January 10 |
Organizational Meeting (APM 7421, 2-3pm) |
January 17 |
No meeting |
January 24 |
Benedict Gross (UC San Diego) -- no pre-talk 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 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) 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 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 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) TBA |
March 14 |
Jake Postema (UC San Diego) TBA |