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Texas A&M University
Mathematics

Number Theory Seminar

Spring 2019

 

Date:February 13, 2019
Time:1:45pm
Location:BLOC 624
Speaker:Maurice Rojas, Texas A&M University
Title:Faster point counting over prime power rings and pseudo-random generators
Abstract:

We discuss a recent advance enabling a simple, randomized polynomial-time 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 one-way functions and pseudo-random generators by Anshel, Goldfeld, and Zuniga-Galindo. 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.


Date:February 20, 2019
Time:1:45pm
Location:BLOC 220
Speaker:Matt Papanikolas, Texas A&M University
Title:Hyperderivative power sums and Carlitz multiplication coefficients
Abstract:We will discuss connections among hyperderivatives of polynomials over finite fields, q-th 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 log-algebraic polynomials for the Carlitz module.

Date:February 27, 2019
Time:1:45pm
Location:BLOC 220
Speaker:Adrián Barquero Sánchez, Universidad de Costa Rica
Title:Faltings heights and a non-abelian Chowla-Selberg formula for CM abelian varieties
Abstract:In this talk we will give an identity which evaluates Faltings heights of CM abelian varieties with complex multiplication by non-abelian CM fields in terms of values of the Barnes multiple Gamma function at algebraic numbers. This identity can be seen as a non-abelian Chowla-Selberg 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.

Date:March 18, 2019
Time:3:00pm
Location:BLOC 220
Speaker:Laura DeMarco, Northwestern University
Title:Heights on P^1 and unlikely intersections
Abstract:I will discuss uniformity questions surrounding Unlikely Intersection problems -- the most famous of which is the Manin-Mumford 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.

Date:March 27, 2019
Time:1:45pm
Location:BLOC 220
Speaker:Wei-Lun Tsai, Texas A&M University
Title:Arithmetic statistics of canonical Hecke L-functions
Abstract: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 L-functions are related to ranks of Gross's elliptic Q-curves. In this talk, we explain how non-trivial bounds for l-torsion 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 L-function 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.

Date:April 3, 2019
Time:1:45pm
Location:BLOC 220
Speaker:Wei-Cheng Huang, Texas A&M University
Title:A t-motivic interpretation of shuffle relations for multizeta values
Abstract:Thakur showed that a product of two Carlitz zeta values, with weights r and s, can be expressed as an F_p-linear 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 t-module 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 t-module 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.

Date:April 17, 2019
Time:1:45pm
Location:BLOC 220
Speaker:Rizwan Khan, University of Mississippi
Title:Non-vanishing of Dirichlet L-functions
Abstract:L-functions are fundamental objects in number theory. At the central point s = 1/2, an L-function 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 Swinnerton-Dyer conjecture), or if its functional equation specialized to s = 1/2 implies that it must. Thus when the central value of an L-function 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 non-vanishing 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% non-vanishing 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 non-vanishing for Dirichlet L-functions.

Date:April 24, 2019
Time:1:45pm
Location:BLOC 220
Speaker:Ian Petrow, ETH Zürich
Title:The Weyl law for algebraic tori
Abstract:A basic but difficult question in the analytic theory of automorphic forms is: given a reductive group G and a representation r of its L-group, 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.