Comparing Two Formulas for the Gross-Stark Units

TL;DR

This paper proves that two conjectured formulas for Gross regulator matrices coincide in the case of cubic totally real fields, advancing understanding of Gross-Stark units and their explicit formulas.

Contribution

It confirms the conjecture that the cohomological and analytic formulas for Gross regulator matrices agree for cubic totally real fields.

Findings

Proof of the conjecture for cubic fields

Validation of the cohomological and analytic formula equivalence

Enhanced understanding of Gross-Stark units in specific fields

Abstract

Let $F$ be a totally real number field. Dasgupta conjectured an explicit $p$-adic analytic formula for the Gross-Stark units of $F$. In a later paper, Dasgupta-Spiess conjectured a cohomological formula for the principal minors and the characteristic polynomial of the Gross regulator matrix associated to a totally odd character of $F$. Dasgupta-Spiess conjectured that these conjectural formulas coincide for the diagonal entries of Gross regulator matrix. In this paper, we prove this conjecture when $F$ is a cubic field.