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 Events
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, 3:30pm-5:15pm

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.
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Avner Segal
UBC
Mon 14 Nov 2016, 3:00pm
Number Theory Seminar
ESB 4127
New-Way Integrals
ESB 4127
Mon 14 Nov 2016, 3:00pm-5:00pm

Abstract

In the theory of automorphic representations the study of L-functions 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 Rankin-Selberg method. In this method an integral representation, with good analytic properties, is attached to the L-function. Many examples of Rankin-Selberg 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 new-way to get Euler products", Krelle, 1988) I. Piatetski-Shapiro and S. Rallis suggested a remarkable mechanism that makes it possible to use integrals containing a "non-unique model" by a slight strengthening of the unramified computations.

In the first part of my talk we will have a crash-course on cuspidal automorphic representations and the new-way mechanism via the classical example of Hecke's integral representation for L-functions 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 Rankin-Selberg integrals representing the standard twisted L-function 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 non-unique model and we apply the new-way mechanism. The unramified computation gives rise to two interesting objects: the generating function of the L-function and its approximations. If time permits, I will discuss the possible poles of this L-function and some applications to the theory of cuspidal representations of G_2.
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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, 3:00pm-5:00pm

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.
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