On \(p\)-class groups and the Fontaine-Mazur conjecture
Let \(p\) be a prime number. Iwasawa theory gives a precise description of the growth behavior of \(p\)-class numbers, while there do exist \(\mathbb Z_{p}\)-extensions with unbounded \(p\)-class numbers. \textit{R. Greenberg}'s conjecture predicts that they should always stay bounded for the cyclotomic \(\mathbb Z_{p}\)-extensions of totally real number fields [Invent. Math. 47, 85--99 (1978; Zbl 0403.12004)]. Contrary to Iwasawa theory, which in general involves extensions with \(p\)-adic analytic Galois groups, there are analogous questions of unramified or more generally tamely ramified pro-\(p\)-extensions. \textit{J.-M. Fontaine} and \textit{B. Mazur} have a conjecture which is still open and says there exist no infinite tamely ramified pro-pextensions with \(p\)-adic analytic Galois group [in: Elliptic curves, modular forms, \& Fermat's last theorem. Proceedings ot the conference on elliptic curves and modular forms, Hong Kong, 1993. Cambridge, MA: International Press, 41--78 (1995; Zbl 0839.14011)]. Related to Fontaine and Mazur's conjucture, Stark proposes the following question: Let \(K'/K\) be an infinite unramified pro-\(p\)-extension of the number field \(k\). Are the \(p\)-class numbers unbounded along the finite subextensions of \(K'/K\) ? [\textit{F. Hajir}, J. Algebra 188, No. 1, 256--271 (1997; Zbl 0879.11069)]. Using group theory arguments, the author of the paper gives a positive answer to Stark's question. The author gives also an equivalent formulation of the Fontaine Mazur's conjecture for \(p\)-extensions unramified outside a finite set of primes not containing any prime above \(p\).
scientific article; zbMATH DE number 6379056