Uniformity in Mordell–Lang for curves
| Autor: | Philipp Habegger, Vesselin Dimitrov, Ziyang Gao |
|---|---|
| Přispěvatelé: | Department of Mathematics [University of Toronto], University of Toronto, Institut de Mathématiques de Jussieu - Paris Rive Gauche (IMJ-PRG (UMR_7586)), Sorbonne Université (SU)-Centre National de la Recherche Scientifique (CNRS)-Université de Paris (UP), University of Basel (Unibas) |
| Jazyk: | angličtina |
| Rok vydání: | 2021 |
| Předmět: |
Pure mathematics
Rank (linear algebra) Mathematics::Number Theory Mordell-Lang 01 natural sciences symbols.namesake Mathematics (miscellaneous) Mathematics::Algebraic Geometry Genus (mathematics) 0103 physical sciences 0101 mathematics [MATH]Mathematics [math] Mathematics Conjecture Degree (graph theory) 010102 general mathematics Algebraic number field uniformity rational points Bounded function Jacobian matrix and determinant symbols Torsion (algebra) height inequality 010307 mathematical physics Statistics Probability and Uncertainty |
| Zdroj: | Annals of Mathematics Annals of Mathematics, Princeton University, Department of Mathematics, 2021, 194 (1), pp.237-298. ⟨10.4007/annals.2021.194.1.4⟩ |
| ISSN: | 0003-486X |
| Popis: | International audience; Consider a smooth, geometrically irreducible, projective curve of genus g ≥ 2 defined over a number field of degree d ≥ 1. It has at most finitely many rational points by the Mordell Conjecture, a theorem of Faltings. We show that the number of rational points is bounded only in terms of g, d, and the Mordell-Weil rank of the curve's Jacobian, thereby answering in the affirmative a question of Mazur. In addition we obtain uniform bounds, in g and d, for the number of geometric torsion points of the Jacobian which lie in the image of an Abel-Jacobi map. Both estimates generalize our previous work for 1-parameter families. Our proof uses Vojta's approach to the Mordell Conjecture, and the key new ingredient is the generalization of a height inequality due to the second-and third-named authors. |
| Databáze: | OpenAIRE |
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