Tenth Algorithmic Number Theory Symposium ANTS-X University of California, San Diego July 9 – 13, 2012 Tenth Algorithmic Number Theory Symposium (ANTS-X) July 9 – 13, 2012

Elliptic factors in Jacobians of hyperelliptic curves with certain automorphism groups

Jennifer Paulhus

Abstract: We decompose the Jacobian variety of hyperelliptic curves up to genus 20, defined over an algebraically closed field of characteristic zero, with reduced automorphism group A_4, S_4, or A_5. Among these curves is a genus 4 curve with Jacobian variety isogenous to E_1^2 x E_2^2 and a genus 5 curve with Jacobian variety isogenous to E^5, for E and E_i elliptic curves. These types of results have some interesting consequences to questions of ranks of elliptic curves and ranks of their twists.

Files available: paper (PDF), slides, errata (July 2015)