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

TL;DR

This paper investigates the statistical distribution of zeros of zeta functions for hyperelliptic curves over finite fields, revealing Gaussian fluctuations and a logarithmic variance growth as the genus increases.

Contribution

It provides a detailed analysis of zero distribution fluctuations, establishing a central limit theorem and variance behavior for large genus hyperelliptic curves.

Findings

Variance of zero counts grows logarithmically with genus

Normalized fluctuations follow a Gaussian distribution

Results hold for shrinking intervals with increasing expected zeros

Abstract

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 for shrinking intervals as long as the expected number of angles 2g|I| tends to infinity.