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

@article{Dimitrov2019UniformBF,
  title={Uniform bound for the number of rational points on a pencil of curves},
  author={Vesselin Dimitrov and Ziyang Gao and Philipp Habegger},
  journal={arXiv: Number Theory},
  year={2019}
}
  • Vesselin Dimitrov, Ziyang Gao, Philipp Habegger
  • Published 2019
  • Mathematics
  • arXiv: Number Theory
  • 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… CONTINUE READING
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