On the mod-p distribution of discriminants of G-extensions

Abstract

This paper was motivated by a recent paper by Krumm and Pollack ([Twists of hyperelliptic curves by integers in progressions modulo p, preprint (2018); https://arXiv.org/abs/1807.00972]) investigating modulo-p behavior of quadratic twists with rational points of a given hyperelliptic curve, conditional on the abc-conjecture. We extend those results to twisted Galois covers with arbitrary Galois groups. The main point of this generalization is to interpret those results as statements about the sets of specializations of a given Galois cover under restrictions on the discriminant. In particular, we make a connection with existing heuristics about the distribution of discriminants of Galois extensions such as the Malle conjecture: our results show in a precise sense the non-existence of "local obstructions" to such heuristics, in many cases essentially only under the assumption that G occurs as the Galois group of a Galois cover defined over Q. This complements and generalizes a similar result in the direction of the Malle conjecture by Debts ([On the Malle conjecture and the self-twisted cover, Israel J. Math. 218(1) (2017) 101-131]).