UCSD Number Theory Seminar (Math 209)

Thursdays 2-3pm PST, APM 7321

We will run an in-person seminar this quarter; the seminar room we previously used for hybrid seminars is not available. In a few cases when the speaker is presenting slides, we may be able to make the talk hybrid over Zoom; these will be notated in the schedule. When available, the Zoom meeting code is 993 5967 5186; the password is the four-digit number of the meeting room.

Most talks will be preceded by a "pre-talk" meant for graduate students and postdocs, starting 40 minutes before the announced time for the main talk and lasting about 30 minutes.

Don't forget to register for Math 209 if you are a UCSD graduate student. Continued department financial support for this seminar is contingent on maintaining sufficient enrollment.

The organizers strive to ensure that all participants in this seminar enjoy a welcoming environment, conducive to the free expression and exchange of ideas. In particular, the pre-talks are meant to provide a safe space for junior researchers to ask questions of the speaker. All participants are expected to cooperate with this effort and encouraged to contact the organizers with any concerns.

To subscribe to our weekly seminar announcement, or to join the number theory group's Zulip discussion server for additional announcements, please contact the organizers. (Thanks to Zulip for providing Sponsored Cloud Hosting for this server.)

As of spring 2020, this site is dynamically generated from researchseminars.org, which see for other seminars worldwide (or for this seminar listed in your local timezone).

Winter Quarter 2024

For previous quarters' schedules, click here.

In addition to the listed talks, we have also scheduled a Southern California Number Theory Day on Saturday, February 24.

January 12, 2:00PM

Jessica Fintzen (Bonn)
Representations of p-adic groups and Hecke algebras

An explicit understanding of the category of all (smooth, complex) representations of p-adic groups provides an important tool in the construction of an explicit and a categorical local Langlands correspondence and also has applications to the study of automorphic forms. The category of representations of p-adic groups decomposes into subcategories, called Bernstein blocks, which are indexed by equivalence classes of so called supercuspidal representations of Levi subgroups. In this talk, I will give an overview of what we know about an explicit construction of supercuspidal representations and about the structure of the Bernstein blocks. In particular, I will discuss a joint project in progress with Jeffrey Adler, Manish Mishra and Kazuma Ohara in which we show that general Bernstein blocks are equivalent to much better understood depth-zero Bernstein blocks. This is achieved via an isomorphism of Hecke algebras and allows to reduce a lot of problems about the (category of) representations of p-adic groups to problems about representations of finite groups of Lie type, where answers are often already known or easier to achieve.

January 18

Eugenia Rosu (Leiden)
A higher degree Weierstrass function

I will discuss recent developments and ongoing work for algebraic and p-adic aspects of L-functions. Interest in p-adic properties of values of L-functions originated with Kummer’s study of congruences between values of the Riemann zeta function at negative odd integers, as part of his attempt to understand class numbers of cyclotomic extensions. After presenting an approach to studying analogous congruences for more general classes of L-functions, I will conclude by introducing ongoing joint work of G. Rosso, S. Shah, and myself (concerning Spin L-functions for GSp_6). I will explain how this work fits into the context of earlier developments, while also indicating where new technical challenges arise. All who are curious about this topic are welcome at this talk, even without prior experience with p-adic L-functions or Spin L-functions.

January 25

John Yin (Wisconsin)
A Chebotarev Density Theorem over Local Fields

I will present an analog of the Chebotarev Density Theorem which works over local fields. As an application, I will use it to prove a conjecture of Bhargava, Cremona, Fisher, and Gajović. This is joint work with Asvin G and Yifan Wei.

February 8, APM 7321 and online

Mckenzie West (Wisconsin-Eau Claire)
A Robust Implementation of an Algorithm to Solve the $S$-Unit Equation

The $S$-unit equation has vast applications in number theory. We will discuss an implementation of an algorithm to solve the $S$-unit equation in the mathematical software Sage. The mathematical foundation for this implementation and some applications will be outlined, including an asymptotic version of Fermat's Last Theorem for totally real cubic number fields with bounded discriminant in which 2 is totally ramified. We will conclude with a discussion on current and future work toward improving the existing Sage functionality.

February 15

Shahed Sharif (CSU San Marcos)
Number theory and quantum computing: Algorithmists, Assemble!

Quantum computing made its name by solving a problem in number theory; namely, determining if factoring could be accomplished efficiently. Since then, there has been immense progress in development of quantum algorithms related to number theory. I'll give a perhaps idiosyncratic overview of the computational tools quantum computers bring to the table, with the goal of inspiring the audience to find new problems that quantum computers can solve.

February 29

Aranya Lahiri (UC San Diego) (paper)
Dagger groups and $p$-adic distribution algebras

Locally analytic representations were introduced by Peter Schneider and Jeremy Teitelbaum as a tool to understand $p$-adic Langlands program. From the very beginning the theory of $p$-valued groups played an instrumental role in the study of locally analytic representations. In a previous work we attached a rigid analytic group to a $\textit{$p$-saturated group}$ (a class of $p$-valued groups that contains uniform pro-$p$ groups and pro-$p$ Iwahori subgroups as examples). In this talk I will outline how to elevate the rigid group to a $\textit{dagger group}$, a group object in the category of dagger spaces as introduced by Elmar Grosse-Klönne. I will further introduce the space of $\textit{overconvergent functions}$ and its strong dual the $\textit{overconvergent distribution algebra}$ on such a group. Finally I will show that in analogy to the locally analytic distribution algebra of compact $p$-adic groups, in the case of uniform pro-$p$ groups the overconvergent distribution algebra is a Fr´echet-Stein algebra, i.e., it is equipped with a desirable algebraic structure. This is joint work with Claus Sorensen and Matthias Strauch.

March 7

René Schoof (Rome 3)