Unramified subextensions of ray class towers
Let \(\ell\) be a prime number, \(k\) be a number field and \(T\) be a finite set of places of \(k\). Let \(k_T\) be the maximal \(\ell\)-extension of \(k\) unramified outside \(T\) and set \(G_{k,T}= \text{Gal}(k_T/k)\). If all finite places of \(T\) have absolute norm congruent to 1 modulo \(\ell\), we write \((T,\ell)=1\). An extension \(L/F\) is said to be a subextension of \(k_T/k\) provided \(k\subseteq F\subset L\subseteq k_T\) with \([F:k]< \infty\). The primary focus of the article is the following question: Suppose that \((T,\ell)=1\) and \(k_T/k\) is infinite. Does \(k_T/k\) admit an infinite unramified subextension? In Theorem 4.1 the authors show that the answer is yes provided that \(|T|\geq 2u+ 4+ 2\sqrt{\ell u+2}+ 2\sqrt{u+1}\) where \(u= r_1+r_2\). Here \(r_1\) and \(2r_2\) denote the number of real and imaginary embeddings of \(k\) into the field of complex numbers. Also, when \(k=Q\), several infinite families of examples are given where \(T= \{p,q,\infty\}\) where \(p\) and \(q\) are odd primes and the question has a positive answer. A specific example is \(T= \{17,101,\infty\}\) with \(\ell=2\). In the final section of the article, the authors formulate a hypothesis regarding subgroup growth of \(G_{k,T}\). A positive answer to this question would not only give a positive answer to the above question, but also positive answers to conjectures of \textit{J.-M. Fontaine} and \textit{B. Mazur} [Proceedings of Conference in Hong Kong, International Press, Ser. Number Theory 1, 41-78 (1995; Zbl 0839.14011)] and \textit{N. Boston} [J. Number Theory 42, 285-291 (1992; Zbl 0768.11044) and 75, 161-169 (1999; Zbl 0928.11050)].
scientific article; zbMATH DE number 1750367
scientific article