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

## TL;DR

This paper generalizes results on the distribution of discriminants of Galois extensions modulo p, connecting them with heuristics like the Malle conjecture and exploring the absence of local obstructions under certain conditions.

## Contribution

It extends previous work on quadratic twists to arbitrary Galois groups, linking discriminant distribution to Galois cover specializations and heuristics.

## Key findings

- Shows non-existence of local obstructions to Malle's conjecture in many cases
- Connects discriminant distribution of Galois extensions with Galois cover specializations
- Generalizes results to arbitrary Galois groups and covers

## 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.