On the equivariant and the non-equivariant main conjecture for imaginary quadratic fields

TL;DR

This paper proves the Iwasawa main conjecture for imaginary quadratic fields for all primes, enhances previous results, and establishes the equivariant conjecture under specific conditions, advancing understanding of number field invariants.

Contribution

It extends the proof of the Iwasawa main conjecture to all primes for imaginary quadratic fields and derives the equivariant version when the -invariant vanishes.

Findings

Main conjecture proved for all primes p

Equivariant main conjecture established under -invariant vanishing

Generalizes previous results by Rubin and Gillard

Abstract

The Iwasawa main conjecture fields has been an important tool to study the arithmetic of special values of $L$-functions of Hecke characters of imaginary quadratic fields. To obtain the finest possible invariants it is important to know the main conjecture for all prime numbers $p$ and also to have an equivariant version at disposal. In this paper we first prove the main conjecture for imaginary quadratic fields for all prime numbers $p$, improving earlier results by Rubin. From this we deduce the equivariant main conjecture in the case that a certain $\mu$-invariant vanishes. For prime numbers $p\nmid 6$ which split in $K$, this is a theorem by a result of Gillard.