# 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 1:20pm.
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 2019

 January 10 Organizational Meeting (APM 7421, 2-3pm) January 17 No meeting January 24 Benedict Gross (UC San Diego) -- no pre-talk The Conjecture of Birch and Swinnerton-Dyer This is an introduction to the Birch and Swinnerton-Dyer Conjecture on L-functions of elliptic curves. The talk is aimed at graduate and undergraduate students who are strongly encouraged to attend. January 31 No meeting February 7 Roman Kitsela (UC San Diego) + pre-talk A Tannaka-Krein reconstruction result for profinite groups The classical Tannaka reconstruction theorem allows one to recover a compact group $G$ (up to isomorphism) from the monoidal category of finite dimensional representations of $G$ over $\mathbb{C}$, $\text{Rep}_{\mathbb{C}}(G)$, as the tensor preserving automorphisms of the forgetful functor $\text{Rep}_{\mathbb{C}}(G) \longrightarrow \text{Vec}_{\mathbb{C}}$. Now let $G$ be a profinite group, $K$ a finite extension of $\mathbb{Q}_p$ and $\text{Ban}_G(K)$ the category of $K$-Banach space representations (of $G$). $\text{Ban}_G(K)$ can be equipped with a (completed) tensor product $(-)\hat\otimes_K(-)$ and has a forgetful functor $\omega : \text{Ban}_G(K) \longrightarrow \text{Ban}(K)$. Using an anti-equivalence of categories between $\text{Ban}_G(K)$ and the category of Iwasawa $G$-modules due to Schneider and Teitelbaum, we prove that a profinite group $G$ can be recovered from $\text{Ban}_G(K)$, in particular $G \cong \text{Aut}^\otimes(\omega)$. February 14 Ananth Shankar (MIT) + pre-talk (starting 1:00pm) Exceptional splitting of abelian surfaces over global function fields Let $A$ denote a non-constant ordinary abelian surface over a global function field (of characteristic $p > 2$) with good reduction everywhere. Suppose that $A$ does not have real multiplication by any real quadratic field with discriminant a multiple of $p$. Then we prove that there are infinitely many places modulo which $A$ is isogenous to the product of two elliptic curves. This is joint work with Davesh Maulik and Yunqing Tang. Note: There will be a preparatory lecture for graduate students and post-docs in the seminar room starting at 1:00pm. February 21 Zavosh Amir Khosravi (Caltech) + pre-talk Special cycles on non-compact Picard modular varieties We'll discuss an extension of the work of Kudla-Millson on the modularity of special cycles on a non-compact Shimura variety associated to U(n,1) over a split CM field. The volume of their intersections with a diagonally embedded Shimura subvariety is related to Fourier coefficients of a Hilbert modular form coming from the restriction of an Eisenstein series on U(n,n). The main new idea is an application of the regularized Siegel-Weil formula of Gan-Qiu-Takeda. February 28 Isabel Vogt (MIT) + pre-talk Low degree points on curves In this talk we will discuss an arithmetic analogue of the gonality of a curve over a number field: the smallest positive integer e such that the points of residue degree bounded by e are infinite. By work of Faltings, Harris--Silverman and Abramovich--Harris, it is well-understood when this invariant is 1, 2, or 3; by work of Debarre--Fahlaoui these criteria do not generalize to e at least 4. We will study this invariant using the auxiliary geometry of a surface containing the curve and devote particular attention to scenarios under which we can guarantee that this invariant is actually equal to the gonality . This is joint work with Geoffrey Smith. March 7 Ila Varma (UC San Diego) + pre-talk Malle's Conjecture for octic $D_4$-fields We consider the family of normal octic fields with Galois group $D_4$, ordered by their discriminant. In forthcoming joint work with Arul Shankar, we verify the strong Malle conjecture for this family of number fields, obtaining the order of growth as well as the constant of proportionality. In this talk, we will discuss and review the combination of techniques from analytic number theory and geometry-of-numbers methods used to prove these results. March 14 Jake Postema (UC San Diego) + pre-talk Higher Smooth Duals for Mod p Representations of Algebraic Groups The Local Langlands program and its variants have lead to the study of smooth, admissible representations of p-adic algebraic groups. The degree to which these are understood depends on the field over which the representations are being taken. Over a field of characteristic p, the usual dual in the category of smooth representations gives less information: in most cases of interest, it is 0! Kohlhaase has defined candidates-the higher duality functors-for a useful replacement. We will go over their properties, and some examples in rank one where they can be computed.