Horizontal factorizations of certain Hasse--Weil zeta functions - a remark on a paper by Taniyama

TL;DR

This paper provides a geometric proof of Taniyama's product formula for the zeta functions of abelian varieties over number fields, expressing them as infinite products of modified Dedekind zeta functions.

Contribution

It offers a new geometric proof of Taniyama's factorization of zeta functions, extending the understanding of their structure beyond l-adic representation arguments.

Findings

Zeta functions can be expressed as infinite products of modified Dedekind zeta functions.

A simple geometric proof confirms Taniyama's product formula.

The approach applies to abelian and more general group schemes.

Abstract

In one of his papers, using arguments about l-adic representations, Taniyama expresses the zeta function of an abelian variety over a number field as an infinite product of modified Artin L-functions. The latter can be further decomposed as products of modified Dedekind zeta functions. After recalling Taniyama's work, we give a simple geometric proof of the resulting product formula for abelian and more general group schemes.