Doubly isogenous genus-2 curves with $D_4$-action

Abstract

We study the zeta functions of curves over finite fields. Suppose C and C are curves over a finite field K, with K-rational base points P and P , and let D and D be the pullbacks (via the Abel-Jacobi map) of the multiplication-by-2 maps on their Jacobians. We say that (C, P) and (C , P ) are doubly isogenous if Jac(C) and Jac(C ) are isogenous over K and Jac(D) and Jac(D ) are isogenous over K. For curves of genus 2 whose automorphism groups contain the dihedral group of order eight, we show that the number of pairs of doubly isogenous curves is larger than naïve heuristics predict, and we provide an explanation for this phenomenon.