TY - JOUR
T1 - Uniform bound for the number of rational points on a pencil of curves
AU - Dimitrov, Vesselin
AU - Gao, Ziyang
AU - Habegger, Philipp
N1 - Publisher Copyright:
© 2021 The Author(s) 2018. Published by Oxford University Press. All rights reserved. For permissions, please e-mail: journals.permission@oup.com.
PY - 2021/1
Y1 - 2021/1
N2 - 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.
AB - 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.
KW - math.NT
KW - math.AG
KW - 11G30, 11G50, 14G05, 14G25
UR - http://www.scopus.com/inward/record.url?scp=85108805672&partnerID=8YFLogxK
U2 - 10.1093/imrn/rnz248
DO - 10.1093/imrn/rnz248
M3 - Article
VL - 2021
SP - 1138
EP - 1159
JO - International Mathematics Research Notices
JF - International Mathematics Research Notices
SN - 1073-7928
IS - 2
ER -