Doubly isogenous genus-2 curves with 𝐷₄-action

Abstract

We study the extent to which curves over finite fields are characterized by their zeta functions and the zeta functions of certain of their covers. Suppose C C and C ′ C’ are curves over a finite field K K , with K K -rational base points P P and P ′ P’ , and let D D and D ′ D’ be the pullbacks (via the Abel–Jacobi map) of the multiplication-by- 2 2 maps on their Jacobians. We say that ( C , P ) (C,P) and ( C ′ , P ′ ) (C’,P’) are doubly isogenous if J a c ( C ) Jac(C) and J a c ( C ′ ) Jac(C’) are isogenous over K K and J a c ( D ) Jac(D) and J a c ( D ′ ) Jac(D’) are isogenous over K K . For curves of genus 2 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.