THEOREM ON THE EXCLUSION OF HIT TRIPLES AND ABC CONJECTURE

The proof of Theorem on the exclusion of hit triples and conjecture is given by the method of mathematical induction.

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

This paper was motivated by a recent paper by Krumm and Pollack ([Twists of hyperelliptic curves by integers in progressions modulo [Formula: see text], preprint (2018); https://arXiv.org/abs/1807.00972] ) investigating modulo-[Formula: see text] 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 [Formula: see text] occurs as the Galois group of a Galois cover defined over [Formula: see text]. This complements and generalizes a similar result in the direction of the Malle conjecture by Dèbes ([On the Malle conjecture and the self-twisted cover, Israel J. Math. 218(1) (2017) 101–131]).

DENSITY RESULTS FOR SPECIALIZATION SETS OF GALOIS COVERS

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.