UCSD Number Theory Seminar (Math 209)

Thursday 2-3pm, AP&M 7321 (this is the usual room but its number has changed)

This quarter, most talks will be preceded by a 30-minute "prep talk" for graduate students and postdocs. These will be in AP&M 7321 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.

Fall Quarter 2019

For previous quarters' schedule, click here.

September 26	NO MEETING
October 3	ORGANIZATIONAL MEETING
October 10	Harry Smit (Utrecht) + pre-talk 1:15-1:45 L-functions and isogenies of abelian varieties Faltings's isogeny theorem states that two abelian varieties over a number field are isogenous precisely when the characteristic polynomials associated to the reductions of the abelian varieties at all prime ideals are equal. This implies that two abelian varieties defined over the rational numbers with the same L-function are necessarily isogenous, but this is false over a general number field. In order to still use the L-function to determine the underlying field, we extract more information from the L-function by twisting: a twist of an L-function is the L-function of the tensor of the underlying representation with a character. We discuss a theorem stating that abelian varieties over a general number field are characterized by their L-functions twisted by Dirichlet characters of the underlying number field.
October 17	NO MEETING
October 24	NO MEETING
October 31	Peter Wear (UCSD) + pre-talk 1:15-1:45 Perfectoid covers of abelian varieties and the weight-monodromy conjecture The theory of perfectoid spaces was initially developed by Scholze to prove new cases of the weight-monodromy conjecture. He constructed perfectoid covers of toric varieties that allowed him to translate results from characteristic p to characteristic 0. We will give an overview of Scholze's method, then explain how to use an analogous construction for abelian varieties to prove the weight-monodromy conjecture for complete intersections in abelian varieties.
November 7, 1pm-3pm	Cristian Popescu/Joe Ferrara (UCSD) no pre-talk An equivariant Tamagawa number formula for Drinfeld modules In this series of lectures we will describe the construction of a $G$-equivariant $L$-function $Theta^E_{K/F}(s)$ associated to an abelian extension $K/F$ of characteristic $p$ global fields of Galois group $G$ and a suitable Drinfeld module $E$ defined over $F$, as well as state and sketch the proof of a theorem linking the special value $Theta^E_{K/F}(0)$ to a quotient of volumes of certain compact topological spaces canonically associated to the pair $(K/F, E)$. In lecture I (1-2pm), Cristian will define the $L$--function, give an arithmetic interpretation of its special value at $s=0$ and state the main theorem. In lecture II (2-3pm), Joe will introduce the main ingredients involved in the proof of the main theorem and sketch the main ideas of proof. These lectures describe joint work of J. Ferrara, N. Green, Z. Higgins and C. Popescu. The results within generalize to the Galois equivariant setting earlier work of L. Taelman on special values of Goss zeta functions associated to Drinfeld modules (Taelman, Annals of Math. 2010).
November 14	NO MEETING
November 21	Vlad Matei (UC Irvine) + pre-talk 1:15-1:45 Average size of the automorphism group of smooth projective hypersurfaces over finite fields We show that the average size of the automorphism group over $\mathbb{F}_q$ of a smooth degree $d$ hypersurface in $\mathbb{P}^{n}_{\mathbb{F}_q}$ is equal to $1$ as $d\rightarrow \infty$. We also discuss some consequences of this result for the moduli space of smooth degree $d$ hypersurfaces in $\mathbb{P}^n$.
November 21: special colloquium (3pm, APM 6402)	Daniel Le (Toronto) Congruences between modular forms Modular forms are holomorphic functions invariant under a certain group action. They have a surprisingamount of number theoretic information. We introduce their basic theory and explain how their connection to Galois theory can be used to study congruences between modular forms.
November 28	NO MEETING (Thanksgiving)
December 5	Joe Kramer-Miller (UC Irvine) + pre-talk 1:15-1:45 p-adic estimates for Artin L-functions on curves Let $C$ be a curve over a finite field and let $\rho$ be a nontrivial representation of $\pi_1(C)$. By the Weil conjectures, the Artin $L$-function associated to $\rho$ is a polynomial with algebraic coefficients. Furthermore, the roots of this polynomial are $\ell$-adic units for $\ell \neq p$ and have Archemedian absolute value $\sqrt{q}$. Much less is known about the $p$-adic properties of these roots, except in the case where the image of $\rho$ has order $p$. We prove a lower bound on the $p$-adic Newton polygon of the Artin $L$-function for any representation in terms of local monodromy decompositions. If time permits, we will discuss how this result suggests the existence of a category of wild Hodge modules on Riemann surfaces, whose cohomology is naturally endowed with an irregular Hodge filtration.