Global Units Modulo Elliptic Units and 2-Ideal Class Groups

TL;DR

This paper investigates the relationship between p-class groups and units modulo elliptic units in certain infinite extensions of imaginary quadratic fields, establishing invariants and ideal coincidences.

Contribution

It proves the equality of Iwasawa invariants and characteristic ideals for p-class groups and units modulo elliptic units in specific Z_p-extensions.

Findings

Same -invariant and -invariant for both groups.

Characteristic ideals of the projective limits coincide up to a constant.

Results hold for p=2,3 in imaginary quadratic fields.

Abstract

Let p\in\{2,3\}, and let k be an imaginary quadratic field in which p decomposes into two distinct primes \mathfrak{p} and \bar{\mathfrak{p}}. Let k_\infty be the unique Z_p-extension of k which is unramified outside of \mathfrak{p}, and let K_\infty be a finite extension of k_\infty, abelian over k. We prove that in K_\infty, the projective limit of the p-class group and the projective limit of units modulo elliptic units share the same \mu-invariant and the same \lambda-invariant. Then we prove that up to a constant, the characteristic ideal of the projective limit of the p-class group coincides with the characteristic ideal of the projective limit of units modulo elliptic units.