|
Online ISSN 1534-7486; Print ISSN 1056-3911
Generalized integral points on abelian varieties and the Geometric Lang–Vojta Conjecture
Author:
Xuan Kien Phung
Journal:
J. Algebraic Geom. 34 (2025), 407-446
DOI:
https://doi.org/10.1090/jag/842
Published electronically:
January 16, 2025
Full-text PDF
Abstract | References | Additional Information
Abstract: Let $f \colon \mathcal {A} \to B$ be a family of abelian varieties over a compact Riemann surface $B$ and fix an effective horizontal divisor $\mathcal {D} \subset \mathcal {A}$. We study $(S, \mathcal {D})$-integral sections $\sigma$ of the family $\mathcal {A}$ where $S \subset B$ is arbitrary. These sections $\sigma$ are algebraic and satisfy the geometric condition $f(\sigma (B) \cap \mathcal {D})\subset S$. Developing the work of Parshin, we establish new quantitative results concerning the finiteness and the polynomial growth of large unions of $(S, \mathcal {D})$-integral sections where $S$ can vary and is required to be finite only in a thin analytic open subset of $B$. Such results are out of the range of purely algebraic methods and imply new evidence and interesting phenomena to the Geometric Lang–Vojta Conjecture.
- Lars V. Ahlfors, Conformal invariants: topics in geometric function theory, McGraw-Hill Series in Higher Mathematics, McGraw-Hill Book Co., New York-Düsseldorf-Johannesburg, 1973. MR 357743
- Joan S. Birman and Caroline Series, Geodesics with bounded intersection number on surfaces are sparsely distributed, Topology 24 (1985), no. 2, 217–225. MR 793185, DOI 10.1016/0040-9383(85)90056-4
- Martin R. Bridson and André Haefliger, Metric spaces of non-positive curvature, Grundlehren der mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 319, Springer-Verlag, Berlin, 1999. MR 1744486, DOI 10.1007/978-3-662-12494-9
- Robert Brody, Compact manifolds and hyperbolicity, Trans. Amer. Math. Soc. 235 (1978), 213–219. MR 470252, DOI 10.1090/S0002-9947-1978-0470252-3
- Robert Brooks, Platonic surfaces, Comment. Math. Helv. 74 (1999), no. 1, 156–170. MR 1677565, DOI 10.1007/s000140050082
- Alexandru Buium, The $abc$ theorem for abelian varieties, Internat. Math. Res. Notices 5 (1994), 219 ff., approx. 15 pp.}, issn=1073-7928, review= MR 1270136, doi=10.1155/S1073792894000255, DOI 10.1155/S1073792894000255
- Peter Buser, Geometry and spectra of compact Riemann surfaces, Progress in Mathematics, vol. 106, Birkhäuser Boston, Inc., Boston, MA, 1992. MR 1183224
- Wei-Liang Chow, Abelian varieties over function fields, Trans. Amer. Math. Soc. 78 (1955), 253–275. MR 67527, DOI 10.1090/S0002-9947-1955-0067527-3
- Wei-Liang Chow, On Abelian varieties over function fields, Proc. Nat. Acad. Sci. U.S.A. 41 (1955), 582–586. MR 71850, DOI 10.1073/pnas.41.8.582
- Brian Conrad, Chow’s $K/k$-image and $K/k$-trace, and the Lang-Néron theorem, Enseign. Math. (2) 52 (2006), no. 1-2, 37–108. MR 2255529
- Pietro Corvaja and Umberto Zannier, Some cases of Vojta’s conjecture on integral points over function fields, J. Algebraic Geom. 17 (2008), no. 2, 295–333. MR 2369088, DOI 10.1090/S1056-3911-07-00489-4
- Pietro Corvaja and Umberto Zannier, Algebraic hyperbolicity of ramified covers of $\Bbb G^2_m$ (and integral points on affine subsets of $\Bbb P_2$), J. Differential Geom. 93 (2013), no. 3, 355–377. MR 3024299
- Laura DeMarco and Niki Myrto Mavraki, Variation of canonical height and equidistribution, Amer. J. Math. 142 (2020), no. 2, 443–473. MR 4084160, DOI 10.1353/ajm.2020.0012
