Geodesic
Divisibility of torsion subgroups of abelian surfaces over number fields
Canadian journal of mathematics, Tome 74 (2022) no. 1, pp. 266-298

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DOI

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.
DOI : 10.4153/S0008414X20000759
Mots-clés : Abelian variety, torsion points, l-adic representation 
Cullinan, John; Yelton, Jeffrey. Divisibility of torsion subgroups of abelian surfaces over number fields. Canadian journal of mathematics, Tome 74 (2022) no. 1, pp. 266-298. doi: 10.4153/S0008414X20000759
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     title = {Divisibility of torsion subgroups of abelian surfaces over number fields},
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     pages = {266--298},
     year = {2022},
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