On the Stark-Shintani conjecture and cyclotomic \({\mathbb{Z}}_ p\)-extensions of class fields over real quadratic fields (Q800420)

Let \(F\subset {\mathbb{R}}\) be a real quadratic field and let M be, in the ray-class field \(M_{{\mathfrak f}} mod {\mathfrak f}\), an abelian extension of F of conductor \({\mathfrak f}\) in which exactly one infinite prime (corresponding to \(F\hookrightarrow {\mathbb{R}})\) splits. For any c of the narrow ray class group mod \({\mathfrak f}\), let \(\zeta_ F(s,c)\) be the usual partial zeta function, and consider \(X_{{\mathfrak f}}(c)=\exp (\zeta_ F\!'(0,c)-\zeta_ F\!'(0,c\nu)), \nu +1\in {\mathfrak f},\quad\nu \gg 0,\) and \(X_{{\mathfrak f}}(c,M)=\prod_{g}X_{{\mathfrak f}}(cg),\quad g\in Gal(M_{{\mathfrak f}}/M)\) (the Stark-Shintani invariant). Then the author shows, among other results, that if the previous M is quadratic over F, the following is true: Let \(M_{\infty}=\cup_{n}M_ n\) be the cyclotomic \({\mathbb{Z}}_ p\)-extension of M (\(p\neq 2\), p does not split in M/F), then the \(X_{{\mathfrak f}_ n}(c,M_ n)\) are units of \(M_ n\) for all n (with evident notations). - It is only known that a power of the \(X_{{\mathfrak f}_ n}\) are units, when the maximal real subfield \(M^+\) of M is abelian over \({\mathbb{Q}}\) [a part of the Stark-Shintani conjectures, proved by \textit{T. Shintani}, ibid. 30, 139-167 (1978; Zbl 0392.12009)]. The paper is mainly based on this paper of Shintani and some papers of Stark.