The Eisenstein cocycle, partial zeta values and Gross--Stark units

TL;DR

This paper introduces an integral Eisenstein cocycle and applies it to prove Gross's conjecture on Stickelberger elements and to construct Gross--Stark units through cohomology, advancing number theory and algebraic number fields.

Contribution

It provides an integral version of the Eisenstein cocycle and uses it to prove conjectures and construct units related to abelian extensions.

Findings

Proves Gross's conjecture on the order of vanishing of Stickelberger elements.

Provides a cohomological construction of Gross--Stark units.

Advances understanding of special values of L-functions and algebraic units.

Abstract

We introduce an integral version of the Eisenstein cocycle. As applications we prove a conjecture of Gross regarding the "order of vanishing" of Stickelberger elements relative to an abelian tower of fields and give a cohomological construction of the conjectural Gross--Stark units.