2021/02/22 by Vishal Arul, Jeremy Booher, Arul, Vishal +13
Computer Science · Mathematics · #11G20 #11M38 #11Y40 #14H25 #14H30 #14H40 #14K02 #14Q05 (Primary) 11G10 #14Q25 (Secondary) #Algebraic Geometry (math.AG) #Algebraic Geometry and Number Theory #Coding theory and cryptography #FOS: Mathematics #Finite Group Theory Research #Number Theory (math.NT)
paper · pdf · doi:10.48550/arxiv.2102.11419
openalex publication_date 2021/02/22 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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.