Uniform Bound for the Number of Rational Points on a Pencil of Curves
| Autor: | Vesselin Dimitrov, Philipp Habegger, Ziyang Gao |
|---|---|
| Rok vydání: | 2019 |
| Předmět: |
Pure mathematics
Conjecture Mathematics - Number Theory Mathematics::Number Theory General Mathematics 010102 general mathematics 02 engineering and technology Algebraic number field 11G30 11G50 14G05 14G25 021001 nanoscience & nanotechnology 01 natural sciences Mathematics - Algebraic Geometry symbols.namesake Mathematics::Algebraic Geometry Bounded function Jacobian matrix and determinant FOS: Mathematics symbols Torsion (algebra) Number Theory (math.NT) 0101 mathematics 0210 nano-technology Algebraic Geometry (math.AG) Pencil (mathematics) Mathematics |
| Zdroj: | International Mathematics Research Notices. 2021:1138-1159 |
| ISSN: | 1687-0247 1073-7928 |
| Popis: | Consider a one-parameter family of smooth, irreducible, projective curves of genus $g\ge 2$ defined over a number field. Each fiber contains 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 the family and the Mordell--Weil rank of the fiber's Jacobian. Our proof uses Vojta's approach to the Mordell Conjecture furnished with a height inequality due to the second- and third-named authors. In addition we obtain uniform bounds for the number of torsion in the Jacobian that lie each fiber of the family. Minor revisions. Accepted to IMRN. Comments are welcome |
| Databáze: | OpenAIRE |
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