UCSD Number Theory Seminar (Math 209)

Thursday 2-3pm, AP&M 7421

This quarter, most talks will be preceded by a 30-minute "prep talk" for graduate students and postdocs. These will be in AP&M 7421 starting at 1:20pm.
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 2018

For previous quarters' schedule, click here.


September 27

ORGANIZATIONAL MEETING

October 4

NO MEETING

October 11

Nolan Wallach (UC San Diego)
Whittaker Theory I: Applications to number theory at the infinite place and the ingredients of Whittaker Plancherel Theorem.

Today the main emphasis in local number theory (i.e the Local Langlands correspondence) is on the finite places. In characteristic 0 the infinite place is the "elephant in the room". This is especially true in the Whittaker Theory in which serious difficulties separate the infinite from the finite places. Whittaker models were developed to help the study of Fourier coefficients at cusps of non-holomophic cusp forms (i.e Maass cusp forms) through representation theory. The first of these lectures will start with an explanation of the role of Whittaker models in the theory of automorphic forms. It will continue with a description of the main results. The second lecture will explain the proof of the Whittaker Plancherel Theorem.

October 18

Nolan Wallach (UC San Diego)
Whittaker Theory II: The proof of the Whittaker Plancherel Theorem.

Today the main emphasis in local number theory (i.e the Local Langlands correspondence) is on the finite places. In characteristic 0 the infinite place is the "elephant in the room". This is especially true in the Whittaker Theory in which serious difficulties separate the infinite from the finite places. Whittaker models were developed to help the study of Fourier coefficients at cusps of non-holomophic cusp forms (i.e Maass cusp forms) through representation theory. The first of these lectures will start with an explanation of the role of Whittaker models in the theory of automorphic forms. It will continue with a description of the main results. The second lecture will explain the proof of the Whittaker Plancherel Theorem.

October 25

Rachel Newton (Reading)
Arithmetic of rational points and zero-cycles on Kummer varieties

In 1970, Manin observed that the Brauer group Br(X) of a variety X over a number field K can obstruct the Hasse principle on X. In other words, the lack of a K-point on X despite the existence of points over every completion of K is sometimes explained by non-trivial elements in Br(X). This so-called Brauer-Manin obstruction may not always suffice to explain the failure of the Hasse principle but it is known to be sufficient for some classes of varieties (e.g. torsors under connected algebraic groups) and conjectured to be sufficient for rationally connected varieties and K3 surfaces. A zero-cycle on X is a formal sum of closed points of X. A rational point of X over K is a zero-cycle of degree 1. It is interesting to study the zero-cycles of degree 1 on X, as a generalisation of the rational points. Yongqi Liang has shown that for rationally connected varieties, sufficiency of the Brauer-Manin obstruction to the Hasse principle for rational points over all finite extensions of K implies sufficiency of the Brauer-Manin obstruction to the Hasse principle for zero-cycles of degree 1 over K. In this talk, I will discuss joint work with Francesca Balestrieri where we extend Liang's result to Kummer varieties.

November 1

NO MEETING

November 8

Daniel Le (Toronto); 1-2pm in APM 7421 -- no pre-talk
Serre weights and affine Grassmannians

A conjecture of Serre (now a theorem of Gross, Edixhoven, and Coleman-Voloch) classifies pairs of weights where one finds modular forms congruent modulo a prime p in terms of local behavior at p. We discuss a generalization of this conjecture in higher rank. A key step in our work is the study of a certain subscheme of Gaitsgory's A^1 affine Grassmannian which shares properties with some affine Springer fibers. This is joint work with B. Le Hung, B. Levin, and S. Morra.

Joe Ferrara (UC San Diego); 2-3pm in APM 7421 -- no pre-talk
A p-adic Stark conjecture in the rank one setting

In the 1970's Stark made precise conjectures about the leading term of the Taylor series at s=0 for Artin L-functions. In the rank one setting when the order vanishing is exactly one, these conjectures relate the derivative of the L-function at s=0 to the logarithm of a unit in an abelian extension of the base field. In this talk, we will define a p-adic L-function and state a p-adic Stark conjecture in the rank one setting when the base field is a quadratic field. We prove our conjecture in the case when the base field is imaginary quadratic and the prime p is split, and discuss numerical evidence in the other cases.

November 15

Nathan Green (UC San Diego)
Logarithms and t-Motivic Multiple Zeta Values

For each function field multiple zeta value (defined by Thakur), we construct a t-module with an attached logarithmic vector such that a specific coordinate of the logarithmic vector is a rational multiple of that multiple zeta value. We then show that the other coordinates of this logarithmic vector contain hyperderivatives of a deformation of these multiple zeta values, which we call t-motivic multiple zeta values. This allows us to give a logarithmic expression for monomials of multiple zeta values. Joint work with Chieh-Yu Chang and Yoshinori Mishiba.

November 22

NO MEETING (Thanksgiving)

November 29

Jacob Tsimerman (Toronto)
TBA

TBA

December 6

Jacob Lurie (Harvard)
TBA

TBA