Abelian varieties and transversal index theorems

TL;DR

This paper interprets the explicit formula for abelian variety zeta functions as a transversal index theorem on a foliated space, extending Deninger's work from elliptic curves to higher dimensions.

Contribution

It generalizes Deninger's index theorem approach from elliptic curves to higher-dimensional abelian varieties over finite fields.

Findings

Explicit formula interpreted as a transversal index theorem.

Extension of Deninger's work to higher dimensions.

Provides a geometric framework for understanding zeta functions.

Abstract

We interpret the "explicit formula" in the sense of analytic number theory for the zeta function of an ordinary abelian variety of dimension g over a finite field as a transversal index theorem on a (2g+1)-dimensional Riemannian foliated space. This generalizes a work of Deninger for elliptic curves.