Annihilation of $\text{tor}_{Z_{p}}(\mathcal G_{K,S}^{ab})$ for real abelian extensions $K/Q$

Authors : Georges GRAS

Pages : 5-34

Doi:10.33434/cams.441035

View : 35 | Download : 9

Publication Date : 2018-09-30

Article Type : Research Paper

Abstract :Let $K$ be a real abelian extension of $\mathbb{Q}$. Let $p$ be a prime number, $S$ the set of $p$-places of $K$ and ${\mathcal G}_{K,S}$ the Galois group of the maximal $S \cup \{\infty\}$-ramified pro-$p$-extension of $K$ insert ignore into journalissuearticles values(i.e., unramified outside $p$ and $\infty$);. We revisit the problem of annihilation of the $p$-torsion group ${\mathcal T}_K := \text{tor}_{Z_{p}}insert ignore into journalissuearticles values(\mathcal G_{K,S}^{ab});$ initiated by us and Oriat then systematized in our paper on the construction of $p$-adic $L$-functions in which we obtained a canonical ideal annihilator of ${\mathcal T}_K$ in full generality insert ignore into journalissuearticles values(1978--1981);. Afterwards insert ignore into journalissuearticles values(1992--2014); some annihilators, using cyclotomic units, were proposed by Solomon, Belliard--Nguyen Quang Do, Nguyen Quang Do--Nicolas, All, Belliard--Martin. In this text, we improve our original papers and show that, in general, the Solomon elements are not optimal and/or partly degenerated. We obtain, whatever $K$ and $p$, an universal non-degenerated annihilator in terms of $p$-adic logarithms of cyclotomic numbers related to $L_p$-functions at $s=1$ of {primitive characters of $K$} insert ignore into journalissuearticles values(Theorem 9.4);. Some computations are given with PARI programs; the case $p=2$ is analyzed and illustrated in degrees $2$, $3$, $4$ to test a conjecture.
Keywords : Abelian p ramification, annihilation of p torsion modules, p adic L functions, Stickelberger`s elements, Cyclotomic units, Class field theory, Cyclotomic units