UCSD Number Theory Seminar (Math 209)

Thursday 2-3pm, AP&M 7321

This quarter, most talks will not be preceded by a 30-minute "prep talk" for graduate students and postdocs, due to a schedule conflict with Gross's course.
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.

Winter Quarter 2020

For previous quarters' schedule, click here.


January 9

ORGANIZATIONAL MEETING

January 16

NO MEETING

January 23

Adam Logan (Carleton University/TIMC)
Automorphism groups of K3 surfaces over nonclosed fields

Using the Torelli theorem for K3 surfaces of Pyatetskii-Shapiro and Shafarevich one can describe the automorphism group of a K3 surface over {\mathbb C} up to finite error as the quotient of the orthogonal group of its Picard lattice by the subgroup generated by reflections in classes of square -2. We will give a similar description valid over an arbitrary field in which the reflection group is replaced by a certain subgroup. We will then illustrate this description by giving several examples of interesting behaviour of the automorphism group, and by showing that the automorphism groups of two families of K3 surfaces that arise from Diophantine problems are finite. This is joint work with Martin Bright and Ronald van Luijk (University of Leiden).

January 23: colloquium (4pm, APM 6402)

Aaron Pollack (Duke)
Modular forms for exceptional groups

By a "modular form" for a reductive group G we mean an automorphic form that has some sort of very nice Fourier expansion. The classic example are the holomorphic Siegel modular forms, which are special automorphic functions for the group Sp_{2g}. Following work of Gan, Gross, Savin, and Wallach, it turns out that there is a notion of modular forms on certain real forms of the exceptional groups. I will define these objects and explain what is known about them.

January 30

Robert Hough (Stony Brook University)
The shape of low degree number fields

In his thesis, M. Bhargava proved parameterizations and identified local conditions which he used to give asymptotic counts for $S_4$ quartic and quintic number fields, ordered by discriminant. This talk will discuss results in an ongoing project to add detail to Bhargava's work by considering in addition to the field discriminant, the lattice shape of the ring of integers in the canonical embedding, and by giving strong rates with lower order terms in the asymptotics. These results build on earlier work of Taniguchi-Thorne, Bhargava-Shankar-Tsimerman and Bhargava-Harron.

February 6

James Upton (UC Irvine)
A trace formula for deformations of p-adic representations

Let X be a smooth affine variety over a finite field of characteristic p. The Dwork-Monsky trace formula is a fundamental tool in understanding the L-functions of p-adic representations of 1(X). We extend this result to the study of representations valued in a higher-dimensional local ring R. The special case R=Zp[[T]] arises naturally in the study of etale Zp-towers over X. Time permitting, we discuss some spectral-halo type results and conjectures describing the p-adic variation of slopes in certain Zp-towers.

February 13

Everett Howe
Unlikely intersections and curves with isogenous covers

Sutherland and Voloch have investigated the idea of distinguishing two curves over a finite field by comparing the zeta functions of the curves and of certain covers of the curves. To test the limits of this method, it is instructive to find examples of nonisomorphic curves that have covers whose Jacobians are unexpectedly isogenous. I will describe a construction that, heuristically, should produce infinitely many examples of pairs of genus-2 curves over finite fields such that the Jacobians of the curves are isogenous and the Jacobians of the (genus-17) maximal unramified abelian 2-covers of the curves are also isogenous. Implementing this construction led to the discovery of a (very!) unlikely intersection in characteristic zero, whose existence proves that indeed there exist infinitely many examples of genus-2 curves as above.

February 20

Edgar Costa (MIT)
Frobenius distributions

In this talk, we will focus on how one can deduce some geometric invariants of an abelian variety or a K3 surface by studying their Frobenius polynomials. In the case of an abelian variety, we show how to obtain the decomposition of the endomorphism algebra, the corresponding dimensions, and centers. Similarly, by studying the variation of the geometric Picard rank, we obtain a sufficient criterion for the existence of infinitely many rational curves on a K3 surface of even geometric Picard rank.

February 20: colloquium (4pm, APM 6402)

Chao Li (Columbia)
From sum of two squares to arithmetic Siegel-Weil formula

Can an integer $n$ be represented as a sum of two squares $n=x^2+y^2$? If so, how many different representations are there? We begin with the answers to these classical questions due to Fermat and Jacobi. We then illustrate Hurwitz's class number formula for binary quadratic forms, and put all these classical formulas under the modern perspective of the Siegel-Weil formula. We explain how the latter perspective led Gross-Keating to discover a new type of identity between arithmetic intersection numbers on modular surfaces andderivatives of certain Eisenstein series. After outlining the influential program of Kudla and Rapoport for generalization to higher dimensions, we report a recent proof (joint with W. Zhang) of the Kudla-Rapoport conjecture and hint at the usage of the uncertainty principle in the proof.

February 22 (Saturday)

Southern California Number Theory Day

February 27

Nathan Green (UCSD)
TBA

TBA

March 6

Jeff Manning (UCLA)
TBA

TBA

March 13

Luciena Xiao (Caltech)
TBA

TBA