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

TL;DR

This paper investigates how genus-2 curves over finite fields with dihedral automorphism groups are characterized by their zeta functions and covers, revealing more doubly isogenous pairs than expected due to specific automorphism structures.

Contribution

It demonstrates that genus-2 curves with D4 automorphism groups have a higher-than-expected number of doubly isogenous pairs, explaining this phenomenon through their automorphism properties.

Findings

More doubly isogenous genus-2 curve pairs than naive heuristics predict

Automorphism groups influence the isogeny relations among curves

Provides an explanation for the abundance of doubly isogenous pairs

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 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 naive heuristics predict, and we provide an explanation for this phenomenon.