A uniform quantitative Manin–Mumford theorem for curves over function fields
-
Nicole Looper
,
Joseph Silverman
Abstract
We prove that any smooth projective geometrically connected non-isotrivial curve of genus
Funding source: National Science Foundation
Award Identifier / Grant number: DMS-1803021
Funding source: Simons Foundation
Award Identifier / Grant number: 712332
Award Identifier / Grant number: 200020_184623
Funding statement: The first author was supported by NSF grant DMS-1803021. The second author was partially supported by Simons Collaboration Grant #712332. The third author was supported by the Swiss National Science Foundation grant “Diophantine Equations: Special Points, Integrality, and Beyond” (nº 200020_184623).
Acknowledgements
The authors would like to thank Matt Baker, Alex Carney, Robin de Jong, Xander Faber, Myrto Mavraki, and Lucia Mocz for their helpful conversations and useful insights on the admissible pairing. We also thank Jiawei Yu for pointing out an improvement in the bound for 𝜖 in Theorem 1.3, as well as the editor for the suggestion of adding Corollary 1.2. We thank the anonymous referees for their very careful reading and useful comments on two drafts of this paper.
References
[1] M. Baker and B. Poonen, Torsion packets on curves, Compos. Math. 127 (2001), no. 1, 109–116. 10.1023/A:1017557230785Search in Google Scholar
[2] M. Baker and R. Rumely, Harmonic analysis on metrized graphs, Canad. J. Math. 59 (2007), no. 2, 225–275. 10.4153/CJM-2007-010-2Search in Google Scholar
[3] A. Buium, Geometry of 𝑝-jets, Duke Math. J. 82 (1996), no. 2, 349–367. 10.1215/S0012-7094-96-08216-2Search in Google Scholar
[4] A. Carney, The arithmetic Hodge index theorem and rigidity of dynamical systems over function fields, Pacific J. Math. 309 (2020), no. 1, 71–102. 10.2140/pjm.2020.309.71Search in Google Scholar
[5] T. Chinburg and R. Rumely, The capacity pairing, J. reine angew. Math. 434 (1993), 1–44. 10.1515/crll.1993.434.1Search in Google Scholar
[6] Z. Cinkir, Zhang’s conjecture and the effective Bogomolov conjecture over function fields, Invent. Math. 183 (2011), no. 3, 517–562. 10.1007/s00222-010-0282-7Search in Google Scholar
[7] Z. Cinkir, Admissible invariants of genus 3 curves, Manuscripta Math. 148 (2015), no. 3–4, 317–339. 10.1007/s00229-015-0759-1Search in Google Scholar
[8] R. F. Coleman, Torsion points on curves and 𝑝-adic abelian integrals, Ann. of Math. (2) 121 (1985), no. 1, 111–168. 10.2307/1971194Search in Google Scholar
[9] R. de Jong, Néron–Tate heights of cycles on Jacobians, J. Algebraic Geom. 27 (2018), no. 2, 339–381. 10.1090/jag/700Search in Google Scholar
[10] R. de Jong and F. Shokrieh, Tropical moments of tropical Jacobians, Canad. J. Math. 75 (2023), no. 4, 1045–1075. 10.4153/S0008414X22000220Search in Google Scholar
[11] W. Fulton, Intersection theory, 2nd ed., Ergeb. Math. Grenzgeb. (3) 2, Springer, Berlin 1998. 10.1007/978-1-4612-1700-8Search in Google Scholar
[12] R. Hartshorne, Algebraic geometry, Grad. Texts in Math. 52, Springer, New York 1977. 10.1007/978-1-4757-3849-0Search in Google Scholar
[13] L. Kühne, Equidistribution in families of abelian varieties and uniformity, preprint (2021), http://arXiv.org/abs/2101.10272. Search in Google Scholar
[14] J. S. Milne, Jacobian varieties, Arithmetic geometry (Storrs 1984), Springer, New York (1986), 167–212. 10.1007/978-1-4613-8655-1_7Search in Google Scholar
[15] A. Moriwaki, A sharp slope inequality for general stable fibrations of curves, J. reine angew. Math. 480 (1996), 177–195. 10.1515/crll.1996.480.177Search in Google Scholar
[16] A. N. Paršin, Algebraic curves over function fields. I, Math. USSR-Izv. 2 (1968), no. 5, 1145–1170. 10.1070/IM1968v002n05ABEH000723Search in Google Scholar
[17] R. Pink and D. Roessler, On 𝜓-invariant subvarieties of semiabelian varieties and the Manin–Mumford conjecture, J. Algebraic Geom. 13 (2004), no. 4, 771–798. 10.1090/S1056-3911-04-00368-6Search in Google Scholar
[18] M. Raynaud, Courbes sur une variété abélienne et points de torsion, Invent. Math. 71 (1983), no. 1, 207–233. 10.1007/BF01393342Search in Google Scholar
[19] T. Scanlon, Diophantine geometry from model theory, Bull. Symb. Log. 7 (2001), no. 1, 37–57. 10.2307/2687822Search in Google Scholar
[20] L. Szpiro, Propriétés numériques du faisceau dualisant relatif, Séminaire sur les pinceaux de courbes de genre au moins deux, Astérisque 86, Société Mathématique de France, Paris (1981), 44–78. Search in Google Scholar
[21] R. Wilms, On arithmetic intersection numbers on self-products of curves, J. Algebraic Geom. 31 (2022), no. 3, 397–424. 10.1090/jag/777Search in Google Scholar
[22] K. Yamaki, Survey on the geometric Bogomolov conjecture, Actes de la conférence “Non-Archimedean analytic geometry: Theory and practice”, Universitaires de Franche-Comté, Besançon (2017), 137–193. 10.5802/pmb.19Search in Google Scholar
[23] X. Yuan, Arithmetic bigness and a uniform Bogomolov-type result, preprint (2021), http://arXiv.org/abs/2108.05625. Search in Google Scholar
[24] S. Zhang, Admissible pairing on a curve, Invent. Math. 112 (1993), no. 1, 171–193. 10.1007/BF01232429Search in Google Scholar
[25] S. Zhang, Small points and adelic metrics, J. Algebraic Geom. 4 (1995), no. 2, 281–300. Search in Google Scholar
[26] S.-W. Zhang, Gross–Schoen cycles and dualising sheaves, Invent. Math. 179 (2010), no. 1, 1–73. 10.1007/s00222-009-0209-3Search in Google Scholar
© 2025 Walter de Gruyter GmbH, Berlin/Boston
Articles in the same Issue
- Frontmatter
-
Inertia groups of
- Annular solutions to the partitioning problem in a ball
- The Hilb-vs-Quot conjecture
- A uniform quantitative Manin–Mumford theorem for curves over function fields
- Precompactness of domains with lower Ricci curvature bound under Gromov–Hausdorff topology
- Minimal surfaces with low genus in lens spaces
- Quantitative long range curvature estimate for mean curvature flow
- 𝑝-adic adelic metrics and quadratic Chabauty I
- Free boundary minimal disks in convex balls
Articles in the same Issue
- Frontmatter
-
Inertia groups of
- Annular solutions to the partitioning problem in a ball
- The Hilb-vs-Quot conjecture
- A uniform quantitative Manin–Mumford theorem for curves over function fields
- Precompactness of domains with lower Ricci curvature bound under Gromov–Hausdorff topology
- Minimal surfaces with low genus in lens spaces
- Quantitative long range curvature estimate for mean curvature flow
- 𝑝-adic adelic metrics and quadratic Chabauty I
- Free boundary minimal disks in convex balls
