DC Field | Value | Language |
---|---|---|
dc.contributor.author | Konig, Joachim | ko |
dc.contributor.author | Legrand, Francois | ko |
dc.date.accessioned | 2022-06-03T05:00:21Z | - |
dc.date.available | 2022-06-03T05:00:21Z | - |
dc.date.created | 2022-06-03 | - |
dc.date.created | 2022-06-03 | - |
dc.date.issued | 2021-09 | - |
dc.identifier.citation | JOURNAL OF THE INSTITUTE OF MATHEMATICS OF JUSSIEU, v.20, no.5, pp.1455 - 1496 | - |
dc.identifier.issn | 1474-7480 | - |
dc.identifier.uri | http://hdl.handle.net/10203/296786 | - |
dc.description.abstract | We provide evidence for this conclusion: given a finite Galois cover f : X -> P-Q(1) of group G, almost all (in a density sense) realizations of G over 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 Q of given group and bounded discriminant. This widely extends a result of Granville on the lack of Q-rational points on quadratic twists of hyperelliptic curves over Q with large genus, under the abc-conjecture (a diophantine reformulation of the case G = Z/2Z 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 P-Q(1) of group G. We also introduce a local{global principle for specializations of Galois covers f : X -> P-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 P-Q(1). On the other hand, it allows to generate conditionally 'many' curves over Q failing the Hasse principle, thus generalizing a recent result of Clark and Watson devoted to the hyperelliptic case. | - |
dc.language | English | - |
dc.publisher | CAMBRIDGE UNIV PRESS | - |
dc.title | DENSITY RESULTS FOR SPECIALIZATION SETS OF GALOIS COVERS | - |
dc.type | Article | - |
dc.identifier.wosid | 000695227600003 | - |
dc.identifier.scopusid | 2-s2.0-85074522006 | - |
dc.type.rims | ART | - |
dc.citation.volume | 20 | - |
dc.citation.issue | 5 | - |
dc.citation.beginningpage | 1455 | - |
dc.citation.endingpage | 1496 | - |
dc.citation.publicationname | JOURNAL OF THE INSTITUTE OF MATHEMATICS OF JUSSIEU | - |
dc.identifier.doi | 10.1017/S1474748019000537 | - |
dc.contributor.localauthor | Konig, Joachim | - |
dc.contributor.nonIdAuthor | Legrand, Francois | - |
dc.description.isOpenAccess | N | - |
dc.type.journalArticle | Article | - |
dc.subject.keywordAuthor | Galois theory | - |
dc.subject.keywordAuthor | specializations | - |
dc.subject.keywordAuthor | the abc-conjecture | - |
dc.subject.keywordAuthor | the Malle conjecture | - |
dc.subject.keywordAuthor | the uniformity conjecture | - |
dc.subject.keywordAuthor | hyperelliptic and superelliptic curves | - |
dc.subject.keywordAuthor | rational points | - |
dc.subject.keywordAuthor | twisted covers | - |
dc.subject.keywordAuthor | the Hasse principle | - |
dc.subject.keywordPlus | QUADRATIC TWISTS | - |
dc.subject.keywordPlus | FIELDS | - |
dc.subject.keywordPlus | RANKS | - |
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