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Divisibility of torsion subgroups of abelian surfaces over number fields

Part of: Arithmetic algebraic geometry Linear algebraic groups and related topics

Published online by Cambridge University Press:  28 October 2020

John Cullinan  and
Jeffrey Yelton
John Cullinan*
Affiliation:
Department of Mathematics, Bard College, Annandale-On-Hudson, NY 12401, USA
Jeffrey Yelton
Affiliation:
Department of Mathematics and Computer Science, Emory University, Atlanta, GA 30322, USA e-mail: jeffrey.yelton@emory.edu
*
e-mail: cullinan@bard.edu
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Abstract

Let A be a two-dimensional abelian variety defined over a number field K. Fix a prime number $\ell $ and suppose $\#A({\mathbf {F}_{\mathfrak {p}}}) \equiv 0 \pmod {\ell ^2}$ for a set of primes ${\mathfrak {p}} \subset {\mathcal {O}_{K}}$ of density 1. When $\ell =2$ Serre has shown that there does not necessarily exist a K-isogenous $A'$ such that $\#A'(K)_{{tor}} \equiv 0 \pmod {4}$. We extend those results to all odd $\ell $ and classify the abelian varieties that fail this divisibility principle for torsion in terms of the image of the mod-$\ell ^2$ representation.

Keywords

Abelian varietytorsion pointsℓ-adic representation

MSC classification

Primary: 11G10: Abelian varieties of dimension $> 1$
Secondary: 20G25: Linear algebraic groups over local fields and their integers

Information

Type
Article
Information
Canadian Journal of Mathematics , Volume 74 , Issue 1 , February 2022 , pp. 266 - 298
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Copyright
© Canadian Mathematical Society 2020

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