# 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.

#### Spring Quarter 2016

 March 31 NO SEMINAR April 7 (plus prep talk) Preston Wake (UCLA) Level structures beyond the Drinfeld case Drinfeld level structures are a key concept in the arithmetic study of the moduli of elliptic curves. They also play an important role in the moduli of 1 dimensional p-divisible groups, and related Shimura varieties studied by Harris and Taylor. I'll explain why Drinfeld level structures (and the related "full set of sections" defined by Katz and Mazur) are not adequate for studying more general Shimura varieties. I'll discuss two examples of a satisfying theory of level structure outside the Drinfeld case: i) full level structures on the group \mu_p x \mu_p; ii) Gamma_1(p^r)-type level structures on an arbitrary p-divisible group (joint work with R. Kottwitz). April 14 (plus prep talk) Gilbert Moss (Oklahoma State University) A local converse theorem and the local Langlands correspondence in families In 2012 it was conjectured by Emerton and Helm that the local Langlands correspondence for GL(n) of a p-adic field should interpolate in \ell-adic families, where \ell is a prime different from p. Recently, Helm showed that the conjecture follows from the existence of an appropriate map from the integral Bernstein center to a Galois deformation ring. In this talk we will present recent work (joint with David Helm) showing the existence of such a map and describing its image. April 21 NO SEMINAR April 28 NO SEMINAR May 5 (plus prep talk) Francesc Castella (UCLA) Lambda-adic Gross-Zagier formula for supersingular primes In 2013, Kobayashi proved an analogue of Perrin-Riou's p-adic Gross-Zagier formula for supersingular primes. In this talk, we will explain an extension of Kobayashi's result to the Lambda-adic setting. The main formula is in terms of plus/minus Heegner points up the anticyclotomic tower, and its proof, rather than on calculations inspired by the original work of Gross-Zagier, is via Iwasawa theory, based on the connection between Heegner points, Beilinson-Flach elements, and different p-adic L-functions. (Joint work in progress with Xin Wan.) May 12 NO SEMINAR May 19 (plus prep talk) Samuele Anni (University of Warwick) Abelian varieties and the inverse Galois problem The inverse Galois problem is one of the greatest open problems in group theory and also one of the easiest to state: is every finite group a Galois group? Hilbert was the first to study it in earnest: Hilbert's irreducibility theorem established a connection between Galois groups over Q and Galois groups over Q[x], and this led him to show that symmetric and alternating groups are Galois realizable over Q. My interest around this problem is connected to the realization of linear groups as Galois groups over Q and over number fields. In this talk I will describe uniform realizations of linear groups using elliptic curves and genus 2 curves. After this introduction, I will describe a joint work with Pedro Lemos and Samir Siksek, concerning the realization of GSp6 (Fℓ) as a Galois group for infinitely many odd primes l. I will also present some generalizations which are work in progress with Vladimir Dokchitser. May 26 (plus prep talk) Zev Klagsbrun (Center for Communications Research) The Joint Distribution Of $Sel_\phi(E/\QQ)$ and $Sel_{\hat\phi}(E^\prime/\QQ)$ in Quadratic Twist Families We show that the $\phi$-Selmer ranks of twists of an elliptic curve $E$ with a point of order two are distributed like the ranks of random groups in a manner consistent with the philosophy underlying the Cohen-Lenstra heuristics. If $E$ has a point of order two, then the distribution of $dim_{\mathbb{F}_2} \Sel_\phi(E^d/\QQ) - dim_{\mathbb{F}_2} Sel_{\hat\phi}(E^{\prime d}/\QQ)$ tends to the discrete normal distribution $\mathcal{N}(0,\frac{1}{2} \log \log X)$ as $X \rightarrow \infty$. We consider the distribution of $dim_{\mathbb{F}_2} \Sel_\phi(E^d/\QQ) - dim_{\mathbb{F}_2} Sel_{\hat\phi}(E^{\prime d}/\QQ)$ has a fixed value $u$. We show that for every $r$, the limiting probability that $dim_{\mathbb{F}_2} \Sel_\phi(E^d/\QQ)= r$ is given by an explicit constant $\alpha_{r,u}$ introduced in Cohen and Lenstra's original work on the distribution of class groups. June 2 (plus prep talk) Robert Lemke Oliver (Stanford) The distribution of consecutive primes While the sequence of primes is very well distributed in the reduced residue classes (mod q), the distribution of pairs of consecutive primes among the permissible pairs of reduced residue classes (mod q) is surprisingly erratic. We propose a conjectural explanation for this phenomenon, based on the Hardy-Littlewood conjectures, which fits the observed data very well. We also study the distribution of the terms predicted by the conjecture, which proves to be surprisingly subtle. This is joint work with Kannan Soundararajan.