Invariants and coinvariants of semilocal units modulo elliptic units

TL;DR

This paper investigates the structure of semi-local units modulo elliptic units in certain number fields, proving finiteness properties of their invariants and coinvariants in the context of Iwasawa theory.

Contribution

It establishes the finiteness of modules of invariants and coinvariants of semi-local units modulo elliptic units in specific Z_p-extensions of imaginary quadratic fields.

Findings

Modules of invariants are finite.

Modules of coinvariants are finite.

Results contribute to understanding Iwasawa modules in number theory.

Abstract

Let p be a prime number, and let k be an imaginary quadratic field in which p decomposes into two 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. Let U_\infty/C_\infty be the projective limit of principal semi-local units modulo elliptic units. We prove that the various modules of invariants and coinvariants of U_\infty/C_\infty are finite.