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arXiv · 2102.11419

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

Vishal Arul·Jeremy Booher·Steven R. Groen·Everett W. Howe·Wanlin Li·Vlad Matei·Rachel Pries·Caleb Springer
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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.

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Vishal Arul, Jeremy Booher, Steven R. Groen, Everett W. Howe, Wanlin Li, Vlad Matei, Rachel Pries, Caleb Springer. 2021-02-22. Doubly isogenous genus-2 curves with $D_4$-action. https://arxiv.org/abs/2102.11419

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