Global units modulo elliptic units and ideal class groups (Q2880323)

Let \(p\) be a prime number, and let \(k\) be an imaginary quadratic number field such that \(p\) splits in \(k\). Let \({\mathfrak p}\) be a prime divisor of \(k\) dividing \(p\). Let \(k_{\infty}\) be the \({\mathbb Z}_p\)--extension of \(k\) which is unramified outside of \({\mathfrak p}\) and let \(K_{\infty}\) be a finite extension of \(k_{\infty}\), abelian over \(k\). Set \(G_{\infty}=\roman{Gal} (K_{\infty}/k)\cong G\times \Gamma\) where \(G\) is the torsion of \(G_{\infty}\) and \(\Gamma \cong {\mathbb Z}_p\). We have that when \(p^n\) is larger that the order of the \(p\)--part of \(G\), \(\Gamma_n:= \Gamma^{p^n}\) does not depend on the choice of \(\Gamma\). Let \(K_n\) be the fixed field of \(\Gamma_n\). Let \({\mathcal E}_n:={\mathbb Z}_p \otimes_{ \mathbb Z} {\mathcal O}_{K_n}^{\ast}\) where \({\mathcal O}_{K_n}\) is the ring of integers of \(K_n\), \({\mathcal C}_n= {\mathbb Z}_p \otimes_{\mathbb Z} C_{K_n}\) where \(C_{K_n}\) is the ideal class group of \(K_n\). Let \({\mathcal U}_{n}\) be the group of semilocal units of \(K_n\), \(A_n\) the \(p\)--Sylow subgroup of \(C_{K_n}\) and \(B_n\) the Galois group of the maximal abelian extension of \(K_n\) unramified outside the prime divisors dividing \({\mathfrak p}\). Finally, let \({\mathcal E}_{\infty}\), \({\mathcal C}_{\infty}\), \({\mathcal U}_{ \infty}\), \(A_{\infty}\) and \(B_{\infty}\) be the respective projective limits, which are \(\Lambda^{\prime}:={\mathbb Z}_p [[\Gamma]]\)--modules in a natural way.NEWLINENEWLINEThe main result of this paper is that the \(\Lambda^{\prime}\)--modules \({\mathcal E}_{\infty}/{\mathcal C}_{\infty}\) and \(A_{\infty}\) share the same \(\lambda\) and \(\mu\) invariants and that the \(\Lambda^{\prime}\)--modules \({\mathcal U}_{\infty}/ {\mathcal C}_{\infty}\) and \(B_{\infty}\) share the same \(\lambda\) and \(\mu\) invariants. This is Theorem 1.1.NEWLINENEWLINETheorem 1.1 was proved by \textit{E. de Shalit} [Iwasawa theory of elliptic curves with complex multiplication. \(p\)-adic \(L\) functions. Perspectives in Mathematics, Vol. 3. Boston etc.: Academic Press (1987; Zbl 0674.12004)] in the case \(K_{\infty}=\cup_{n=0}^{\infty} k({\mathfrak f}{\mathfrak p}^n)\) where \(k({\mathfrak f}{\mathfrak p}^n)\) is the ray class field of \(k\) (\(\bmod (\mathfrak {fp}^n)\)) for some nonzero ideal \({\mathfrak f}\) of \({\mathcal O}_k\). The proof of de Shalit relies on \textit{R. Gillard}'s Theorem on the nullity of the \(\mu\)-invariant of \(B_{\infty}\) [J. Reine Angew. Math. 358, 76--91 (1985; Zbl 0551.12011)], which does not cover the case \(p\in\{2,3\}\). The proof provided here is inspired by the cyclotomic version of Theorem 1.1 proved by \textit{J.-R. Belliard} [Can. J. Math. 61, No. 3, 518--533 (2009; Zbl 1205.11115)] and it is valid for every prime.NEWLINENEWLINEAn important consequence of Theorem 1.1 is a formulation of the main conjecture that was still unproved for \(p\in\{2,3\}\).