- Schémas en groupes. I: Propriétés générales des schémas en groupes, Lecture Notes in Mathematics, Vol. 151, Springer-Verlag, Berlin-New York, 1970 (French). Séminaire de Géométrie Algébrique du Bois Marie 1962/64 (SGA 3); Dirigé par M. Demazure et A. Grothendieck. MR 274458
- Charles Ehresmann, Les connexions infinitésimales dans un espace fibré différentiable, Colloque de topologie (espaces fibrés), Bruxelles, 1950, Georges Thone, Liège, 1951, pp. 29–55 (French). MR 42768
- Mark L. Green, Holomorphic maps to complex tori, Amer. J. Math. 100 (1978), no. 3, 615–620. MR 501228, DOI 10.2307/2373842
- Jacques Tits, Appendix to: “Groups of polynomial growth and expanding maps” [Inst. Hautes Études Sci. Publ. Math. No. 53 (1981), 53–73] by M. Gromov, Inst. Hautes Études Sci. Publ. Math. 53 (1981), 74–78. MR 623535
- A. Grothendieck, Séminaire de Géométrie Algébrique du Bois Marie (1965–66) – Cohomologie l-adique et Fonctions L – (SGA 5) (French), Lecture Notes in Mathematics, vol. 589, Springer-Verlag, Berlin, New York, 1977, \PrintDOI{10.1007/BFb0096802}.
- M. Hindry and J. H. Silverman, The canonical height and integral points on elliptic curves, Invent. Math. 93 (1988), no. 2, 419–450. MR 948108, DOI 10.1007/BF01394340
- Marc Hindry and Joseph H. Silverman, Diophantine geometry, Graduate Texts in Mathematics, vol. 201, Springer-Verlag, New York, 2000. An introduction. MR 1745599, DOI 10.1007/978-1-4612-1210-2
- Igor Rivin, Simple curves on surfaces, Geom. Dedicata 87 (2001), no. 1-3, 345–360. MR 1866856, DOI 10.1023/A:1012010721583
- Richard Kershner, The number of circles covering a set, Amer. J. Math. 61 (1939), 665–671. MR 43, DOI 10.2307/2371320
- Shoshichi Kobayashi, Invariant distances on complex manifolds and holomorphic mappings, J. Math. Soc. Japan 19 (1967), 460–480. MR 232411, DOI 10.2969/jmsj/01940460
- Shoshichi Kobayashi, Hyperbolic complex spaces, Grundlehren der mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 318, Springer-Verlag, Berlin, 1998. MR 1635983, DOI 10.1007/978-3-662-03582-5
- S. Lang and A. Néron, Rational points of abelian varieties over function fields, Amer. J. Math. 81 (1959), 95–118. MR 102520, DOI 10.2307/2372851
- Serge Lang, Hyperbolic and Diophantine analysis, Bull. Amer. Math. Soc. (N.S.) 14 (1986), no. 2, 159–205. MR 828820, DOI 10.1090/S0273-0979-1986-15426-1
- G. A. Margulis, Applications of ergodic theory to the investigation of manifolds of negative curvature, Funct. Anal. Appl. 3 (1969), 335–336.
- David Minda, Estimates for the hyperbolic metric, Kodai Math. J. 8 (1985), no. 2, 249–258. MR 790414, DOI 10.2996/kmj/1138037050
- Maryam Mirzakhani, Growth of the number of simple closed geodesics on hyperbolic surfaces, Ann. of Math. (2) 168 (2008), no. 1, 97–125. MR 2415399, DOI 10.4007/annals.2008.168.97
- Junjiro Noguchi and Jörg Winkelmann, Bounds for curves in abelian varieties, J. Reine Angew. Math. 572 (2004), 27–47. MR 2076118, DOI 10.1515/crll.2004.052
- A. P. Ogg, Cohomology of abelian varieties over function fields, Ann. of Math. (2) 76 (1962), 185–212. MR 155824, DOI 10.2307/1970272
- A. N. Parshin, Finiteness theorems and hyperbolic manifolds, The Grothendieck Festschrift, Vol. III, Progr. Math., vol. 88, Birkhäuser Boston, Boston, MA, 1990, pp. 163–178. MR 1106914, DOI 10.1007/978-0-8176-4576-2_{6}
- X. K. Phung, Points entiers généralisés sur les variétés abéliennes, PhD diss., University of Strasbourg, 2020, tel-02515788.
