# Density results for specialization sets of Galois covers

## TL;DR

This paper investigates the rarity of realizing certain Galois groups as specializations of covers over , showing that, under conjectures like abc and Malle, most such realizations do not occur, and explores local-global principles and failures of the Hasse principle.

## Contribution

It extends Granville's result to general Galois groups under conjectures, introduces a local-global principle for Galois cover specializations, and constructs new examples of curves failing the Hasse principle.

## Key findings

- Most Galois groups do not appear as specializations of given covers under conjectures.
- A local-global principle for Galois cover specializations often fails with many branch points.
- Constructs many curves over  that fail the Hasse principle conditionally.

## Abstract

We provide evidence for this conclusion: given a finite Galois cover $f: X \rightarrow \mathbb{P}^1_\mathbb{Q}$ of group $G$, almost all (in a density sense) realizations of $G$ over $\mathbb{Q}$ do not occur as specializations of $f$. We show that this holds if the number of branch points of $f$ is sufficiently large, under the abc-conjecture and, possibly, the lower bound predicted by the Malle conjecture for the number of Galois extensions of $\mathbb{Q}$ of given group and bounded discriminant. This widely extends a result of Granville on the lack of $\mathbb{Q}$-rational points on quadratic twists of hyperelliptic curves over $\mathbb{Q}$ with large genus, under the abc-conjecture (a diophantine reformulation of the case $G=\mathbb{Z}/2\mathbb{Z}$ of our result). As a further evidence, we exhibit a few finite groups $G$ for which the above conclusion holds unconditionally for almost all covers of $\mathbb{P}^1_\mathbb{Q}$ of group $G$. We also introduce a local-global principle for specializations of Galois covers $f: X \rightarrow \mathbb{P}^1_\mathbb{Q}$ and show that it often fails if $f$ has abelian Galois group and sufficiently many branch points, under the abc-conjecture. On the one hand, such a local-global conclusion underscores the "smallness" of the specialization set of a Galois cover of $\mathbb{P}^1_\mathbb{Q}$. On the other hand, it allows to generate conditionally "many" curves over $\mathbb{Q}$ failing the Hasse principle, thus generalizing a recent result of Clark and Watson devoted to the hyperelliptic case.