Algebra & Number Theory Seminar Spring 2016

Fridays from 12:00-1:00pm

McHenry Library room 4130
For more information please contact Professor Samit Dasgupta or call the Mathematics Department at 831-459-2969

Friday, April 8, 2016

"On p-adic Regulators"

Samit Dasgupta, University of California, Santa Cruz

In this purely expository talk, I will define the p-adic regulators of Leopoldt and Gross associated to characters of totally real fields. I will describe the importance of these regulators via their relation to Iwasawa theory and the special values of p-adic L-functions.

Friday, April 22, 2016

"The fundamental weights of vector-valued modular forms"

Geoff Mason, University of California, Santa Cruz

The fundamental weights of a finite-dimensional representation $\rho$ of the modular group are an important set of numerical invariants thatarise naturally in the study of $\rho$ and its associated modular forms. They can be studied geometrically (vector bundles) and algebraically (rings of differential operators). They have surprising properties, some ofwhich remain conjecture. I will describe what we know and what we would like to know about them

Friday, April 29, 2016

"Slopes of modular forms and the ghost conjecture"

John Bergdall, Boston University

In this talk we will discuss the p-adic properties of the Atkin-Lehner Up operator acting on spaces of cuspforms as the weight varies. Specifically we will construct a completely explicit and elementary two-variable Fredholm series over Zp, one of the variables being the weight, whose Newton polygons, weight-by-weight, we conjecture to be computing the so-called slopes of Up in the Buzzard regular case. Time permitting we will discuss the evidence for our conjecture and consequences. This is joint work with Robert Pollack.

Friday, May 6, 2016

"Motives with Galois group type of G_2 - construction of Gross and Savin revisited"

Sug Woo Shin, University of California, Berkeley

Serre asked whether there exists a motive (over Q) with Galois group G_2. Put it in another way, the question is to find (a compatible family of) ell-adic Galois representations whose image has Zariski closure G_2. This has been answered affirmatively since 2010 by Dettweiler and Reiter, Khare-Larsen-Savin, Yun, and Patrikis (including generalizations to exceptional groups other than G_2). In this talk I revisit the construction of Gross-Savin (which was conditional when proposed in 1998) which aims to realize such a motive in the cohomology of a Siegel modular variety of genus 3 via exceptional theta correspondence between G_2 and PGSp_6. Then I will explain that the construction is now unconditional due to my recent work with Arno Kret on the construction of GSpin(2n+1)-valued Galois representations in the cohomology of Siegel modular varieties, closing with some open questions raised by Gross and Savin.

Friday, May 13, 2016

"Galois representations attached to elliptic curves, and torsion subgroups"

Álvaro Lozano-Robledo, University of Connecticut

In this talk we will discuss what is known about the images of Galois representations attached to elliptic curves (mostly over $\mathbb{Q}$), and what consequences we can deduce about the field of definition of their torsion subgroups. In particular, we will discuss applications of recent results of Rouse and Zureick-Brown, and Sutherland and Zywina, about 2-adic images, and mod-p images of Galois representations, respectively. For instance, we will show sharp divisibility bounds (explicit) for the degree of the field of definition of any 2-primary torsion structure of an elliptic curve defined over $\mathbb{Q}$.

Friday, May 20, 2016
** Note Room Change: McHenry 1257 **

"The distribution of consecutive primes"

Robert Lemke Oliver, Stanford University

While the sequence of primes is very well distributed in the reduced residue classes (mod q), the distribution of pairs of consecutive primes among the permissible pairs of reduced residue classes (mod q) is surprisingly erratic. We propose a conjectural explanation for this phenomenon, based on the Hardy-Littlewood conjectures, which fits the observed data very well. We also study the distribution of the terms predicted by the conjecture, which proves to be surprisingly subtle. This is joint work with Kannan Soundararajan.

Friday, May 27, 2016

"Half-integer weight modular forms"

Richard Gottesman, University of California, Santa Cruz

This expository talk is aimed at graduate students. Half-integer weight modular forms play prominent roles in both number theory and geometry. The Dedekind eta function and the Jacobi theta function are both modular forms of weight 1/2. In this talk, I will give an introduction to the rich theory of half-integer weight modular forms. No previous background with half-integer weight modular forms will be assumed.

Friday, June 3, 2016

"Algebraic tori and a computational inverse Galois problem"

David Roe, University of Pittsburgh

Algebraic tori play a central role in the structure theory and representation theory of algebraic groups. I will describe an ongoing project to investigate algebraic tori over p-adic fields. The project naturally divides into two parts: finding finite subgroups of GL(n,Z) and listing all p-adic fields with a given Galois group. I will summarize existing work on the first part, and present a new algorithm for the second problem.