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

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

2008-04-17

arxiv.org/abs/0804.2828v2

Mathematics

Number Theory

38 pages, completely revised version, inaccuracies and typos fixed

Scientific paper

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.

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