Determining optimal test functions for $2$-level densities

Abstract

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 $\Phi :{\mathbb{R}}^2\to [0,\infty )$ with compactly supported Fourier transform. We study a restricted version of this optimization problem, imposing that our test functions take the form $\varphi (x)\psi (y)$ for some fixed admissible $\psi (y)$ and $\mathrm{supp}(\hat{\varphi })\subseteq [-1,1]$. 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 $\varphi$ for appropriately chosen fixed test function $\psi$. 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.

arXiv comment18 pages

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Cited by in corpus (1)

• On the moments of one-level densities in families of holomorphic cusp forms in the level aspect