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 na¨ıve heuristics predict, and we
provide an explanation for this phenomenon
