Traces of High Powers of the Frobenius Class in the Moduli Space of Hyperelliptic Curves

TL;DR

This paper investigates the behavior of high powers of Frobenius matrices associated with hyperelliptic curves over finite fields, revealing average trace values and biases as the genus grows large.

Contribution

It computes the expected traces of high powers of Frobenius matrices over the hyperelliptic moduli space, extending understanding of their statistical properties in large genus limits.

Findings

Expected trace values of Frobenius powers computed

Expected number of points on curves analyzed

Biases in trace distributions for small n identified

Abstract

The Zeta function of a curve $C$ over a finite field may be expressed in terms of the characteristic polynomial of a unitary matrix $\Theta_C$. Following the work of Rudnick, we compute the expected value of $\mbox{tr}(\Theta_C^n)$ over the moduli space of hyperelliptic curves of genus $g$, over a fixed finite field $\mathbb{F}_q$, in the limit of large genus. As an application, we compute the expected value of the number of points on $C$ in $\mathbb{F}_{q^n}$ as the genus tends to infinity. We also look at biases in both expected values for small values of $n$.