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Published in: Annales Mathématiques Blaise Pascal, 24(2) (2017), 235--291.; Let k be a totally real number field ant let k∞ be its cyclotomic Zp-extension for a prime p>2. We give (Theorem 3.2) 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∞) 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 >> 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≤i≤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 ≥ (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.

Dans cet article, nous nous intéressons aux dimensions de certaines variétés qui ont été introduites récemment par Kisin pour démontrer la modularité de certaines représentations galoisiennes. Nous étudions plus spécialement un cas particulier pour lequel nous donnons une estimation de la dimension en question, puis, en nous basant sur ce résultat, nous énonçons une conjecture dans le cas général. In this paper, we study dimensions of some varieties, that were introduced recently by Kisin in order to prove modularity of some Galois representations. In fact, we mainly consider a special case for which we obtain an estimation of the dimension we are interested in. Then, based on this result, we state a conjecture for the general case.

This article completes the results of our previous paper [B.91] on interaction of consecutive strata for the vanishing cycles. In the same context that loc. cit., we treat here the tangling phenomenon for a given cohomology class of the Milnor’s fiber of f at the origin in the case where its nilpotency order for the given eigenvalue e −2iπ.u � of the monodromy is not necessarily bigger or equal than the nilpotency order of the monodromy for this eigenvalue at the generic point of the big stratum. We show that this tangling phenomenon can always be detected on the poles of the meromorphic extension of the distribution R X |f | 2λ ˜.

We transpose Delon’s analysis of types in valued fields to unramified henselian valued fields of mixed characteristic, by using coefficient maps of order n. This yields the Ax-Kochen-Ershov transfer principle for the independence property in this class of valued fields.