## Non-density of small points on divisors on Abelian varieties and the Bogomolov conjecture

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- by Kazuhiko Yamaki
- J. Amer. Math. Soc.
**30**(2017), 1133-1163 - DOI: https://doi.org/10.1090/jams/874
- Published electronically: December 19, 2016
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## Abstract:

The Bogomolov conjecture for a curve claims finiteness of algebraic points on the curve which are small with respect to the canonical height. Ullmo has proved that this conjecture holds over number fields, and Moriwaki generalized it to the assertion over finitely generated fields over $\mathbb {Q}$ with respect to arithmetic heights. As for the case of function fields with respect to the geometric heights, Cinkir has proved the conjecture over function fields of characteristic $0$ and of transcendence degree $1$. However, the conjecture has been open over other function fields.

In this paper, we prove that the Bogomolov conjecture for curves holds over any function field. In fact, we show that any non-special closed subvariety of *dimension* $1$ in an abelian variety over function fields has only a finite number of small points. This result is a consequence of the investigation of non-density of small points of closed subvarieties of abelian varieties of *codimension* $1$. As a by-product, we show that the geometric Bogomolov conjecture, which is a generalization of the Bogomolov conjecture for curves over function fields, holds for any abelian variety of dimension at most $3$. Combining this result with our previous works, we see that the geometric Bogomolov conjecture holds for all abelian varieties for which the difference between its nowhere degeneracy rank and the dimension of its trace is not greater than $3$.

