the functional analytic arguments of Iwaniec, Luo and Sarnak and the differential equations method introduced by Freeman and Miller, we explicitly solve for the optimal $ϕ$ for appropriately chosen fixed test function $ψ$. The solution allows us to deduce strong estimates for the proportion of newforms of rank $0$ or $2$ in the case of $\mathrm{SO}(\mathrm{even})$, rank $1$ or $3$ in the case of $\mathrm{SO}(\mathrm{odd})$, and rank at most $2$ for $\mathrm{O}$, $\mathrm{Sp}$, and $\mathrm{U}$; our estimates are a significant strengthening of the best known estimates obtained with the $1$-level density. We conclude by discussing further improvements on estimates by the method of iteration. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2011_10140 |
| institution | arXiv |
| publishDate | 2020 |
| record_format | arxiv |
| spellingShingle | Determining optimal test functions for $2$-level densities Bołdyriew, Elżbieta Chen, Fangu VI, Charles Devlin Miller, Steven J. Zhao, Jason Number Theory Katz and Sarnak conjectured a correspondence between the $n$-level density statistics of zeros from families of $L$-functions with eigenvalues from random matrix ensembles. In many cases the sums of smooth test functions, whose Fourier transforms are finitely supported, over scaled zeros in a family converge to an integral of the test function against a density $W_{n, G}$ depending on the symmetry $G$ of the family (unitary, symplectic or orthogonal). This integral bounds the average order of vanishing at the central point of the corresponding family of $L$-functions. We can obtain better estimates on this vanishing by finding better test functions to minimize the integral. We pursue this problem when $n=2$, minimizing \[ \frac{1}{Φ(0, 0)} \int_{{\mathbb R}^2} W_{2,G} (x, y) Φ(x, y) dx dy \] over test functions $Φ\colon {\mathbb R}^2 \to [0, \infty)$ with compactly supported Fourier transform. We study a restricted version of this optimization problem, imposing that our test functions take the form $ϕ(x) ψ(y)$ for some fixed admissible $ψ(y)$ and $\mathrm{supp}({\hat ϕ}) \subseteq [-1, 1]$. Extending results from the $1$-level case, namely the functional analytic arguments of Iwaniec, Luo and Sarnak and the differential equations method introduced by Freeman and Miller, we explicitly solve for the optimal $ϕ$ for appropriately chosen fixed test function $ψ$. The solution allows us to deduce strong estimates for the proportion of newforms of rank $0$ or $2$ in the case of $\mathrm{SO}(\mathrm{even})$, rank $1$ or $3$ in the case of $\mathrm{SO}(\mathrm{odd})$, and rank at most $2$ for $\mathrm{O}$, $\mathrm{Sp}$, and $\mathrm{U}$; our estimates are a significant strengthening of the best known estimates obtained with the $1$-level density. We conclude by discussing further improvements on estimates by the method of iteration. |
| title | Determining optimal test functions for $2$-level densities |
| topic | Number Theory |
| url | https://arxiv.org/abs/2011.10140 |