Annihilation of tor\_p(G\_S^ab) for real abelian extensions

TL;DR

This paper improves previous results on annihilating the p-torsion part of Galois groups of real abelian extensions, providing a universal non-degenerated annihilator using p-adic logarithms and testing conjectures computationally.

Contribution

It introduces a new universal annihilator for the p-torsion group in real abelian extensions, surpassing previous cyclotomic unit-based methods.

Findings

The Solomon elements are not always optimal or degenerate.

A universal non-degenerated annihilator is constructed using p-adic logarithms.

Computational tests for p=2 in degrees 2, 3, 4 support the theoretical results.

Abstract

Preprint of a paper to appear in "Communications in Advanced Mathematical Sciences". Let K be a real abelian extension of Q. Let p be a prime number, S the set of p-places of K and G\_K,S the Galois group of the maximal S-ramified pro-p-extension of K (i.e., unramified outside p and infinity). We revisit the problem of annihilation of the p-torsion group T\_K:=tor\_Z\_p(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 T\_K in full generality (1978--1981). Afterwards (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 its primitive characters of K (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.