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

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 $K$ (i.e., unramified outside $p$ and $\infty$). We revisit the problem of annihilation of the $p$-torsion group ${\mathcal T}_K := \text{tor}_{Z_{p}}(\mathcal G_{K,S}^{ab})$ 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 (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 {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.