Currently available as a preprint.
We study the zeta functions of curves over finite fields. Suppose C1 and C2 are curves over a finite field K, with K-rational base points P1 and P2, and let D1 and D2 be the pullbacks (via the Abel–Jacobi map) of the multiplication-by-2 maps on their Jacobians. We say that (C1, P1) and (C1, P2) are doubly isogenous if Jac(C1) and Jac(C2) are isogenous over K and Jac(D1) and Jac(D2) 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.
We used Magma to gather data and to test our heuristics. The code we used, and the data it produced for doubly-isogenous curves, is available here: