Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 
 

 

Torsion sections of Abelian fibrations


Author: Siman Wong
Journal: Proc. Amer. Math. Soc. 143 (2015), 4133-4141
MSC (2010): Primary 11G35; Secondary 11G05, 14G05
DOI: https://doi.org/10.1090/proc12736
Published electronically: June 16, 2015
MathSciNet review: 3373914
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: Let $ K$ be a number field, and let $ B$ be a smooth projective curve over $ K$ of genus $ \le 1$ such that $ B(K)$ is infinite. Let $ A$ be an Abelian variety defined over the function field $ K(B)$. Suppose there are infinitely many non-trivial, pairwise disjoint extensions $ L/K$ of bounded degree such that $ B(L)$ is infinite and that $ A_P(L)_{\rm tor}\not =0 $ for every point $ P\in B(L)$ at which the specialization $ A_P$ is smooth. We show that $ A(K(B))_{\rm tor}\not =0 $. If $ A/K(B)$ is not a constant Abelian variety over $ K$, the extensions $ L/K$ need not have bounded degree, and we can replace $ B(K)$ being infinite by $ B(K)\not =\emptyset $. This provides evidence in support of a question of Graber-Harris-Mazur-Starr on rational pseudo-sections of arithmetic surjective morphisms.


References [Enhancements On Off] (What's this?)

  • [1] Amir Akbary, Dragos Ghioca, and V. Kumar Murty, Reductions of points on elliptic curves, Math. Ann. 347 (2010), no. 2, 365-394. MR 2606941 (2011d:11147), https://doi.org/10.1007/s00208-009-0433-6
  • [2] Anna Cadoret and Akio Tamagawa, Uniform boundedness of $ p$-primary torsion of abelian schemes, Invent. Math. 188 (2012), no. 1, 83-125. MR 2897693, https://doi.org/10.1007/s00222-011-0343-6
  • [3] Jordan S. Ellenberg, Chris Hall, and Emmanuel Kowalski, Expander graphs, gonality, and variation of Galois representations, Duke Math. J. 161 (2012), no. 7, 1233-1275. MR 2922374, https://doi.org/10.1215/00127094-1593272
  • [4] Tom Graber, Joe Harris, Barry Mazur, and Jason Starr, Arithmetic questions related to rationally connected varieties, The legacy of Niels Henrik Abel, Springer, Berlin, 2004, pp. 531-542. MR 2077583 (2005g:14097)
  • [5] G. H. Hardy and E. M. Wright, An introduction to the theory of numbers, 5th ed., The Clarendon Press, Oxford University Press, New York, 1979. MR 568909 (81i:10002)
  • [6] Marc Hindry and Joseph H. Silverman, Diophantine geometry, An introduction. Graduate Texts in Mathematics, vol. 201, Springer-Verlag, New York, 2000. MR 1745599 (2001e:11058)
  • [7] Dale Husemoller, Elliptic curves, With an appendix by Ruth Lawrence. Graduate Texts in Mathematics, vol. 111, Springer-Verlag, New York, 1987. MR 868861 (88h:11039)
  • [8] Serge Lang, Fundamentals of Diophantine geometry, Springer-Verlag, New York, 1983. MR 715605 (85j:11005)
  • [9] Nicholas M. Katz, Galois properties of torsion points on abelian varieties, Invent. Math. 62 (1981), no. 3, 481-502. MR 604840 (82d:14025), https://doi.org/10.1007/BF01394256
  • [10] M. Ram Murty, On Artin's conjecture, J. Number Theory 16 (1983), no. 2, 147-168. MR 698163 (86f:11087), https://doi.org/10.1016/0022-314X(83)90039-2
  • [11] Joseph H. Silverman, The arithmetic of elliptic curves, Graduate Texts in Mathematics, vol. 106, Springer-Verlag, New York, 1986. MR 817210 (87g:11070)
  • [12] Joseph H. Silverman, Advanced topics in the arithmetic of elliptic curves, Graduate Texts in Mathematics, vol. 151, Springer-Verlag, New York, 1994. MR 1312368 (96b:11074)

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC (2010): 11G35, 11G05, 14G05

Retrieve articles in all journals with MSC (2010): 11G35, 11G05, 14G05


Additional Information

Siman Wong
Affiliation: Department of Mathematics and Statistics, University of Massachusetts, Amherst, Massachusetts 01003-9305
Email: siman@math.umass.edu

DOI: https://doi.org/10.1090/proc12736
Received by editor(s): February 27, 2014
Published electronically: June 16, 2015
Additional Notes: This work was supported in part by the NSA and by NSF grant DMS-0901506
Communicated by: Matthew A. Papanikolas
Article copyright: © Copyright 2015 American Mathematical Society

American Mathematical Society