p-adic approach of Greenberg's conjecture (p-split totally real case)
Approche p-adique de la conjecture de Greenberg (cas totalement réel p-décomposé)
Résumé
Let k be a totally real number field ant let k∞ be its cyclotomic Zp-extension for a prime p > 2. We give (Theorem 3.4) a sufficient condition of nullity of the Iwasawa invariants λ, µ, when p totally splits in k, and we obtain important tables giving quadratic fields and various p for which we can conclude that λ = µ = 0. We show that the number of ambiguous p-classes of kn (the nth stage in k∞) becomes equal to the order of the torsion group Tk , of the Galois group of the maximal Abelian p-ramified prop extension of k (Theorem 4.7), for all n ≥ e, where pe is the exponent of U*k /Ek (local units of norm 1 modulo global units). Thus we recover some classical results of Fukuda, Greenberg, Inatomi, Komatsu, Sumida, Taya,. .. Then we establish analogs of Chevalley's formula for a family (Λi) 0≤i≤mn of subgroups of k$\times$ , prime to p, containing Ek , in which any x is norm of an ideal of kn (Theorem 4.9, Corollary 4.12). This family is attached to the subgroups of the classical filtration of the p-class group of kn giving the theoretical algorithm computing its order in mn steps. We show that mn ≥ (λ · n + µ · p n + ν)/vp(#Tk) and that the condition mn = O(1) (i.e., λ = µ = 0) depends essentially on the p-adic valuations, for p|p, of the Fermat quotient of xi ∈ Λi, so that Greenberg's conjecture seems strongly related to the (tricky) properties of Fermat quotients of suitable elements of k$\times$. A statistical analysis of these Fermat quotients (Section 7) shows that they follow natural probabilities, whatever the value of n, showing that, almost surely, λ = µ = 0 (see the main Heuristic 7.3). This would imply that for a proof of Greenberg's conjecture, some deep p-adic results are necessary before referring to the purely algebraic Iwasawa theory.
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