Number Theory Seminar
Rachel Davis
Univ of Wisconsin-Madison
On the Images of Metabelian Galois Representations Associated to Elliptic Curves
Abstract: For $\ell$-adic Galois representations associated to elliptic curves, there are theorems
concerning when the images are surjective. For example, Serre proved that for a fixed non-CM elliptic
curve $E/{\mathbb Q}$, for all but finitely many primes $\ell$, the $\ell$-adic Galois representation is surjective.
Grothendieck and others have developed a theory of outer Galois representations. These are representations
from the absolute Galois group to an outer automorphism group of a free pro-$\ell$-group. In this case,
there is less known about the size of the images.
The goal of this research is to understand more tangibly Galois representations to automorphism
groups of non-abelian groups. Let $E$ be a semistable elliptic curve over ${\mathbb Q}$ with good supersingular
reduction at $2$. Associated to $E$, there is a Galois representation to a subgroup of the automorphism
group of a metabelian group. I conjecture that there is a Galois representation surjecting to this
subgroup (with the right ramification) and give evidence for this conjecture. Then, I compute some
conjugacy invariants for the images of the Frobenius elements. This will give rise to new arithmetic
information analogous to the traces of Frobenius for the $\ell$-adic representation.
Tuesday October 23, 2012 at 3:00 PM in SEO 512