# Elliptic curves with maximal Galois action on torsion points

Seminar

Speaker

David Corwin (Massachusetts Institute of Technology)

Date

13/05/2015 - 11:30 - 10:30Add to Calendar

`2015-05-13 10:30:00``2015-05-13 11:30:00``Elliptic curves with maximal Galois action on torsion points``Let E be an elliptic curve over a number field K with algebraic closure K'. For an integer n, the set of n-torsion points in E(K') forms a group isomorphic to Z/n X Z/n, which carries an action of G_K=Gal(K'/K). A result of Serre shows that if K=Q, then the associated homomorphism from G_K to GL_2(Z/n) cannot be surjective for all n. The result is false, however, over other number fields. The 2010 PhD thesis of Greicius found the first counterexample, over a very special non-Galois cubic extension of Q. In this talk I will describe the above background and then describe more recent results of the speaker and others allowing one to find such elliptic curves over a more general class of number fields. As time allows, I may describe other results from our paper about finding elliptic curves with maximal (but not surjective) Galois action given certain constraints, or a forthcoming paper doing similar work for abelian surfaces.``Third floor seminar room``אוניברסיטת בר-אילן - Department of Mathematics``mathoffice@math.biu.ac.il``Asia/Jerusalem``public`Place

Third floor seminar room

Abstract

Let E be an elliptic curve over a number field K with algebraic closure K'. For an integer n, the set of n-torsion points in E(K') forms a group isomorphic to Z/n X Z/n, which carries an action of G_K=Gal(K'/K).

A result of Serre shows that if K=Q, then the associated homomorphism from G_K to GL_2(Z/n) cannot be surjective for all n. The result is false, however, over other number fields. The 2010 PhD thesis of Greicius found the first counterexample, over a very special non-Galois cubic extension of Q.

In this talk I will describe the above background and then describe more recent results of the speaker and others allowing one to find such elliptic curves over a more general class of number fields. As time allows, I may describe other results from our paper about finding elliptic curves with maximal (but not surjective) Galois action given certain constraints, or a forthcoming paper doing similar work for abelian surfaces.

Last Updated Date : 26/04/2015