A note on asymptotically good extensions in which infinitely many primes split completely
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Archiv der Mathematik
A note on asymptotically good extensions in which infinitely many primes split completely Oussama Hamza and Christian Maire Abstract. Let p be a prime number, and let K be a number field. For p = 2, assume moreover that K is totally imaginary. In this note, we prove the existence of asymptotically good extensions L/K of cohomological dimension 2 in which infinitely many primes split completely. Our result is inspired by a recent work of Hajir, Maire, and Ramakrishna. Mathematics Subject Classification. 11R37, 11R29. Keywords. Pro-p extensions with restricted ramification, Asymptotically good extensions, Mild pro-p extensions.
Let K be a number field, and let L/K be an infinite unramified extension. Denote by SL/K the set of prime ideals of K that split completely in L/K. In [8], Ihara proved that logN (p) < ∞, N (p) p∈S L/K
where N (p) := |OK /p|; and this result raised the following interesting question: are there L/K for which SL/K is infinite? This question was recently answered in the positive by Hajir, Maire, and Ramakrishna in [7]. Infinite unramified extensions L/K are special cases of infinite extensions for which the root discriminants rdF := |DiscF |1/[F :Q] are bounded, where the field F ranges over the finite-dimensional subextensions of L/K and DiscF is the discriminant of F . Such extensions are called asymptotically good, and it is now The authors thank Karim Belabas, Baptiste Cercl´e, Farshid Hajir, and Alexander Schmidt for useful comments; and Philippe Lebacque and Jan Min´ aˇ c for their interests in this work. They also want to thank the anonymous referee for his/her careful reading of the paper. CM was partially supported by the ANR project FLAIR (ANR-17-CE40-0012), and by the EIPHI Graduate School (ANR-17-EURE-0002).
O. Hamza and C. Maire
Arch. Math.
well-known that in such extensions, the inequality of Ihara involving SL/K still holds (see for example [13,16]). Pro-p extensions of number fields with restricted ramification allow us to exhibit asymptotically good extensions. Let p be a prime number, and let S be a finite set of prime ideals of K coprime to p (more precisely, each p ∈ S is such that N (p) ≡ 1(mod p)); the set S is called tame. Let KS be the maximal pro-p extension of K unramified outside S, put GS := Gal(KS /K). In KS /K, the root discriminants are bounded by some constant depending on the discriminant of K and the norm of the places of S (see for example [6, Lemma 5]). Moreover, thanks to the Golod–Shafarevich criterion, it is wellknown that KS /K is infinite when |S| is large in comparison to the degree of K over Q (see for example [14, Chapter X, §10, Theorem 10.10.1]), and therefore asymptotically good. For instance, if p > 2, QS /Q is infinite when |S| ≥ 4. In [7], the authors showed that when S is large, there exists an infinite subextension L/K of KS /K for which the set SL/K is infinite, without providing any information on Gal(L/K). Here we prove: Theorem A. Let p be a prime number, and let K be a number field. For p = 2, assume that K is totally imaginary.
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