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Article Dans Une Revue Communications in Advanced Mathematical Sciences Année : 2018

Annihilation of tor_p(G_S^ab) for real abelian extensions K/Q

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_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.
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Dates et versions

hal-01807711 , version 1 (05-06-2018)
hal-01807711 , version 2 (04-10-2018)

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Citer

Georges Gras. Annihilation of tor_p(G_S^ab) for real abelian extensions K/Q. Communications in Advanced Mathematical Sciences, 2018, Communications in Advanced Mathematical Sciences, 1 (1), pp.5 - 34. ⟨10.33434/cams.441035⟩. ⟨hal-01807711v2⟩
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