Nombres de Weil, sommes de Gauss et annulateurs Galoisiens

TL;DR

This paper investigates the structure of class groups in certain abelian number fields containing roots of unity, using Jacobi sums and Gauss sums to parametrize annihilators and Fitting ideals.

Contribution

It introduces a new parametrization of Z_p[G(K/Q)]-annihilators of class group parts via modules of Jacobi sums and relates Fitting ideals to twisted Gauss sums.

Findings

Parametrization of class group annihilators using Jacobi sums.

Explicit determination of Fitting ideals via Gauss sums.

Application of reflection theorem and Bloch-Kato reciprocity law.

Abstract

For an abelian number field K containing a primitive p-th root of unity (p an odd prime) and satisfying certain technical conditions, we parametrize the Z_p[G(K/Q)]-annihilators of the "minus" part A_K^- of the p-class group by means of modules of Jacobi sums. Using a reflection theorem and Bloch-Kato's reciprocity law, we then determine the Fitting ideal of the "plus" part A_K^+ in terms of "twisted" Gauss sums.