Computational Algebra Seminar: Costa -- Computing isogeny classes of genus 2 Jacobians over Q

Speaker: Edgar Costa (MIT)
Title: Computing isogeny classes of genus 2 Jacobians over Q
Time & Place: 15.00-16.00, Thursday 27 November, SMRI Seminar Rm
Abstract:
Given a genus 2 curve C over Q, its Jacobian J(C) is a 2-dimensional analogue 
of an elliptic curve. Just as for elliptic curves, one can look at the isogeny 
class of J(C): all abelian surfaces over Q that are linked to J(C) by an isogeny.

In this talk I will describe a practical algorithm which, starting from a genus 
2 curve C/Q whose Jacobian has trivial geometric endomorphism ring, computes all 
genus 2 curves D/Q whose Jacobians are isogenous to J(C) over Q. I will outline the 
main ideas behind the method, which use information from Galois representations 
attached to J(C) together with a mix of analytic and algebraic tools to prove or 
rule out the existence of isogenies, and I will illustrate the algorithm with 
numerical examples.

This is joint work with Raymond van Bommel, Shiva Chidambaram, and Jean Kieffer.