Annihilation of tor_p(G_S^ab) for real abelian extensions
Résumé
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_S the Galois group of the maximal Su{∞}-ramified pro-p-extension of K. We revisit the problem of annihilation of the p-torsion group T:=tor_p(G_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 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 (Theorem 9.4), an universal non-degenerated annihilator in terms of p-adic logarithms of cyclotomic numbers related to L_p-functions at s=1. Some computations are given with PARI programs; the case p=2 is analyzed and illustrated in degrees 2, 3, 4 to test a conjecture.
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