The Main Conjecture(s) of Iwasawa theory are an essential tool for studying the arithmetical properties of special values of \(L\)-functions attached to motives. In this paper, the authors treat the \(MC\) for an imaginary quadratic field \(K\) both in the character-wise and the equivariant (i.e., taking into account the Galois action of an abelian extension \(F/K)\) setting. We don’t recall the statements of the \(MC's,\) except to say that they relate “above \(K_\infty\)” certain \(p\)-adic analytic objects (here, “zeta elements” built from Euler systems of elliptic units) to certain algebraic objects (here, the cohomology groups of certain perfect complexes \(\displaystyle{\mathop R_{=}}\;\Gamma ({\mathcal O}_K [1/p], \Lambda (\eta) (1))\) and \(\displaystyle{\mathop R_{=}}\;\Gamma ({\mathcal O}_K [1/p\;{\mathfrak f}], \;\Omega (1)).\)
Notation : \(K_\infty\) = the maximal multiple \({\mathbb{Z}}_p\)-extension of \(K,\) \(\Gamma = \text{Gal} (K_\infty/K) \simeq {\mathbb{Z}}^2_p\), \(\Lambda = \Lambda (\Gamma) \simeq {\mathbb{Z}}_p [\![ S, T ]\!]\), \({\mathfrak{f}}\) = a non zero ideal of \({\mathcal O}_K\), \(\eta\) = a character \(\text{Gal}(K({\mathfrak f}) / K) \to {\mathbb{C}}^\ast\), \({\mathcal G}_{\mathfrak f} = \text{Gal}(K ({\mathfrak f} p^\infty) / K)\), \(\Delta\) = the torsion subgroup of \({\mathcal G}_{\mathfrak f}\), \(\Omega = \Lambda ({\mathcal G}_{\mathfrak f}) \simeq {\mathbb{Z}}_p [\Delta] \;[\![ S, T]\!]\).
The character-wise (resp. equivariant) \(MC\) will be called \(\Lambda\)-\(MC\) (resp. \(\Omega\)-MC). The \(\Lambda\)-\(MC\) (in a more classical language) was proved by K. Rubin [Invent. Math. 103, No. 1, 25–68 (1991; Zbl 0737.11030)], but modulo a semi-simplicity assumption, i.e., \(p\) does not divide the orders of the characters involved. In the first part of the present paper, the authors reprove the \(\Lambda\)-\(MC\) for all primes. To get rid of the problem of non semi-simplicity, they follow the scheme of proof developed by A. Huber and G. Kings [Duke Math. J. 119, No. 3, 393–464 (2003; Zbl 1044.11095)]: instead of decomposing the classical Iwasawa modules under character-wise projectors (which may not be integral), they use Galois cohomology with coefficients in the Galois representations defined by the characters. Following K. Kato’s treatment of the Euler system of elliptic units [Cohomologies \(p\)-adiques et applications arithmétiques. III. Paris: Société Mathématique de France. Astérisque 295, 117–290 (2004; Zbl 1142.11336)] and “going up in the direction of \(K_\infty\)”, the usual machinery yields “half” of the \(MC,\) namely certain divisibility relations. To get the reverse divisibility, the authors use a precise relation between elliptic units and zeta elements to reduce the problem to the Tamagawa Number Conjecture at \(s = 0\) (a generalization of the analytic class number formula), proved for abelian fields e.g. in Huber-Kings [op. cit.].
In the second part of the paper, the authors follow the scheme of proof of M. Witte over \({\mathbb Q}\) [Acta Arith. 122, No. 3, 275–296 (2006; Zbl 1098.11055)] to derive the \(\Omega\)-\(MC\) from the \(\Lambda\)-\(MC\) by assuming the (now familiar) “\(\mu = 0\)” hypothesis, which implies the vanishing of the localized \(H^2\) at the so called singular primes. Note that a result of R. Gillard [J. Reine Angew. Math. 358, 76–91 (1985; Zbl 0551.12011)] implies “\(\mu = 0\)” for all primes \(p\nmid 6\) which split in \(K\).