# Past Number Theory Seminar

12 June 2014
16:00
Christopher Lazda
Abstract
If X/F is a smooth and proper variety over a global function field of characteristic p, then for all l different from p the co-ordinate ring of the l-adic unipotent fundamental group is a Galois representation, which is unramified at all places of good reduction. In this talk, I will ask the question of what the correct p-adic analogue of this is, by spreading out over a smooth model for C and proving a version of the homotopy exact sequence associated to a fibration. There is also a version for path torsors, which enables me to define an function field analogue of the global period map used by Minhyong Kim to study rational points.
• Number Theory Seminar
5 June 2014
16:00
Andrew Granville
Abstract
For the last few years Soundararajan and I have been developing an alternative "pretentious" approach to analytic number theory. Recently Harper established a more intuitive proof of Halasz's Theorem, the key result in the area, which has allowed the three of us to provide new (and somewhat simpler) proofs to several difficult theorems (like Linnik's Theorem), as well as to suggest some new directions. We shall review these developments in this talk.
• Number Theory Seminar
29 May 2014
16:00
Miguel Walsh
Abstract
We will discuss some connections between the polynomial method, sieve theory, inverse problems in arithmetic combinatorics and the estimation of rational points on curves. Our motivating questions will be to classify those sets that are irregularly distributed in residue classes and to understand how many rational points of bounded height can a curve of fixed degree have.
• Number Theory Seminar
15 May 2014
16:00
Abstract
I'll sketch a construction which associates a canonical p-adic L-function with a 'non-critically refined' cohomological cuspidal automorphic representation of GL(2) over an arbitrary number field F, generalizing and unifying previous results of many authors. These p-adic L-functions have good interpolation and growth properties, and they vary analytically over eigenvarieties. When F=Q this reduces to a construction of Pollack and Stevens. I'll also explain where this fits in the general picture of Iwasawa theory, and I'll point towards the iceberg of which this construction is the tip.
• Number Theory Seminar
14 May 2014
15:00
Abstract

This is a report on joint work (still in progress) with Ellen Eischen, Jian-Shu Li,
and Chris Skinner.  I will describe the general structure of our construction of p-adic L-functions
attached to families of ordinary holomorphic modular forms on Shimura varieties attached to
unitary groups.  The complex L-function is studied by means of the doubling method;
its p-adic interpolation applies adelic representation theory to Ellen Eischen's Eisenstein
measure.

• Number Theory Seminar
8 May 2014
16:00
Abstract
We extend Kisin's construction of integral canonical models of Hodge-type Shimura varieties to p=2, using Dieudonné display theory.
• Number Theory Seminar
1 May 2014
16:00
Ilya Vinogradov
Abstract
Let $G=SL(2,\R)\ltimes R^2$ and $\Gamma=SL(2,Z)\ltimes Z^2$. Building on recent work of Strombergsson we prove a rate of equidistribution for the orbits of a certain 1-dimensional unipotent flow of $\Gamma\G$, which projects to a closed horocycle in the unit tangent bundle to the modular surface. We use this to answer a question of Elkies and McMullen by making effective the convergence of the gap distribution of $\sqrt n$ mod 1.
• Number Theory Seminar