UCSD Number Theory Seminar (Math 209)

Wednesdays 4-5pm PST, APM 7321 and online

We will run a hybrid seminar this quarter, with talks simultaneously streamed over Zoom. 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 at 3:00pm and lasting about 30 minutes (in-person only, in the same room as the seminar). In any case there will be tea and cookies in the department common room (adjacent to the seminar room) at 3:30pm.

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).

Fall Quarter 2024

For previous quarters' schedules, click here.


October 2

Organizational meeting

October 16

Jon Aycock (UC San Diego)
Jacobians of Graphs via Edges and Iwasawa Theory

The Jacobian (or sandpile group) is an algebraic invariant of a graph that plays a similar role to the class group in classical number theory. There are multiple recent results controlling the sizes of these groups in Galois towers of graphs that mimic the classical results in Iwasawa theory, though the connection to the values of the Ihara zeta function often requires some adjustment. In this talk we will give a new way to view the Jacobian of a graph that more directly centers the edges of the graph, construct a module over the relevant Iwasawa algebra that nearly corresponds to the interpolated zeta function, and discuss where the discrepancy comes from.

October 23
+pre-talk

Jennifer Johnson-Leung (University of Idaho)
Rationality of certain power series attached to paramodular Siegel modular forms

The Euler product expression of the Dirichlet series of Fourier coefficients of an elliptic modular eigenform follows from a formal identity in the Hecke algebra for GL(2) with full level. In the case of Siegel modular forms of degree two with paramodular level, the situation is more delicate. In this talk, I will present two rationality results. The first concerns the Dirichlet series of radial Fourier coefficients for an eigenform of paramodular level divisible by the square of a prime. This result is an application of the theory of stable Klingen vectors (joint work with Brooks Roberts and Ralf Schmidt). While we are able to calculate the action of certain Hecke operators on eigenforms, the structure of the Hecke algebra of deep level is not known in general. However, in the case of prime level, there is a robust description of the local Hecke algebra which yields a rationality result for a formal power series of Hecke operators (joint work with Joshua Parker and Brooks Roberts). In both cases, we obtain the expected local L-factor as the denominator of the rational function.

October 30
+pre-talk

Chengyang Bao (UCLA)
Computing crystalline deformation rings via the Taylor-Wiles-Kisin patching method

Crystalline deformation rings play an important role in Kisin's proof of the Fontaine-Mazur conjecture for GL2 in most cases. One crucial step in the proof is to prove the Breuil-Mezard conjecture on the Hilbert-Samuel multiplicity of the special fiber of the crystalline deformation ring. In pursuit of formulating a horizontal version of the Breuil-Mezard conjecture, we develop an algorithm to compute arbitrarily close approximations of crystalline deformation rings. Our approach, based on reverse-engineering the Taylor-Wiles-Kisin patching method, aims to provide detailed insights into these rings and their structural properties, at least conjecturally.

November 6
+pre-talk

Brandon Alberts (Eastern Michigan)
Inductive methods for counting number fields

We will discuss an inductive approach to determining the asymptotic number of G-extensions of a number field with bounded discriminant, and outline the proof of Malle's conjecture in numerous new cases. This talk will include discussions of several examples demonstrating the method.

November 13

Chris Xu (UC San Diego)
TBA

November 20

Linli Shi (Connecticut)
On higher regulators of Picard modular surfaces

The Birch and Swinnerton-Dyer conjecture relates the leading coefficient of the L-function of an elliptic curve at its central critical point to global arithmetic invariants of the elliptic curve. Beilinson’s conjectures generalize the BSD conjecture to formulas for values of motivic L-functions at non-critical points. In this talk, I will relate motivic cohomology classes, with non-trivial coefficients, of Picard modular surfaces to a non-critical value of the motivic L-function of certain automorphic representations of the group GU(2,1).

December 4

Yu Fu (Caltech)
TBA