# UCSD Number Theory Seminar (Math 209)

### Thursday 2-3pm, AP&M 7421

This quarter, most talks will be preceded by a 30-minute "prep talk" for graduate students and postdocs. These will be in AP&M 7421 starting at 1:15pm.
Don't forget to register for Math 209 if you are a graduate student. We are eligible for department funding as long as we maintain sufficient enrollment.

#### Winter Quarter 2017

 January 12 (plus prep talk) Ozlem Ejder (USC) Torsion subgroups of elliptic curves in elementary abelian 2-extensions Let $E$ be an elliptic curve defined over $\Q$. The torsion subgroup of $E$ over the compositum of all quadratic extensions of $\Q$ was studied by Michael Laska, Martin Lorenz, and Yasutsugu Fujita. Laska and Lorenz described a list of $31$ possible groups and Fujita proved that the list of $20$ different groups is complete. In this talk, we will generalize the results of Laska, Lorenz and Fujita to the elliptic curves defined over a quadratic cyclotomic field i.e. $\Q(i)$ and $\Q(\sqrt{-3})$. January 19 (plus prep talk) Brian Hwang (Cornell) An application of (harmonic (families of)) automorphic forms to Galois theory A number of questions in Galois theory can be phrased in the following way: how large (in various senses) can the Galois group G of an extension of the rational numbers be, if the extension is only allowed to ramify at a small set of primes? If we assume that G is abelian, class field theory provides a complete answer, but the question is open is almost every nonabelian case, since there is no known way to systematically and explicitly construct such extensions in full generality. However, there have been some programs that are gaining ground on this front. While the problem above is natural and the objects are classical, we will see that to answer certain questions about of this Galois group, it seems necessary to use techniques involving automorphic forms and their representation-theoretic avatars. In particular, it will turn out that some recent results on "harmonic" families of automorphic forms translate to the fact that such number fields, despite not being explicitly constructible by known methods, turn out to "exist in abundance" and allow us to find bounds on the sizes of such Galois groups. January 26 Peter Stevenhagen (Leiden) TBA February 2 Michiel Kosters (UC Irvine) TBA February 9 (plus prep talk) Jennifer Balakrishnan (Boston University) TBA February 16 James Maynard (Oxford) TBA February 23 Serin Hong (Caltech) TBA March 2 Nathan Kaplan (UC Irvine) TBA March 9 TBA TBA March 16 TBA TBA