Let K be an imaginary quadratic field in which the prime p splits, say \(p={\mathfrak pp}\bar {\;}\), and let \(K_{\infty}/K\) be the unique \({\mathbb{Z}}_ p\)-extension unramified outside \({\mathfrak p}\). Let F be an abelian extension of K and put \(F_{\infty}:=FK_{\infty}\). Finally, let \(M_{\infty}\) be the maximal abelian p-extension of \(F_{\infty}\) unramified outside \({\mathfrak p}\) and set \(X_{\infty,F}:=Gal(M_{\infty}/F_{\infty}).\)
The author proves a result which can be viewed as an analogue of the classical Hurwitz genus formula, relating the \(\lambda\)-invariants of the Iwasawa modules \(X_{\infty,F}\) and \(X_{\infty,H}\), where \(H\supset F\supset K\) is a tower of abelian extensions such that \(| Gal(H/F)|\) is a p-power and \(p\nmid | Gal(F/K)|\). This result is motivated by a corresponding formula of Y. Kida [J. Number Theory 12, 519-528 (1980; Zbl 0455.12007)] in the cyclotomic case, and the proof follows the one by W. M. Sinnott [Compos. Math. 53, 3-17 (1984; Zbl 0545.12011)] of Kida’s formula using p-adic L-functions.