Curves, dynamical systems and weighted point counting

TL;DR

This paper demonstrates that weighted point counting via all Dirichlet L-series uniquely determines a curve over a finite field, extending classical results and linking spectral data to geometric isomorphism.

Contribution

It establishes that equality of all Dirichlet L-series implies the curves are isomorphic up to Frobenius twists, thus solving an arithmetic analogue of the isospectrality problem.

Findings

Weighted point counting determines the curve.

L-series equality implies curve isomorphism up to Frobenius twists.

The method connects class field theory with dynamical systems.

Abstract

Suppose X is a (smooth projective irreducible algebraic) curve over a finite field k. Counting the number of points on X over all finite field extensions of k will not determine the curve uniquely. Actually, a famous theorem of Tate implies that two such curves over k have the same zeta function (i.e., the same number of points over all extensions of k) if and only if their corresponding Jacobians are isogenous. We remedy this situation by showing that if, instead of just the zeta function, all Dirichlet L-series of the two curves are equal via an isomorphism of their Dirichlet character groups, then the curves are isomorphic up to "Frobenius twists", i.e., up to automorphisms of the ground field. Since L-series count points on a curve in a "weighted" way, we see that weighted point counting determines a curve. In a sense, the result solves the analogue of the isospectrality problem for curves over finite fields (also know as the "arithmetic equivalence problem"): it says that a curve is determined by "spectral" data, namely, eigenvalues of the Frobenius operator of k acting on the cohomology groups of all l-adic sheaves corresponding to Dirichlet characters. The method of proof is to shown that this is equivalent to the respective class field theories of the curves being isomorphic as dynamical systems, in a sense that we make precise.