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Divisibility of torsion subgroups of abelian surfaces over number fields

Published online by Cambridge University Press:  28 October 2020

John Cullinan*
Affiliation:
Department of Mathematics, Bard College, Annandale-On-Hudson, NY12401, USA
Jeffrey Yelton
Affiliation:
Department of Mathematics and Computer Science, Emory University, Atlanta, GA 30322, USA e-mail: jeffrey.yelton@emory.edu

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.

Type
Article
Copyright
© Canadian Mathematical Society 2020

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