# Number Theory

## Class Numbers of Certain Quadratic Fields

Speaker:
Kalyan Chakraborty
Date:
Thu, Jul 5, 2018
Location:
PIMS, University of Calgary
Conference:
PIMS CRG in Explicit Methods for Abelian Varieties
Abstract:
Class number of a number field is one of the fundamental and mysterious objects in algebraic number theory and related topics. I will discuss the class numbers of some quadratic fields. More precisely, I will discuss some results concerning the divisibility of the class numbers of certain families of real (respectively, imaginary) quadratic fields in both qualitative and quantitative aspects. I will also look at the 3-rank of the ideal class groups of certain imaginary quadratic fields. The talk will be based on some recent works done along with my collaborators.

## Some specialization problems in Geometry and Number Theory

Speaker:
Umberto Zannier
Date:
Fri, Mar 2, 2018
Location:
PIMS, University of British Columbia
Conference:
PIMS-UBC Math Distinguished Colloquium
Abstract:
We shall survey over the general issue of `specializations which preserve a property', for a parametrized family of algebraic varieties. We shall limit ourselves to a few examples. We shall start by recalling typical contexts like Bertini and Hilbert Irreducibility theorems, illustrating some new result. Then we shall jump to much more recent instances, related to algebraic families of abelian varieties. ** Please note, this video was recorded using an older in room system and has substantially diminished video quality.**

## Multivariate (phi, Gamma)-modules

Speaker:
Kiran Kedlaya
Date:
Thu, May 18, 2017
Location:
PIMS, University of British Columbia
Conference:
Focus Period on Representations in Arithmetic
Abstract:
The classical theory of (phi, Gamma)-modules relates continuous p-adic representations of the Galois group of a p-adic field with modules over a certain mildly noncommutative ring. That ring admits a description in terms of a group algebra over Z_p which is crucial for Colmez's p-adic local Langlands correspondence for GL_2(Q_p). We describe a method for applying a key property of perfectoid spaces, the analytic analogue of Drinfeld's lemma, to the construction of "multivariate (phi, Gamma)-modules" corresponding to p-adic Galois representations in more exotic ways. Based on joint work with Annie Carter and Gergely Zabradi.

## On the local Langlands conjectures

Speaker:
Rachel Ollivier
Date:
Fri, Sep 30, 2016
Location:
PIMS, University of British Columbia
Conference:
UBC-PIMS Mathematical Sciences Faculty Award
Abstract:

### 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.

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

Speaker:
Clifton Cunningham
Date:
Thu, Feb 18, 2016
Location:
PIMS, University of Calgary
Conference:
PIMS CRG in Explicit Methods for Abelian Varieties
Abstract:
The Langlands program predicts that for every n-dimensional Abelian variety over Q there is an automorphic representation of GSpin(2n+1) over Q whose L-function coincides with the L-function coming from the Galois representation on the Tate module of the Abelian variety. Recently, Gross has refined this prediction by identifying specific properties that one should find in a vector in the automorphic representation. In joint work with Lassina Dembele, we have found some examples of automorphic representations of GSpin(2n+1) over Q whose L-functions match those coming from certain n-dimensional Abelian varieties over Q, all built from certain Hilbert modular forms. We are in the process of checking if these examples contain vectors with the properties predicted by Gross. In this talk I will explain the lifting procedure we are using to manufacture GSpin automorphic representations and describe the examples we are focusing on as we hunt for the predicted vectors in the representation space.

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

Speaker:
Drew Sutherland
Date:
Tue, Jan 12, 2016
Location:
PIMS, University of Washington
Conference:
PIMS CRG in Explicit Methods for Abelian Varieties
Abstract:
Let A be an abelian variety of dimension g over a number field K. The Sato-Tate group ST(A) is a compact subgroup of the unitary symplectic group USp(2g) that can be defined in terms of the l-adic Galois representation associated to A. Under the generalized Sato-Tate conjecture, the Haar measure of ST(A) governs the distribution of various arithmetic statistics associated to A, including the distribution of normalized Frobenius traces at primes of good reduction. The Sato-Tate groups that can and do arise for g=1 and g=2 have been completely determined, but the case g=3 remains open. I will give a brief overview of the classification for g=2 and then discuss the current state of progress for g=3.

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

Speaker:
Jens Bauch
Date:
Thu, Dec 10, 2015
Location:
PIMS, Simon Fraser University
Conference:
PIMS CRG in Explicit Methods for Abelian Varieties
Abstract:
Let $A$ be a Dedekind domain, $K$ the fraction field of $A$, and $f\in A[x]$ a monic irreducible separable polynomial. Denote by $\theta\in K^{\mathrm{sep}}$ a root of $f$ and let $F=K(\theta)$ be the finite separable extension of $K$ generated by $\theta$. We consider $\mathcal{O}$ the integral closure of $A$ in $L$. For a given non-zero prime ideal $\mathfrak{p}$ of $A$ the Montes algorithm determines a parametrization (OM representation) for every prime ideal $\mathfrak{P}$ of $\mathcal{O}$ lying over $\mathfrak{p}$. For a field $k$ and $f\in k[t,x]$ this yields a new representation of places of the function field $F/k$ determined by $f$. In this talk we summarize some applications which improve the arithmetic in the divisor class group of $F$ using this new representation.

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

Speaker:
Bianca Viray
Date:
Thu, Nov 12, 2015
Location:
PIMS, University of Washington
Conference:
PIMS CRG in Explicit Methods for Abelian Varieties
Abstract:
Igusa class polynomials are the genus 2 analogue of Hilbert class polynomials; their roots are invariants of genus 2 curves that have complex multiplication by a fixed order. The coefficients of Igusa class polynomials are rational, but, unlike in genus 1, are not integral. An exact formula, or tight upper bound, for these denominators is needed to compute Igusa class polynomials and has applications to cryptography. In this talk, we explain how to obtain a formula for the arithmetic intersection number G1.CM(K) and how this results in a bound for denominators of Igusa class polynomials. We also explain how the formula for G1.CM(K) leads us to a generalization of Gross and Zagier's formula for differences of CM j-invariants. This is joint work with Kristin Lauter.

## Local-global principles for quadratic forms

Author:
Raman Parimala
Date:
Fri, Oct 30, 2015
Location:
PIMS, University of British Columbia
Conference:
PIMS/UBC Distinguished Colloquium
Abstract:
The classical theorem of Hasse-Minkowski asserts that a quadratic form over a number field represents zero nontrivially provided it represents zero nontrivially over its completions at all its places. We discuss analogous local global principles over function fields of p-adic curves. Such local-global principles in the general setting for homogeneous spaces have implications to the understanding of the arithmetic of these fields.
Notes:

## Undecidability in Number Theory

Speaker:
Bjorn Poonen
Date:
Mon, May 26, 2014
Location:
PIMS, University of British Columbia
Conference:
2014 Niven Lecture
Abstract:

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.