UCSD Number Theory Seminar (Math 209)

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
+pre-talk

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
+pre-talk

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
+pre-talk

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)
Greenberg’s $\lambda=0$ conjecture

Recent and not so recent computations by Mercuri and Paoluzi have verified Greenberg’s $\lambda=0$ conjecture in Iwasawa theory in many cases. We discuss the conjecture and the computations.