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

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.