确定2层密度的最优测试函数
Determining optimal test functions for $2$-level densities
AI总结：
本文研究2层密度中使正交、辛或酉对称密度积分最小的测试函数，在乘积形式与Fourier支集限制下显式求出最优ϕ，并强化newform秩比例估计。
摘要：
Katz和Sarnak猜想，L函数族零点的n层密度统计量与随机矩阵系综的特征值之间存在对应关系。在许多情形中，Fourier变换具有有限支集的光滑测试函数在一个族中经过尺度化的零点上的求和，会收敛于该测试函数关于密度W_{n,G}的积分，其中W_{n,G}取决于该族的对称性G（酉、辛或正交）。这一积分界定了相应L函数族在中心点处消失的平均阶。通过寻找更好的测试函数来最小化该积分，我们可以获得关于这种消失的更好估计。我们在n=2时研究这一问题，即在Fourier变换具有紧支集的测试函数Φ: R^2→[0,∞)中最小化[1/Φ(0,0)]∫_{R^2} W_{2,G}(x,y)Φ(x,y)dxdy。我们研究该优化问题的一个受限版本，要求测试函数取ϕ(x)ψ(y)的形式，其中ψ(y)是某个固定的容许测试函数，并且supp(ϕ̂)⊆[-1,1]。通过推广1层情形的结果，即Iwaniec、Luo和Sarnak的泛函分析论证，以及Freeman和Miller引入的微分方程方法，我们针对适当选取的固定测试函数ψ显式求解出最优ϕ。该解使我们能够推出关于newform比例的强估计：在SO(even)情形下秩为0或2，在SO(odd)情形下秩为1或3，而在O、Sp和U情形下秩至多为2；我们的估计显著强化了利用1层密度所得的最佳已知估计。最后，我们讨论如何通过迭代方法进一步改进这些估计。
英文摘要：
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.
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