Algebra Seminar, Jeremy Rouse (Wake Forest): 'Elliptic curves over $\mathbb{Q}$ and 2-adic images of Galois'

Friday, April 3, 2015
1:10 PM - 2:00 PM (ET)
ESC 638
Event Type
Lecture
Contact
Christopher Rasmussen,x2315
Department
Academic
Link
https://eaglet.wesleyan.edu/MasterCalendar/EventDetails.aspx?EventDetailId=48380

Abstract: 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 numbersand 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.

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