Sur une forme faible de la conjecture de Greenberg II

For a totally real number field which is abelian over Q, in which an odd prime [Formula: see text] is totally split, and which verifies certain rather mild conditions on the cohomology of the circular units, we show that a weak form of Greenberg’s conjecture for [Formula: see text] holds true. This fills a gap — pointed out by René Schoof — in a previous proof (see Sur la conjecture faible de Greenberg dans le cas abélien [Formula: see text]-décomposé, Int. J. Number Theory 2(1) (2006) 49–64), and also extends the original result (semi-simplicity is no longer required).

Cyclotomic units and class groups in p-extensions of real abelian fields

For a real abelian number field F and for a prime p we study the relation between the p-parts of the class groups and of the quotients of global units modulo cyclotomic units along the cyclotomic p-extension of F. Assuming Greenberg's conjecture about the vanishing of the λ-invariant of the extension, a map between these groups has been constructed by several authors, and shown to be an isomorphism if p does not split in F. We focus in the split case, showing that there are, in general, non-trivial kernels and cokernels.

On the Vanishing of μ-Invariants of Elliptic Curves over ℚ

Let E/ℚ be an elliptic curve with good ordinary reduction at a prime p > 2. It has a welldefined Iwasawa μ-invariant μ(E)p which encodes part of the information about the growth of the Selmer group ) as Kn ranges over the subfields of the cyclotomic Zp-extension K∞/ℚ. Ralph Greenberg has conjectured that any such E is isogenous to a curve E′ with μ(E′)p = 0. In this paper we prove Greenberg's conjecture for infinitely many curves E with a rational p-torsion point, p = 3 or 5, no two of our examples having isomorphic p-torsion. The core of our strategy is a partial explicit evaluation of the global duality pairing for finite flat group schemes over rings of integers.