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