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

Volume 178 / 2017

Timothy All Acta Arithmetica 178 (2017), 273-299 MSC: Primary 11R80. DOI: 10.4064/aa8537-2-2017 Published online: 26 April 2017

Abstract

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.

Authors

  • Timothy AllDepartment of Mathematics
    Rose-Hulman Institute of Technology
    5500 Wabash Ave
    Terre Haute, IN 47803, U.S.A.
    e-mail

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