# Approche p-adique de la conjecture de Greenberg (cas totalement réel p-décomposé)

Abstract : 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.
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Cited literature [41 references]

https://hal.archives-ouvertes.fr/hal-01404933
Contributor : Georges Gras <>
Submitted on : Saturday, April 1, 2017 - 1:22:06 PM
Last modification on : Monday, April 9, 2018 - 12:20:07 PM
Document(s) archivé(s) le : Sunday, July 2, 2017 - 12:54:10 PM

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• HAL Id : hal-01404933, version 3
• ARXIV : 1611.09592

### Citation

Georges Gras. Approche p-adique de la conjecture de Greenberg (cas totalement réel p-décomposé). 2017. ⟨hal-01404933v3⟩

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