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

Publikation: Beitrag in Fachzeitschrift › Artikel › Forschung › Peer-Review

Autorschaft

  • Vesselin Dimitrov
  • Ziyang Gao
  • Philipp Habegger

Externe Organisationen

  • University of Cambridge
  • Centre national de la recherche scientifique (CNRS)
  • Universität Basel
Forschungs-netzwerk anzeigen

Details

OriginalspracheEnglisch
Seiten (von - bis)1138–1159
Seitenumfang22
FachzeitschriftInternational Mathematics Research Notices
Jahrgang2021
Ausgabenummer2
Frühes Online-Datum10 Dez. 2019
PublikationsstatusVeröffentlicht - Jan. 2021
Extern publiziertJa

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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Uniform bound for the number of rational points on a pencil of curves. / Dimitrov, Vesselin; Gao, Ziyang; Habegger, Philipp.
in: International Mathematics Research Notices, Jahrgang 2021, Nr. 2, 01.2021, S. 1138–1159.

Publikation: Beitrag in Fachzeitschrift › Artikel › Forschung › Peer-Review

Dimitrov V, Gao Z, Habegger P. Uniform bound for the number of rational points on a pencil of curves. International Mathematics Research Notices. 2021 Jan;2021(2):1138–1159. Epub 2019 Dez 10. doi: 10.1093/imrn/rnz248
Dimitrov, Vesselin ; Gao, Ziyang ; Habegger, Philipp. / Uniform bound for the number of rational points on a pencil of curves. in: International Mathematics Research Notices. 2021 ; Jahrgang 2021, Nr. 2. S. 1138–1159.
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