On the Malle conjecture and the self-twisted cover

2014/04/15 by Dèbes, Pierre
#11R44 #11Rxx #12E25 #12F12 #12Fxx #14H30 #FOS: Mathematics #Number Theory (math.NT) #Primary 11R58 #Secondary 14Gxx

paper · doi:10.48550/arxiv.1404.4074

Abstract

We show that for a large class of finite groups G, the number of Galois extensions E/Q of group G and discriminant |d_E|≤ y grows like a power of y (for some specified exponent). The groups G are the regular Galois groups over Q and the extensions E/Q that we count are obtained by specialization from a given regular Galois extension F/Q(T). The extensions E/Q can further be prescribed any unramified local behavior at each suitably large prime p≤ log (y)/δ for some δ≥ 1. This result is a step toward the Malle conjecture on the number of Galois extensions of given group and bounded discriminant. The local conditions further make it a notable constraint on regular Galois groups over Q. The method uses the notion of self-twisted cover that we introduce.

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