Approche $p$-adique de la conjecture de Greenberg pour les corps totalement réels ($p$-adic approach of Greenberg’s conjecture for totally real fields)

摘要

Let $k$ be a totally real number field ant let $k_infty $ be its cyclotomic $mathbb{Z}_p$-extension for a prime $p>2$. We give (Theorem 3.4) 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 $k_n$ ($n$th stage in $k_infty $) is equal to the order of the torsion group ${mathcal{T}}_k$, of the Galois group of the maximal Abelian $p$-ramified pro-$p$-extension of $k$ (Theorem 4.7), for all $n ge e$, where $p^e$ is the exponent of $U_k^*/ overline{E}_k$ (in terms of local and global units). Then we establish analogs of Chevalley’s formula using a family $(Lambda _i^n)_{0le i le m_n}$ of subgroups of $k^imes $ containing $E_k$, in which any $x$ is norm of an ideal of $k_n$. This family is attached to the classical filtration of the $p$-class group of $k_n$ defining the algorithm of computation of its order in $m_n$ steps. From this, we prove (Theorem 6.3) that $m_n ge (lambda cdot n + mu cdot p^n + u )/{v_p(#{mathcal{T}}_k)}$ and that the condition $m_n = O(1)$ (i.e., $lambda = mu =0$) essentially depends on the ${mathfrak{p}}$-adic valuations of the $rac{x^{p-1}-1}{p}$, $x in Lambda _i^n$, for ${mathfrak{p}} mid p$, so that Greenberg’s conjecture is strongly related to “Fermat quotients” in $k^imes $. Heuristics and statistical analysis of these Fermat quotients (Sections 6, 7, 8) show that they follow natural probabilities, linked to ${mathcal{T}}_k$ whatever $n$, suggesting that $lambda = mu =0$ (Heuristics 7.5, 7.6, 7.10).