Gauss Sums, Stickelberger's Theorem, and the Gras Conjecture for Ray Class Groups

TL;DR

This paper proves a ray class version of the Gras Conjecture for real abelian fields, relating units and ray class groups via Galois characters, and constructs explicit Galois annihilators similar to Stickelberger's theorem.

Contribution

It establishes a new relation between units and ray class groups in real abelian fields and constructs explicit Galois annihilators, extending classical results like Stickelberger's theorem.

Findings

Proves a ray class version of the Gras Conjecture under ramification conditions.

Constructs explicit Galois annihilators for ray class groups.

Extends classical Gauss sum and Stickelberger techniques to real fields.

Abstract

Let $k$ be a real abelian number field and $p$ an odd prime not dividing $[k:\mathbb{Q}]$. For a natural number $d$, let $E_d$ denote the group of units of $k$ congruent to $1$ modulo $d$, $C_d$ the subgroup of $d$-circular units of $E_d$, and $\mathfrak{C}(d)$ the ray class group of modulus $d$. Let $\rho$ be an irreducible character of $G=\mathrm{Gal}(k/\mathbb{Q})$ over $\mathbb{Q}_p$ and $e_{\rho} \in \mathbb{Z}_p[G]$ the corresponding idempotent. We show that if the ramification index of $p$ in $k$ is less than $p-1$, then $|e_{\rho} \mathrm{Syl}_p(E_d/C_d) | = |e_{\rho} \mathrm{Syl}_p(\mathfrak{C}_d)|$ where $\mathfrak{C}_d$ is the part of $\mathfrak{C}(d)$ where $G$ acts non-trivially. This is a ray class version of the Gras Conjecture. In the case when $p \mid [k:\mathbb{Q}]$, similar but slightly less precise results are obtained. In particular, beginning with what could be considered a Gauss sum for real fields, we construct explicit Galois annihilators of $\mathrm{Syl}_p(\mathfrak{C}_{\mathfrak{a}})$ akin to the classical Stickelberger Theorem.