GREENBERG'S CONJECTURE AND UNITS IN MULTIPLE Zp-EXTENSIONS

Let A be the inverse limit of the p-part of the ideal class groups in a Zpextension K∞/K. Greenberg conjectures that if r is maximal, then A is pseudo-null as a module over the Iwasawa algebra Λ (that is, has codimension at least 2). We prove this conjecture in the case that K is the field of p-th roots of unity, p has index of irregularity 1, satisfies Vandiver’s conjecture, and satisfies a mild additional hypothesis on units. We also show that if K is the field of p-th roots of unity and r is maximal, Greenberg’s conjecture for K implies that the maximal p-ramified pro-pextension of K cannot have a free pro-p quotient of rank r unless p is regular (see also [LN]). Finally, we prove a generalization of a theorem of Iwasawa in the case r = 1 concerning the Kummer extension of K∞ generated by p-power roots of p-units. We show that the Galois group of this extension is torsion-free as a Λ-module if there is only one prime of K above p and K∞ contains all the p-power roots of unity. Let K be a number field, let p be an odd prime number, and let A(K) be the p-Sylow subgroup of the ideal class group of K. In 1956, Iwasawa introduced the idea of studying the behaviour of A(F ) as F varies over all intermediate fields in a Zp-extension K∞/K. Greenberg [G2] gives a comprehensive account of the subsequent development of Iwasawa theory; we recall here some of the basic ideas. The inverse limit A = lim ←−F A(F ) is a finitely generated torsion module over the Iwasawa algebra Λ = Zp[[Gal(K∞/K)]] = lim ←−F Zp[Gal(F/K)]. This algebra has a particularly simple structure: given a topological generator γ of Γ, there is an isomorphism Λ ' Zp[[T ]] taking 1+γ to T . Iwasawa showed that there is a pseudoisomorphism (a map with finite kernel and cokernel) from A to an elementary module E = ∑ i Λ/(fi); the power series f = ∏ i fi is the characteristic power series of A. In [G1], Greenberg conjectured that f = 1 when K is totally real. It is natural to consider the generalization of these ideas to a compositum of Zpextensions of K, that is, an extension K∞/K whose galois group Γ = Gal(K∞/K) is isomorphic to Zp for some positive integer r. The corresponding Iwasawa algebra, Λ = lim ←−F Zp[Gal(F/K)], is isomorphic to a power series ring in r variables over Zp. If K is totally real and satisfies Leopoldt’s conjecture, then the only Zp-extension is the cyclotomic one and Greenberg’s conjecture that f = 1 in this case is equivalent to the statement that A is finite, and hence has annihilator of height 2. More generally, Greenberg has made the following conjecture [G2]. This research was supported in part by an AMS Bicentennial Fellowship and a National Science Foundation grant DMS-9624219.