
Date Time 
Location  Speaker 
Title – click for abstract 

02/13 1:45pm 
BLOC 624 
Maurice Rojas Texas A&M University 
Faster point counting over prime power rings and pseudorandom generators
We discuss a recent advance enabling a simple, randomized polynomialtime algorithm to count the roots of any polynomial in one variable over the prime power ring Z/(p^k). (The best previous general algorithms were exponential.) We also show how this implies a fast algorithm for computing certain Igusa zeta functions. These zeta functions, along with several other families of zeta functions, have been proposed as a method for generating oneway functions and pseudorandom generators by Anshel, Goldfeld, and ZunigaGalindo. We'll review the latter connections as well.
This is joint work with Yuyu Zhu, and build upon earlier joint work with Qi Cheng, Shuhong Gao, Leann Kopp, Natalie Randall, and Daqing Wan.
Abstract 

02/20 1:45pm 
BLOC 220 
Matt Papanikolas Texas A&M University 
Hyperderivative power sums and Carlitz multiplication coefficients
We will discuss connections among hyperderivatives of polynomials over finite fields, qth powers of polynomials, and specializations of Vandermonde matrices. From these relations we will construct formuls for Carlitz multiplication coefficients using hyperderivatives and symmetric polynomials, and we will obtain precise identities for hyperderivative power sums in terms of the inverse of the Vandermonde matrix. We will further discuss a new proof of a theorem of Thakur that gives exact formulas for Anderson's special logalgebraic polynomials for the Carlitz module. Abstract 

02/27 1:45pm 
BLOC 220 
Adrián Barquero Sánchez Universidad de Costa Rica 
Faltings heights and a nonabelian ChowlaSelberg formula for CM abelian varieties
In this talk we will give an identity which evaluates Faltings heights of CM abelian varieties with complex multiplication by nonabelian CM fields in terms of values of the Barnes multiple Gamma function at algebraic numbers. This identity can be seen as a nonabelian ChowlaSelberg formula for CM abelian varieties. The proof uses recent joint work with Riad Masri and Frank Thorne on the Colmez conjecture. This is joint work with Riad Masri and Wei Lun Tsai. Abstract 

03/18 3:00pm 
BLOC 220 
Laura DeMarco Northwestern University 
Heights on P^1 and unlikely intersections
I will discuss uniformity questions surrounding Unlikely Intersection problems  the most famous of which is the ManinMumford Conjecture (proved by Raynaud, 1983)  and a new result about elliptic curves and the geometry of their torsion points, joint with Holly Krieger and Hexi Ye. Our results hold over C, the field of complex numbers, but the proofs are carried out first over the field of algebraic numbers (and involve an analysis of certain height functions on P^1). Our strategy is based on equidistribution results for dynamical systems on P^1.
Joint with Groups and Dynamics Seminar.Abstract 

03/27 1:45pm 
BLOC 220 
WeiLun Tsai Texas A&M University 
Arithmetic statistics of canonical Hecke Lfunctions
The canonical Hecke characters in the sense of Rohrlich form a set of algebraic Hecke characters with important arithmetic properties. For example, the central values of their corresponding Lfunctions are related to ranks of Gross's elliptic Qcurves. In this talk, we explain how nontrivial bounds for ltorsion in class groups of number fields can be used to prove that for an asymptotic density of 100 percent of CM fields E within certain general families, the number of canonical Hecke characters of E whose Lfunction has a nonvanishing central value is >> disc(E)^{delta} for some absolute constant delta > 0. This is joint work with B. D. Kim and Riad Masri. Abstract 

04/03 1:45pm 
BLOC 220 
WeiCheng Huang Texas A&M University 
A tmotivic interpretation of shuffle relations for multizeta values
Thakur showed that a product of two Carlitz zeta values, with weights r and s, can be expressed as an F_plinear combination of single and double zeta values of weight equal to r+s where F_p is the finite field of p elements and p is a prime number. Such an expression is called shuffle relation by Thakur. Fixing positive integers r, s, we construct a tmodule E. To determine whether an (r+s)tuple C in F_q(\theta)^{r+s} gives a shuffle relation, we relate it to the F_q[t]torsion property of the integral point v_C in the tmodule constructed with respect to the given (r+s)tuple C. We also provide an effective criterion for deciding the F_q[t]torsion property of the point v_C. Abstract 

04/17 1:45pm 
BLOC 220 
Rizwan Khan University of Mississippi 
Nonvanishing of Dirichlet Lfunctions
Lfunctions are fundamental objects in number theory. At the central point s = 1/2, an Lfunction L(s) is expected to vanish only if there is some deep arithmetic reason for it to do so (such as in the Birch and SwinnertonDyer conjecture), or if its functional equation specialized to s = 1/2 implies that it must. Thus when the central value of an Lfunction is not a "special value", and when it does not vanish for trivial reasons, it is conjectured to be nonzero. In general it is very difficult to prove such nonvanishing conjectures. For example, nobody knows how to prove that L(1/2, chi) is nonzero for all primitive Dirichlet characters chi. In such situations, analytic number theorists would like to prove 100% nonvanishing in the sense of density, but achieving any positive percentage is still valuable and can have important applications. In this talk, I will discuss recent work on establishing such positive proportions of nonvanishing for Dirichlet Lfunctions. Abstract 

04/24 1:45pm 
BLOC 220 
Ian Petrow ETH Zürich 
The Weyl law for algebraic tori
A basic but difficult question in the analytic theory of automorphic forms is: given a reductive group G and a representation r of its Lgroup, how many automorphic representations of bounded analytic conductor are there? In this talk I will present an answer to this question in the case that G is a torus over a number field. Abstract 