Sideways equidistribution of function field L-functions

Abstract: In the first part of this talk, we will explain a concise description of the asymptotic distributions of eigenvalues of Haar-random orthogonal matrices using a new $\sigma$-moment generating function that replaces the usual exponential with the plethystic exponential of symmetric function theory. Similar descriptions can be obtained also for compact symplectic, unitary, and symmetric groups.

In the second part of the talk, we will explain how to use point-counting techniques to compute, for a fixed finite field $\mathbb{F}_q$, the distribution of the zeroes of the $L$-function of a random smooth degree $d$ surface in $\mathbb{P}^3_{\mathbb{F}_q}$ as $d \rightarrow \infty$. The result is a simple description of the asymptotic $\sigma$-moment generating function. Comparing this with our description of the asymptotic distribution of the eigenvalues of a Haar-random orthogonal matrix, we obtain an equidistribution result that is "sideways" compared to the equidistribution results obtained by Katz and Sarnak, i.e. where the order of the limits in $d$ and $q$ have been exchanged. This sideways equidistribution is finer in that it sees the stable cohomology of local systems in all degrees instead of just the zeroth degree needed to compute monodromy.

The techniques used are robust and apply also to the L-functions of more general smooth hypersurface sections, as well as some simple Dirichlet characters that were previously studied by Bergström-Diaconu-Petersen-Westerland. Time permitting, we will briefly discuss further generalizations and related work in progress joint with Bertucci / Bilu / Bilu and Das.

algebraic geometrynumber theory

Audience: researchers in the topic

( paper )

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MIT number theory seminar

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