Abstract
Consider a smooth, geometrically irreducible, projective curve of genus $g\ge 2$ defined over a number field of degree $d \ge 1$. It has 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 $g$, $d$, and the Mordell–Weil rank of the curve’s Jacobian, thereby answering in the affirmative a question of Mazur. In addition we obtain uniform bounds, in $g$ and $d$, for the number of geometric torsion points of the Jacobian which lie in the image of an Abel–Jacobi map. Both estimates generalize our previous work for one-parameter families. Our proof uses Vojta’s approach to the Mordell Conjecture, and the key new ingredient is the generalization of a height inequality due to the second- and third-named authors.
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author = {Stoll, Michael},
title = {Uniform bounds for the number of rational points on hyperelliptic curves of small {M}ordell-{W}eil rank},
journal = {J. Eur. Math. Soc. (JEMS)},
fjournal = {Journal of the European Mathematical Society (JEMS)},
volume = {21},
year = {2019},
number = {3},
pages = {923--956},
issn = {1435-9855},
mrclass = {11G30 (14G05 14H25 14H40)},
mrnumber = {3908770},
mrreviewer = {Sajad Salami},
doi = {10.4171/JEMS/857},
url = {https://doi.org/10.4171/JEMS/857},
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series = {Cambridge Stud. Adv. Math.},
volume = {76},
note = {translated from the French original by Leila Schneps},
publisher = {Cambridge Univ. Press, Cambridge},
year = {2002},
pages = {x+322},
isbn = {0-521-80260-1},
mrclass = {32J25 (14C30 14D07 32G20)},
mrnumber = {1967689},
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fjournal = {Annals of Mathematics. Second Series},
volume = {133},
year = {1991},
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mrclass = {11G30 (11J68 14G25)},
mrnumber = {1109352},
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url = {https://doi.org/10.2307/2944318},
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author = {Waldschmidt, Michel},
title = {Nombres transcendants et groupes algébriques},
edition = {Second},
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journal = {Astérisque},
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volume = {69-70},
year = {1987},
pages = {218},
issn = {0303-1179},
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zblnumber = {0621.10022},
} -
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title = {Some Problems of Unlikely Intersections in Arithmetic and Geometry},
series = {Ann. of Math. Stud.},
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note = {with appendixes by David Masser},
publisher = {Princeton Univ. Press, Princeton, NJ},
year = {2012},
pages = {xiv+160},
isbn = {978-0-691-15371-1},
mrclass = {11G35 (11G15 11G18 11G50 14G05)},
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} -
[ZhangEquidist]
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@ARTICLE{ZhangEquidist,
author = {Zhang, Shou-Wu},
title = {Equidistribution of small points on abelian varieties},
journal = {Ann. of Math. (2)},
fjournal = {Annals of Mathematics. Second Series},
volume = {147},
year = {1998},
number = {1},
pages = {159--165},
issn = {0003-486X},
mrclass = {14G40 (11G10 14G05 14K15)},
mrnumber = {1609518},
mrreviewer = {Yuichiro Takeda},
doi = {10.2307/120986},
url = {https://doi.org/10.2307/120986},
zblnumber = {0991.11034},
}