Papers › Computing isogeny classes of typical principally polarized abelian surfaces over the rationals
Computing isogeny classes of typical principally polarized abelian surfaces over the rationals
Raymond van Bommel, Shiva Chidambaram, Edgar Costa, Jean Kieffer
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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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