To every abelian extension \(K\) of an imaginary quadratic field \(k\) and to every prime number \(p\) one may attach two Iwasawa modules: the projective limit of \(p\)-parts of class groups in the doubly-infinite tower \(K_\infty\) which is obtained by composing \(K\) with the unique \(\mathbb Z_p^2\)-extension of \(k\), and the projective limit of the quotient of units modulo elliptic units in the same tower. The 2-variable main conjecture then says that these two Iwasawa modules should have the same characteristic series. This main conjecture was proved, under some restrictions, in the ground-breaking paper of K. Rubin [Invent. Math. 103, No. 1, 25–68 (1991; Zbl 0737.11030)], using the machinery of Euler systems, which comes from work of Kolyvagin and Thaine.
Subsequently some of the restrictions were removed. One of the remaining restrictions says that \(p\) should not divide the degree \([K:k]\). This hypothesis is now eliminated in the paper under review, which however needs to make the assumptions \(p\geq 5\) and \(p\) splits in the imaginary quadratic field \(k\). The ingenuous technique of Euler systems is used in many papers on the main conjecture, and the present paper is no exception. An important technical point in working with primes \(p\) that divide \([K:k]\) seems to be the following: One proves the desired equalities of characteristic series in a \(\mathbb Q\)-tensored version, that is, only up to a priori unknown \(p\)-power factors, but one is able to remove them in the end just by showing that there are none of these factors on either side. This comes under the general label “\(\mu=0\)”. The author is able to draw on many results that were established previously by K. Rubin [Invent. Math. 93, No. 3, 701–713 (1988; Zbl 0673.12004)], R. Gillard [J. Reine Angew. Math. 358, 76–91 (1985; Zbl 0551.12011)] and others.
The Main Conjecture has been crucial in the theory of elliptic curves with CM, so it is nice to see that another of the few remaining cases is now settled.