On the Gross-Stark Conjecture

TL;DR

This paper proves Gross's conjecture, which relates the leading term of a $p$-adic $L$-function at zero to a $p$-adic regulator of units in a totally real field extension.

Contribution

The paper provides a proof of Gross's conjecture, establishing a fundamental link between $p$-adic $L$-functions and algebraic units in number fields.

Findings

Proof of Gross's conjecture completed

Connection between $p$-adic $L$-functions and units established

Advances understanding of special values of $p$-adic $L$-functions

Abstract

In 1980, Gross conjectured a formula for the expected leading term at $s=0$ of the Deligne--Ribet $p$-adic $L$-function associated to a totally even character $\psi$ of a totally real field $F$. The conjecture states that after scaling by $L(\psi \omega^{-1}, 0)$, this value is equal to a $p$-adic regulator of units in the abelian extension of $F$ cut out by $\psi \omega^{-1}$. In this paper, we prove Gross's conjecture.