On iterated extensions of number fields arising from quadratic polynomial maps

Abstract

A post-critically finite rational map ϕ of prime degree p and a base point β yield a tower of finitely ramified iterated extensions of number fields, and sometimes provide an arboreal Galois representation with a p-adic Lie image. In this paper, we take ϕ to be the monic Chebyshev polynomial x2−2, and we examine the size of the 2-part of the ideal class group of extensions in the resulting tower. In some cases, we prove an analogue of Greenberg's conjecture from Iwasawa theory. A key tool is a general theorem on p-indivisibility of class numbers of relative cyclic extensions of degree p2.