We provide evidence for this conclusion: given a finite Galois cover $f:X\rightarrow \mathbb{P}_{\mathbb{Q}}^{1}$ 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 $\mathbb{P}_{\mathbb{Q}}^{1}$ of group $G$. We also introduce a local–global principle for specializations of Galois covers $f:X\rightarrow \mathbb{P}_{\mathbb{Q}}^{1}$ 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 $\mathbb{P}_{\mathbb{Q}}^{1}$. 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.

中文翻译：

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专业化 GALOIS 盖板的密度结果

我们为这个结论提供了证据：给定有限的伽罗瓦覆盖$f:X\rightarrow \mathbb{P}_{\mathbb{Q}}^{1}$组的$G$，几乎所有（在密度意义上）实现$G$超过$\mathbb{Q}$不作为专业化出现$f$. 我们证明，如果分支点的数量$f$足够大，在 abc 猜想下，并且可能是 Malle 猜想预测的 Galois 扩展数的下限$\mathbb{Q}$给定组和有界判别式。这广泛扩展了 Granville 缺乏的结果$\mathbb{Q}$-超椭圆曲线二次扭曲上的有理点$\mathbb{Q}$具有大属，在 abc 猜想下（该案例的丢番图重新表述$G=\mathbb{Z}/2\mathbb{Z}$我们的结果）。作为进一步的证据，我们展示了一些有限群$G$对于几乎所有的覆盖，上述结论无条件成立$\mathbb{P}_{\mathbb{Q}}^{1}$组的$G$. 我们还为伽罗瓦覆盖的专业化引入了一个局部-全局原则$f:X\rightarrow \mathbb{P}_{\mathbb{Q}}^{1}$并表明它经常失败，如果$f$在 abc 猜想下，有阿贝尔伽罗瓦群和足够多的分支点。一方面，这样的局部-全局结论强调了伽罗瓦覆盖的专业化集的“小”$\mathbb{P}_{\mathbb{Q}}^{1}$. 另一方面，它允许有条件地生成“许多”曲线$\mathbb{Q}$不符合 Hasse 原理，从而概括了 Clark 和 Watson 最近致力于超椭圆情况的结果。