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

Abstract

This paper was motivated by a recent paper by Krumm and Pollack investigating modulo-$p$ behaviour 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 $\mathbb{Q}$. This complements and generalizes a similar result in the direction of the Malle conjecture by D\`ebes.