Quebec Mathematical Sciences Colloquium

January 23, 2009 from 16:00 to 18:00 (Montreal/Miami time) On location

Statistics for the Zeroes and Traces of Zeta Functions over Finite Fields

Colloquium presented by Chantal David (Concordia University)

In recent years, much research has been devoted to the understanding of the distribution of zeroes of zeta functions and L-functions. The seminal work of Katz and Sarnak showed that the zeroes of zeta functions of curves over the finite fields $\F_q$ in various families are distributed as eigenvalues of random symplectic matrices as $q$ tends to infinity. Very recently, Kurlberg and Rudnick studied distributions of zeroes in similar families of curves over finite fields, but from a different perspective, fixing the finite field $\F_q$ and varying the genus of the curve. For the family of zeta functions of the hyperelliptic curves $y^2=F(x)$, they showed that the limiting distribution of the trace is that of a sum of $q$ independent random variables. We will explain in this talk how to build zeta functions of curves over finite fields, and what is the significance of their zeroes. We will also present a natural generalisation of the trace distribution from hyperelliptic curves to cyclic trigonal curves $y^3=F(x)$. In this case, the limiting distribution of the trace is no longer given by independent random variables, but involves bias and mixed probabilities.



UdeM, Pav. André-Aisenstadt, 2920, ch. de la Tour, room 6214