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arXiv · 1904.07268

Uniform bound for the number of rational points on a pencil of curves

Vesselin Dimitrov·Ziyang Gao·Philipp Habegger
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Abstract

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.

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Vesselin Dimitrov, Ziyang Gao, Philipp Habegger. 2019-04-15. Uniform bound for the number of rational points on a pencil of curves. https://arxiv.org/abs/1904.07268

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