Zeta statistics and Hadamard functions

TL;DR

This paper introduces the Hadamard topology on the Witt ring of rational functions, proposing a unifying framework for arithmetic and motivic statistics through the concept of Hadamard functions and conjectural convergence results.

Contribution

It defines the Hadamard topology on the Witt ring and formulates a meta-conjecture linking convergence of zeta functions in multiple topologies, unifying arithmetic and motivic statistics.

Findings

Hadamard convergence holds for many zero-cycle statistics.

Convergence also observed for the motivic height zeta function in toric varieties.

Proposes a conjectural unification of motivic and arithmetic results.

Abstract

We introduce the Hadamard topology on the Witt ring of rational functions, giving a simultaneous refinement of the weight and point-counting topologies. Zeta functions of algebraic varieties over finite fields are elements of the rational Witt ring, and the Hadamard topology allows for a conjectural unification of results in arithmetic and motivic statistics: The completion of the Witt ring for the Hadamard topology can be identified with a space of meromorphic functions which we call Hadamard functions, and we make the meta-conjecture that any "natural" sequence of zeta functions which converges to a Hadamard function in both the weight and point-counting topologies converges also in the Hadamard topology. For statistics arising from Bertini problems, zero-cycles or the Batyrev-Manin conjecture, this yields an explicit conjectural unification of existing results in motivic and arithmetic statistics that were previously connected only by analogy. As evidence for our conjectures, we show that Hadamard convergence holds for many natural statistics arising from zero-cycles, as well as for the motivic height zeta function associated to the motivic Batyrev-Manin problem for split toric varieties.