Canonical Equivariant Cohomology Classes Generating Zeta Values of Totally Real Fields

TL;DR

This paper extends classical results relating special values of Dirichlet L-functions to Bernoulli numbers to the setting of totally real fields, using equivariant cohomology and Shintani generating classes.

Contribution

It introduces a canonical Shintani generating class in equivariant cohomology that captures zeta values of Lerch type for totally real fields.

Findings

Constructed a canonical Shintani generating class.

Linked derivatives of the class at torsion points to zeta values.

Provided a higher-dimensional cohomological framework for zeta values.

Abstract

It is known that the special values at nonpositive integers of a Dirichlet $L$-function may be expressed using the generalized Bernoulli numbers, which are defined by a canonical generating function. The purpose of this article is to consider the generalization of this classical result to the case of Hecke $L$-functions of totally real fields. Hecke $L$-functions may be expressed canonically as a finite sum of zeta functions of Lerch type. By combining the non-canonical multivariable generating functions constructed by Shintani, we newly construct a canonical class, which we call the Shintani generating class, in the equivariant cohomology of an algebraic torus associated to the totally real field. Our main result states that the specializations at torsion points of the derivatives of the Shintani generating class give values at nonpositive integers of the zeta functions of Lerch type. This result gives the insight that the correct framework in the higher dimensional case is to consider higher equivariant cohomology classes instead of functions.