manuscripta mathematica

, Volume 160, Issue 3–4, pp 385–389 | Cite as

The triviality of Brauer–Manin obstruction for subvarieties of semi-abelian varieties over function fields of characteristic zero

  • Chia-Liang SunEmail author


For every semi-abelian variety over a function field K of characteristic zero, we show that any subgroup of its S-integral K-valued points is discrete in the product of its local points over an infinite set of places of K, where S is a finite set of places of K.

Mathematics Subject Classification

Primary: 11G35 Secondary: 14G25 14K12 



This research is motivated by a reviewing question on Research Plan 107-2115-M-001-013-MY2 of Ministry of Science and Technology in Taiwan, During the period of writing up this paper, I am supported by both the above plan and Research Plan 104-2115-M-001-012-MY3 of Ministry of Science and Technology in Taiwan.


  1. 1.
    Bombieri, E., Gubler, W.: Heights in Diophantine Geometry. New Mathematical Monographs, vol. 4. Cambridge University Press, Cambridge (2006)zbMATHGoogle Scholar
  2. 2.
    Harari, D.: The Brauer–Manin obstruction for integral points on curves. Math. Proc. Camb. Philos. Soc. 149(3), 413–421 (2010)MathSciNetCrossRefGoogle Scholar
  3. 3.
    Lang, Serge: Fundamentals of Diophantine Geometry. Springer, New York (1983)CrossRefGoogle Scholar
  4. 4.
    Poonen, B., Voloch, J.F.: The Brauer–Manin obstruction for subvarieties of abelian varieties over function fields. Ann. of Math. (2) 171(1), 511–532 (2010)MathSciNetCrossRefGoogle Scholar
  5. 5.
    Rosen, M.: Number Theory in Function Fields. Graduate Texts in Mathematics, vol. 210. Springer, New York (2002)CrossRefGoogle Scholar
  6. 6.
    Scharaschkin, Victor: Local-global problems and the Brauer–Manin obstruction. ProQuest LLC, Ann Arbor, MI, 1999. Thesis (Ph.D.)–University of MichiganGoogle Scholar
  7. 7.
    Serre, Jean-Pierre: Lie algebras and Lie groups, Lecture Notes in Mathematics, 2nd ed., vol 1500. Springer, Berlin (1992). 1964 lectures given at Harvard UniversityGoogle Scholar
  8. 8.
    Stoll, Michael: Finite descent obstructions and rational points on curves. Algebra Number Theory 1(4), 349–391 (2007)MathSciNetCrossRefGoogle Scholar
  9. 9.
    Sun, Chia-Liang: Product of local points of subvarieties of almost isotrivial semi-abelian varieties over a global function field. Int. Math. Res. Not. IMRN 19, 4477–4498 (2013)MathSciNetCrossRefGoogle Scholar
  10. 10.
    Sun, Chia-Liang: Local-global principle of affine varieties over a subgroup of units in a function field. Int. Math. Res. Not. IMRN 11, 3075–3095 (2014)MathSciNetCrossRefGoogle Scholar

Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Institute of Mathematics, Academia SinicaTaipeiTaiwan

Personalised recommendations