Elliptic curves with maximal Galois action on torsion points

Wed, 13/05/2015 - 10:30
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Seminar: 
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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.