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

For previous quarters' schedule, click here.

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)


March 14

Jake Postema (UC San Diego)