UBC

Thu 10 Nov 2016, 3:30pm
Number Theory Seminar
MATH 126

Counting irreducible divisors and irreducibles in progressions

MATH 126
Thu 10 Nov 2016, 3:30pm5:15pm
Abstract
Let K/\mathbb{Q} be a number field with ring of integers \mathbb{Z}_K. If K has class number one, the set of irreducible elements of \mathbb{Z}_K coincides with the set of prime elements; in general, this need not be the case. One is led to wonder: Do statements about primes in \mathbb{Z} have analogues for irreducibles in \mathbb{Z}_K, for a general choice of K? This talk concerns two instances where the answer is yes. We will discuss the maximal order of the number of irreducible divisors of an element of \mathbb{Z}_K, and we will provide an asymptotic formula for the number of irreducible elements of norm up to x belonging to a given arithmetic progression.
hide

UBC

Mon 14 Nov 2016, 3:00pm
Number Theory Seminar
ESB 4127

NewWay Integrals

ESB 4127
Mon 14 Nov 2016, 3:00pm5:00pm
Abstract
In the theory of automorphic representations the study of Lfunctions plays a key role. A common method to study the analytic behavior of such functions (and, in fact, proving that they are meromorphic functions) is the RankinSelberg method. In this method an integral representation, with good analytic properties, is attached to the Lfunction. Many examples of RankinSelberg integrals were studied along the years. However most examples rely on the uniqueness of certain models of the representation (most popular in use is the Whittaker model but many other, such as the Peterson bilinear form and Bessel model, are used). In a pioneering paper ("A newway to get Euler products", Krelle, 1988) I. PiatetskiShapiro and S. Rallis suggested a remarkable mechanism that makes it possible to use integrals containing a "nonunique model" by a slight strengthening of the unramified computations.
In the first part of my talk we will have a crashcourse on cuspidal automorphic representations and the newway mechanism via the classical example of Hecke's integral representation for Lfunctions of cuspidal representations of GL_2.
In the second part of my talk I will present a joint work with N. Gurevich in which we proved that a family of RankinSelberg integrals representing the standard twisted Lfunction of a cuspidal representation of the exceptional group of type G_2. In its unfolded form (a term which will be explained in the talk), the integrals contain a nonunique model and we apply the newway mechanism. The unramified computation gives rise to two interesting objects: the generating function of the Lfunction and its approximations. If time permits, I will discuss the possible poles of this Lfunction and some applications to the theory of cuspidal representations of G_2.
hide

UBC

Mon 28 Nov 2016, 3:00pm
Number Theory Seminar
ESB 4127

The modular method and Fermat's Last Theorem

ESB 4127
Mon 28 Nov 2016, 3:00pm5:00pm
Abstract
Fermat's Last Theorem states that the equation x^n + y^n = z^n for n > 2 has no integer solutions such that xyz \neq 0. It's proof was completed in 1995 by the groundbreaking work of Andrew Wiles on the modularity of semistable elliptic curves over Q. From its proof a new revolutionary method to attack Diophantine equations was born. This method, now known as the modular method, builds on the work of Frey, Serre, Ribet, Mazur and makes use of the Galois representations attached to modular forms and elliptic curves.
In the first part of this talk, guided by the proof of FLT, we will introduce the tools and sketch the basic strategy behind the modular method. In the second part, we will discuss the main obstacles that arise when we try to apply the method to other type of equations or over number fields.
hide

Seminar Information Pages
