Expected Value of High Powers of Trace of Frobenius of Biquadratic Curves Over a Finite Field

TL;DR

This paper calculates the average trace of high powers of Frobenius classes for biquadratic curves over finite fields, extending previous work on hyperelliptic and cyclic curves to a broader class.

Contribution

It provides the first explicit determination of the expected trace of Frobenius powers for biquadratic curves, generalizing earlier results on hyperelliptic and cyclic curves.

Findings

Expected value of Tr(Θ_C^n) for biquadratic curves computed

Extends previous results from hyperelliptic and cyclic curves

Results hold as genus g tends to infinity with fixed q

Abstract

Denote $\Theta_C$ as the Frobenius class of a curve $C$ over the finite field $\mathbb{F}_q$. In this paper we determine the expected value of Tr$(\Theta_C^n)$ where $C$ runs over all biquadratic curves when $q$ is fixed and $g$ tends to infinity. This extends work done by Rudnick and Chinis who separately looked at hyperelliptic curves and Bucur, Costa, David, Guerreiro and Lowry-Duda who looked at $\ell$-cyclic curves, for $\ell$ a prime, as well as cubic non-Galois curves.