# 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**

**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**

**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**

**Friday, May 27, 2016**

**"Half-integer weight modular forms"**

**Richard Gottesman, University of California, Santa Cruz**

**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.