Computing isogeny classes of typical principally polarized abelian surfaces over the rationals

2023/01/24 by van Bommel, Raymond, Chidambaram, Shiva, Costa, Edgar +1
#Algebraic Geometry (math.AG) #FOS: Mathematics #Number Theory (math.NT)

paper · doi:10.48550/arxiv.2301.10118

Abstract

We describe an efficient algorithm which, given a principally polarized (p.p.) abelian surface A over ℚ with geometric endomorphism ring equal to ℤ, computes all the other p.p. abelian surfaces over ℚ that are isogenous to A. This algorithm relies on explicit open image techniques for Galois representations, and we employ a combination of analytic and algebraic methods to efficiently prove or disprove the existence of isogenies. We illustrate the practicality of our algorithm by applying it to 1 440 894 isogeny classes of Jacobians of genus 2 curves.

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