## References

- F. A. Bogomolov,
*Points of finite order on an abelian variety*, Izv. Akad. Nauk SSSR Ser. Mat.**44**(1980), no. 4, 782–804, 973 (Russian). MR**587337** - Zubeyir Cinkir,
*Zhang’s conjecture and the effective Bogomolov conjecture over function fields*, Invent. Math.**183**(2011), no. 3, 517–562. MR**2772087**, DOI 10.1007/s00222-010-0282-7 - X. W. C. Faber,
*The geometric Bogomolov conjecture for curves of small genus*, Experiment. Math.**18**(2009), no. 3, 347–367. MR**2555704**, DOI 10.1080/10586458.2009.10129049 - William Fulton,
*Intersection theory*, 2nd ed., Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge. A Series of Modern Surveys in Mathematics [Results in Mathematics and Related Areas. 3rd Series. A Series of Modern Surveys in Mathematics], vol. 2, Springer-Verlag, Berlin, 1998. MR**1644323**, DOI 10.1007/978-1-4612-1700-8 - Walter Gubler,
*Local and canonical heights of subvarieties*, Ann. Sc. Norm. Super. Pisa Cl. Sci. (5)**2**(2003), no. 4, 711–760. MR**2040641** - Walter Gubler,
*The Bogomolov conjecture for totally degenerate abelian varieties*, Invent. Math.**169**(2007), no. 2, 377–400. MR**2318560**, DOI 10.1007/s00222-007-0049-y - Walter Gubler,
*Equidistribution over function fields*, Manuscripta Math.**127**(2008), no. 4, 485–510. MR**2457191**, DOI 10.1007/s00229-008-0198-3 - Jean-Pierre Jouanolou,
*Théorèmes de Bertini et applications*, Progress in Mathematics, vol. 42, Birkhäuser Boston, Inc., Boston, MA, 1983 (French). MR**725671** - Serge Lang,
*Abelian varieties*, Springer-Verlag, New York-Berlin, 1983. Reprint of the 1959 original. MR**713430**, DOI 10.1007/978-1-4419-8534-7 - Serge Lang,
*Fundamentals of Diophantine geometry*, Springer-Verlag, New York, 1983. MR**715605**, DOI 10.1007/978-1-4757-1810-2 - Hideyuki Matsumura,
*Commutative ring theory*, Cambridge Studies in Advanced Mathematics, vol. 8, Cambridge University Press, Cambridge, 1986. Translated from the Japanese by M. Reid. MR**879273** - Atsushi Moriwaki,
*Bogomolov conjecture for curves of genus $2$ over function fields*, J. Math. Kyoto Univ.**36**(1996), no. 4, 687–695. MR**1443744**, DOI 10.1215/kjm/1250518448 - Atsushi Moriwaki,
*Bogomolov conjecture over function fields for stable curves with only irreducible fibers*, Compositio Math.**105**(1997), no. 2, 125–140. MR**1440719**, DOI 10.1023/A:1000139117766 - Atsushi Moriwaki,
*Relative Bogomolov’s inequality and the cone of positive divisors on the moduli space of stable curves*, J. Amer. Math. Soc.**11**(1998), no. 3, 569–600. MR**1488349**, DOI 10.1090/S0894-0347-98-00261-6 - Atsushi Moriwaki,
*Arithmetic height functions over finitely generated fields*, Invent. Math.**140**(2000), no. 1, 101–142. MR**1779799**, DOI 10.1007/s002220050358 - David Mumford,
*Abelian varieties*, Tata Institute of Fundamental Research Studies in Mathematics, vol. 5, Published for the Tata Institute of Fundamental Research, Bombay by Oxford University Press, London, 1970. MR**0282985** - D. Mumford, J. Fogarty, and F. Kirwan,
*Geometric invariant theory*, 3rd ed., Ergebnisse der Mathematik und ihrer Grenzgebiete (2) [Results in Mathematics and Related Areas (2)], vol. 34, Springer-Verlag, Berlin, 1994. MR**1304906**, DOI 10.1007/978-3-642-57916-5 - Richard Pink and Damian Roessler,
*On $\psi$-invariant subvarieties of semiabelian varieties and the Manin-Mumford conjecture*, J. Algebraic Geom.**13**(2004), no. 4, 771–798. MR**2073195**, DOI 10.1090/S1056-3911-04-00368-6 - Michel Raynaud,
*Faisceaux amples sur les schémas en groupes et les espaces homogènes*, Lecture Notes in Mathematics, Vol. 119, Springer-Verlag, Berlin-New York, 1970 (French). MR**0260758** - M. Raynaud,
*Courbes sur une variété abélienne et points de torsion*, Invent. Math.**71**(1983), no. 1, 207–233 (French). MR**688265**, DOI 10.1007/BF01393342 - M. Raynaud,
*Sous-variétés d’une variété abélienne et points de torsion*, Arithmetic and geometry, Vol. I, Progr. Math., vol. 35, Birkhäuser Boston, Boston, MA, 1983, pp. 327–352 (French). MR**717600** - Michel Raynaud and Laurent Gruson,
*Critères de platitude et de projectivité. Techniques de “platification” d’un module*, Invent. Math.**13**(1971), 1–89 (French). MR**308104**, DOI 10.1007/BF01390094 - Thomas Scanlon,
*Diophantine geometry from model theory*, Bull. Symbolic Logic**7**(2001), no. 1, 37–57. MR**1836475**, DOI 10.2307/2687822 - Thomas Scanlon,
*A positive characteristic Manin-Mumford theorem*, Compos. Math.**141**(2005), no. 6, 1351–1364. MR**2185637**, DOI 10.1112/S0010437X05001879 - Emmanuel Ullmo,
*Positivité et discrétion des points algébriques des courbes*, Ann. of Math. (2)**147**(1998), no. 1, 167–179 (French). MR**1609514**, DOI 10.2307/120987 - P. Vojta, Nagata’s embedding theorem, arXiv:0706.1907.
- Kazuhiko Yamaki,
*Geometric Bogomolov’s conjecture for curves of genus 3 over function fields*, J. Math. Kyoto Univ.**42**(2002), no. 1, 57–81. MR**1932737**, DOI 10.1215/kjm/1250284711 - Kazuhiko Yamaki,
*Effective calculation of the geometric height and the Bogomolov conjecture for hyperelliptic curves over function fields*, J. Math. Kyoto Univ.**48**(2008), no. 2, 401–443. MR**2436745**, DOI 10.1215/kjm/1250271420 - Kazuhiko Yamaki,
*Geometric Bogomolov conjecture for abelian varieties and some results for those with some degeneration (with an appendix by Walter Gubler: the minimal dimension of a canonical measure)*, Manuscripta Math.**142**(2013), no. 3-4, 273–306. MR**3117164**, DOI 10.1007/s00229-012-0599-1 - Kazuhiko Yamaki,
*Strict supports of canonical measures and applications to the geometric Bogomolov conjecture*, Compos. Math.**152**(2016), no. 5, 997–1040. MR**3505646**, DOI 10.1112/S0010437X15007721 - Kazuhiko Yamaki, Trace of abelian varieties over function fields and the geometric Bogomolov conjecture, J. Reine Angew. Math. (2016) (online), DOI: 10.1515/crelle-2015-0086.
- Shouwu Zhang,
*Admissible pairing on a curve*, Invent. Math.**112**(1993), no. 1, 171–193. MR**1207481**, DOI 10.1007/BF01232429 - Shou-Wu Zhang,
*Equidistribution of small points on abelian varieties*, Ann. of Math. (2)**147**(1998), no. 1, 159–165. MR**1609518**, DOI 10.2307/120986 - Shou-Wu Zhang,
*Gross-Schoen cycles and dualising sheaves*, Invent. Math.**179**(2010), no. 1, 1–73. MR**2563759**, DOI 10.1007/s00222-009-0209-3

## Bibliographic Information

**Kazuhiko Yamaki**- Affiliation: Institute for Liberal Arts and Sciences, Kyoto University, Kyoto, 606-8501, Japan
- MR Author ID: 703158
- Email: yamaki.kazuhiko.6r@kyoto-u.ac.jp
- Received by editor(s): June 5, 2015
- Received by editor(s) in revised form: September 18, 2016, and November 2, 2016
- Published electronically: December 19, 2016
- Additional Notes: The author was partly supported by KAKENHI 26800012.
- © Copyright 2016 American Mathematical Society
- Journal: J. Amer. Math. Soc.
**30**(2017), 1133-1163 - MSC (2010): Primary ~14G40; Secondary ~11G50
- DOI: https://doi.org/10.1090/jams/874
- MathSciNet review: 3671938