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Torsion sections of Abelian fibrations


Author: Siman Wong
Journal: Proc. Amer. Math. Soc. 143 (2015), 4133-4141
MSC (2010): Primary 11G35; Secondary 11G05, 14G05
Published electronically: June 16, 2015
MathSciNet review: 3373914
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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.


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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