Expected value of high powers of trace of frobenius of biquadratic curves over a finite field

Abstract

AbstractDenote ΘCas the Frobenius class of a curveCover the finite field 𝔽q. In this paper we determine the expected value of Tr(ΘCn) whereCruns over all biquadratic curves whenqis fixed andgtends to infinity. This extends work done by Rudnick [15] and Chinis [5] who separately looked at hyperelliptic curves and Bucur, Costa, David, Guerreiro and Lowry-Duda [1] who looked at ℓ-cyclic curves, for ℓ a prime, as well as cubic non-Galois curves.