p-adic approach of Greenberg's conjecture (p-split totally real case)

Abstract

Let k be a totally real number field ant let k$\infty$ be its cyclotomic Zp-extension for a prime p\textgreater{}2. We give (Theorem 3.2) a sufficient condition of nullity of the Iwasawa invariants lambda, mu, when p totally splits in k, and we obtain important tables of quadratic fields and p for which we can conclude that lambda = mu=0.We show that the number of ambiguous p-classes of kn (nth stage in k$\infty$) is equal to the order of the torsion group T, of the Galois group of the maximal Abelian p-ramified pro-p-extension of k (Theorem 4.2), for all n \textgreater{}\textgreater{} e, where p^e is the exponent of U*/ adh(E) (in terms of local and global units of k). Then we establish analogs of Chevalley's formula using a family (Lambda\_i^n)\_{0$\le$i$\le$m\_n} of subgroups of k* containing E, in which any x is norm of an ideal of kn. This family is attached to the classical filtration of the p-class group of kn defining the algorithm of computation of its order in m\_n steps. From this, we prove (Theorem 6.1) that m\_n $\ge$ (lambda.n + mu.p^n + nu)/v\_p(T\_k) and that the condition m\_n = O(1) (i.e., lambda = mu=0) essentially depends on the P-adic valuations of the (x^(p-1)-1)/p, x in Lambda\_i^n, for P I p, so that Greenberg's conjecture is strongly related to "Fermat quotients" in k*. Heuristics and statistical analysis of these Fermat quotients (Sections 6, 7, 8) show that they follow natural probabilities, linked to T\_k whatever n, suggesting that lambda = mu=0 (Heuristics 7.1, 7.2, 7.3). This would imply that, for a proof of Greenberg's conjecture, some deep p-adic results (probably out of reach now), having some analogy with Leopoldt's conjecture, are necessary before referring to the sole algebraic Iwasawa theory.