- Xuan Kien Phung, Large unions of generalized integral sections on elliptic surfaces, Trans. Amer. Math. Soc. 378 (2025), no. 1, 45–65. MR 4840299, DOI 10.1090/tran/9325
- Xuan Kien Phung, Finiteness criteria and uniformity of integral sections in some families of abelian varieties, Geom. Dedicata 217 (2023), no. 3, Paper No. 59, 19. MR 4581522, DOI 10.1007/s10711-023-00799-7
- Jean-Pierre Serre, Cohomologie galoisienne, 5th ed., Lecture Notes in Mathematics, vol. 5, Springer-Verlag, Berlin, 1994 (French). MR 1324577, DOI 10.1007/BFb0108758
- I. R. Šafarevič, Principal homogeneous spaces defined over a function field, Trudy Mat. Inst. Steklov. 64 (1961), 316–346 (Russian). MR 162806
- Tetsuji Shioda, On elliptic modular surfaces, J. Math. Soc. Japan 24 (1972), 20–59. MR 429918, DOI 10.2969/jmsj/02410020
- Joseph H. Silverman, A quantitative version of Siegel’s theorem: integral points on elliptic curves and Catalan curves, J. Reine Angew. Math. 378 (1987), 60–100. MR 895285, DOI 10.1515/crll.1987.378.60
- Joseph H. Silverman, Advanced topics in the arithmetic of elliptic curves, Graduate Texts in Mathematics, vol. 151, Springer-Verlag, New York, 1994. MR 1312368, DOI 10.1007/978-1-4612-0851-8
- M. Spivak, A Comprehensive Introduction to Differential Geometry, Vol. II, Publish of Perish, Houston, 1999.
- Toshiyuki Sugawa and Matti Vuorinen, Some inequalities for the Poincaré metric of plane domains, Math. Z. 250 (2005), no. 4, 885–906. MR 2180380, DOI 10.1007/s00209-005-0782-0
- Amos Turchet, Fibered threefolds and Lang-Vojta’s conjecture over function fields, Trans. Amer. Math. Soc. 369 (2017), no. 12, 8537–8558. MR 3710634, DOI 10.1090/tran/6968
- V. A. Vassiliev, Introduction to topology, Student Mathematical Library, vol. 14, American Mathematical Society, Providence, RI, 2001. Translated from the 1997 Russian original by A. Sossinski. MR 1816237, DOI 10.1090/stml/014
- Paul Vojta, A higher-dimensional Mordell conjecture, Arithmetic geometry (Storrs, Conn., 1984) Springer, New York, 1986, pp. 341–353. MR 861984
- Paul Vojta, Diophantine approximations and value distribution theory, Lecture Notes in Mathematics, vol. 1239, Springer-Verlag, Berlin, 1987. MR 883451, DOI 10.1007/BFb0072989
- Umberto Zannier, Some problems of unlikely intersections in arithmetic and geometry, Annals of Mathematics Studies, vol. 181, Princeton University Press, Princeton, NJ, 2012. With appendixes by David Masser. MR 2918151
Additional Information
Xuan Kien Phung
Affiliation:
IRMA, Université de Strasbourg, 67084 Strasbourg Cedex, France
Address at time of publication:
Département d’informatique et de recherche opérationnelle, Université de Montréal, Montréal, Québec H3T 1J4, Canada; and Département de mathématiques et de statistique, Université de Montréal, Montréal, Québec H3T 1J4, Canada
MR Author ID:
1334211
ORCID:
0000-0002-4347-8931
Email:
phungxuankien1@gmail.com
Received by editor(s):
March 12, 2022
Received by editor(s) in revised form:
October 7, 2024
Published electronically:
January 16, 2025
Article copyright:
© Copyright 2025
University Press, Inc.