On the classical main conjecture for imaginary quadratic fields

TL;DR

This paper proves a key conjecture relating units and class groups in certain imaginary quadratic fields using Euler systems, with full results for p not in {2,3} and divisibility results otherwise.

Contribution

It establishes the classical main conjecture for imaginary quadratic fields for primes p not in {2,3} using Euler systems, extending previous partial results.

Findings

Characteristic ideals of global units modulo elliptic units and p-class groups coincide for p not in {2,3}.

For p in {2,3}, a divisibility relation is established up to a constant.

The approach employs Euler systems, building on Rubin's work.

Abstract

Let p be a prime number which is split in an imaginary quadratic field k. Let \mathfrak{p} be a place of k above p. Let k_\infty be the unique Z_p-extension of k which unramified outside of \mathfrak{p}, and let K_\intfy be a finite extension of k_\infty, abelian over k. In case p \notin {2,3}, we prove that the characteristic ideal of the projective limit of global units modulo elliptic units coincides with the characteristic ideal of the projective limit of the p-class groups. Our approach uses Euler systems, which were first used in this context by K.Rubin. If p \in {2,3}, we obtain a divisibility relation, up to a certain constant.