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

TL;DR

This paper investigates Greenberg's conjecture for totally real fields using p-adic methods, providing conditions for vanishing Iwasawa invariants, and analyzing Fermat quotients to support the conjecture's validity.

Contribution

It offers a new p-adic framework linking Iwasawa invariants, class groups, and Fermat quotients, advancing understanding of Greenberg's conjecture in the totally real case.

Findings

Sufficient conditions for lambda and mu invariants to be zero.

Tables of quadratic fields where lambda=mu=0.

Heuristic evidence supporting Greenberg's conjecture based on Fermat quotients.

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.