Gauss sums, Stickelberger’s theorem and the Gras conjecture for ray class groups

Timothy All¹

¹ Department of Mathematics Rose-Hulman Institute of Technology 5500 Wabash Ave Terre Haute, IN 47803, U.S.A.

Acta Arithmetica, Tome 178 (2017) no. 3, pp. 273-299

Résumé

Let $k$ be a real abelian number field and $p$ an odd prime not dividing\break $[k:\mathbb{Q}]$. 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 $G=\operatorname{Gal}(k/\mathbb{Q})$ 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 $|e_{\rho} \operatorname{Syl}_p(E_d/C_d) | = |e_{\rho} \operatorname{Syl}_p(\mathfrak{C}_d)|$ 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 $p \,|\, [k:\mathbb{Q}]$, 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 $\operatorname{Syl}_p(\mathfrak{C}_{\mathfrak{a}})$ akin to the classical Stickelberger Theorem.

Détail

DOI : 10.4064/aa8537-2-2017

Keywords: real abelian number field odd prime dividing break mathbb natural number denote group units congruent modulo nbsp subgroup d circular units mathfrak ray class group modulus rho irreducible character operatorname gal mathbb mathbb rho mathbb corresponding idempotent ramification index p rho operatorname syl c rho operatorname syl mathfrak where mathfrak part mathfrak where acts non trivially ray class version gras conjecture mathbb similar slightly precise results obtained particular beginning what could considered gauss sum real fields construct explicit galois annihilators operatorname syl mathfrak mathfrak akin classical stickelberger theorem