Zeta statistics and Hadamard functions

Statistics of the zeros of zeta functions in families of hyperelliptic curves over a finite field

We study the fluctuations in the distribution of zeros of zeta functions of a family of hyperelliptic curves defined over a fixed finite field, in the limit of large genus. According to the Riemann Hypothesis for curves, the zeros all lie on a circle. Their angles are uniformly distributed, so for a curve of genus g a fixed interval I will contain 2g|I| angles as the genus grows. We show that for the variance of number of angles in I is asymptotically a constant multiple of log(2g|I|) and prove a central limit theorem: The normalized fluctuations are Gaussian. These results continue to hold fo…

Non-Abelian Zeta Function, Fokker-Planck Equation and Projectively Flat Connection

Over the moduli space of rank $n$ semi-stable lattices is a universal family of tori. Along the fibers, there are natural differential operators and differential equations, particularly, the heat equations and the Fokker-Planck equations in statistical mechanics. In this paper, we explain why, by taking averages over the moduli spaces, all these are connected with the zeros of rank $n$ non-abelian zeta functions of the field of rationals, which are known lie on the central line except a finitely many if $n\geq 2$. Certainly, when $n=1$, our current work recovers that of Armitage, which from th…

Motivic Euler products in motivic statistics

We formulate and prove an analog of Poonen's finite-field Bertini theorem with Taylor conditions that holds in the Grothendieck ring of varieties. This gives a broad generalization of the work of Vakil-Wood, who treated the case of smooth hypersurface sections. In fact, our techniques give analogs in motivic statistics of all known results in arithmetic statistics that have been proven using Poonen's sieve, including work of Bucur-Kedlaya on complete intersections and Erman-Wood on semi-ample Bertini theorems. A key ingredient is the use of motivic Euler products, as introduced by the first au…

Hurwitz 空间、Nichols 代数与 Igusa zeta 函数

Hurwitz spaces, Nichols algebras, and Igusa zeta functions

本文构造次数 4、5 的 Hurwitz 空间拟映射紧化，建立算术统计、Nichols 代数与 Igusa zeta 函数的精确联系，并回答相关计数与拓扑起源问题。

有限域$\u211d_2$上的三次四折形普查

A census of cubic fourfolds over $\mathbb{F}_2$

本研究完成了有限域$\u211d_2$上三次四折形的完整同构类计算，统计了其各类不变量，改进了轨道枚举方法以提升计算效率，并关联了K3曲面zeta函数实现问题与判别式补渐近框架。