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

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.