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# Jeremy Rouse : Elliptic curves over $\mathbb{Q}$ and 2-adic images of Galois

Given an elliptic curve $E/\mathbb{Q}$, let $E[2^k]$ denote the set of points on $E$ that have order dividing $2^k$. The coordinates of these points are algebraic numbers and using them, one can build a Galois representation $\rho : G_{\mathbb{Q}} \to \GL_{2}(\mathbb{Z}_{2})$. We give a classification of all possible images of this Galois representation. To this end, we compute the 'arithmetically maximal' tower of 2-power level modular curves, develop techniques to compute their equations, and classify the rational points on these curves.

**Category**: Number Theory**Duration**: 01:34:46**Date**: March 2, 2016 at 1:25 PM**Views**: 107-
**Tags:**seminar, Number Theory Seminar

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