# Number Theory

## On the local Langlands conjectures

### Abstract

The Langlands program, initiated in the 1960s, is a set of conjectures predicting a unification of number theory and the representation theory of groups. More precisely, the Langlands correspondence provides a way to interpret results in number theory in terms of group theory, and vice versa.

In this talk we sketch a few aspects of the local Langlands correspondence using elementary examples. We then comment on some questions raised by the emerging "mod p" Langlands program.

### Biography

Professor Ollivier works in the Langlands Programme, a central theme in pure mathematics which predicts deep connections between number theory and representation theory. She has made profound contributions in the new branches of the "p-adic" and "mod-p" Langlands correspondence that emerged from Fontaine's work on studying the p-adic Galois representation, and is one of the pioneers shaping this new field. The first results on the mod-p Langlands correspondence were limited to the group GL2(Qp); but Dr. Ollivier has proved that this is the only group for which this holds, a surprising result which has motivated much subsequent research.

She has also made important and technically challenging contributions in the area of representation theory of p-adic groups, in particular, in the study of the Iwahori-Hecke algebra. In joint work with P. Schneider, Professor Ollivier used methods of Bruhat-Tits theory to make substantial progress in understanding these algebras. She has obtained deep results of algebraic nature, recently defining a new invariant that may shed light on the special properties of the group GL2(Qp).

Rachel Ollivier received her PhD from University Paris Diderot (Paris 7), and then held a research position at ENS Paris. She subsequently was an assistant professor at the University of Versailles and then Columbia University, before joining the UBC Department of Mathematics in 2013.

Rachel is the recepient of the 2015 UBC Mathematics and Pacific Institute for the Mathematical Sciences Faculty Award.

More information on this event is available on the event webpage

.## Lifts of Hilbert modular forms and application to modularity of Abelian varieties

## Abelian Varieties Multi-Site Seminar Series: Drew Sutherland

## OM representation of prime ideals and applications in function fields

## An arithmetic intersection formula for denominators of Igusa class polynomials

## Local-global principles for quadratic forms

## Undecidability in Number Theory

Hilbert’s Tenth Problem asked for an algorithm that, given a multivariable polynomial equation with integer coefficients, would decide whether there exists a solution in integers. Around 1970, Matiyasevich, building on earlier work of Davis, Putnam, and Robinson, showed that no such algorithm exists. However, the answer to the analogous question with integers replaced by rational numbers is still unknown, and there is not even agreement among experts as to what the answer should be.

## The rank of elliptic curves

After quadratic equations in two variables come cubic equations, or elliptic curves. The set of rational points on an elliptic curve has the structure of a finitely generated abelian group. I will recall the basic theory of elliptic curves, then discuss the conjecture of Birch and Swinnerton-Dyer, which attempts to predict the rank of the group of rational points from the number of solutions (mod p) for all primes p. I will also discuss some recent results on the average rank, due to Manjul Bhargava and his collaborators. (PIMS-UBC Distinguished Colloquium)

## Native American Mathematics

One sometimes hears that the indigenous peoples of the Americas are for some reason not predisposed to be able to do mathematics. This belief is surprising, since the mathematical traditions of the Western Hemisphere prior to European contact were already rich and extensive. This talk will focus on some of those traditions, primarily Central American but with some information about mathematical traditions in Algonkian cultures such as the Blackfoot. Almost all of this talk will be accessible to any interested listener, with perhaps five minutes in the middle using a small amount of very elementary number theory. Along the way any listener who has ever eaten an 18 Rabbits granola bar will learn why doing so celebrates indigenous mathematics.

**ABOUT THE RICHARD AND LOUISE GUY LECTURE SERIES:**

The Richard & Louise Guy lecture series celebrates the joy of discovery and wonder in mathematics for everyone. Indeed, the lecture series was a 90th birthday present from Louise Guy to Richard in recognition of his love of mathematics and his desire to share his passion with the world. Richard Guy is the author of over 100 publications including works in combinatorial game theory, number theory and graph theory. He strives to make mathematics accessible to all. The other contributions to the lecture series have been made by Elwyn Berlekamp (2006), John Conway (2007), Richard Nowakowski (2008), William Pulleyblank (2009), Erik Demaine (2010), Noam Elkies (2011), Ravi Vakil (2012) and Carl Pomerance (2013).