Collaborative Number Theory Seminar at the CUNY Graduate Center

Co-organizers: Gautam Chinta, Brooke Feigon, Krzysztof Klosin, Maria Sabitova, Lucien Szpiro.

The seminar currently meets Fridays 2:00 to 3:30 PM in Room 3209. The CUNY Graduate Center is located on Fifth Avenue, on the east side of the street, between 34th and 35th Streets in midtown Manhattan. For further information, please contact Maria Sabitova.

Fall 2014 Schedule:

September 19: Lloyd West (CUNY)

Title: Moduli spaces of arithmetic dynamical systems: invariants and fields of definition

Abstract: I shall describe how to use the classical invariant theory of binary forms to construct the moduli space, $M_d$, of degree $d$ arithmetic dynamical systems on the projective line (i.e. degree $d$ rational maps). In particular, I shall give an explicit description in the case $d=3$. I shall then discuss the issue of field of moduli verses field of definition, and show how to solve this problem explicitly for the spaces $M_d$.

October 10: Alexei Entin (IAS)

Title: On the Bateman-Horn conjecture for polynomials over large finite fields


October 31: Alex Kemarsky (Technion Israel Institute of Technology)

Title: Gamma factors of GL_n(R)-distinguished representations of GL_n(C)

Abstract: An irreducible representation (\pi,V) of GL_n(C) is called GL_n(R)-distinguished if there exists a non-zero continuous GL_n(R)-invariant functional L:V \to C. In the talk we give a necessary condition for GL_n(R)-distinction. As a corollary, we prove that the Rankin-Selberg gamma factors of \pi \times \pi' at s=1/2 for \pi,\pi' distinguished representations of GL_m(C), GL_n(C) respectively equals 1.

November 14: Thomas Tucker (University of Rochester)

Title: Proportions of periodic points modulo primes

Abstract: The birthday problem heuristic suggests that given a map g:S-->S on a typical finite space, most points in S are not periodic under g.  That is, for most s, there is no positive n such that g^n(s)=s.  We will show that if one fixes a typical polynomial g and varies over the reductions g_p sending the finite field F_p to itself then this heuristic is borne out and the proportion of periodic points for g_p goes to zero as p goes to infinity.  This is joint work with Jamie Juul, Par Kurlberg, and Kalyani Madhu.

November 21: Jonah Leshin (CUNY)

Title: The Malle-Bhargava principle and local conditions on Cohen-Lenstra heuristics

Abstract: We study the effect of imposing local constraints on Cohen-Lenstra heuristics for unramified extensions of number fields. In particular, we prove, under mild hypotheses, that a principle of Malle and Bhargava implies that the Cohen-Lenstra heuristics are robust to local constraints. We also verify that in the case of unramified quadratic extensions of non-Galois cubic fields, this predicted local invariance is correct.

December 12: Xiaoqing Li (SUNY Buffalo)

Title: A standard zero free region for Rankin-Selberg L-functions on GL(n)

Abstract: In this talk, we will derive a standard zero free region for Rankin-Selberg L-function L(s, fxf) where f is a tempered Maass form on GL(n). The method is based on the theory of Eisenstein series generalizing a work of Sarnak. This is a joint work with D. Goldfeld.