Divisibility of torsion subgroups of abelian surfaces over number fields
Keyword(s):
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.
2013 ◽
Vol 13
(3)
◽
pp. 517-559
◽
Keyword(s):
2017 ◽
Vol 13
(04)
◽
pp. 991-1001
Keyword(s):
2016 ◽
Vol 102
(3)
◽
pp. 316-330
◽
2012 ◽
Vol 08
(01)
◽
pp. 255-264
Keyword(s):
2015 ◽
Vol 67
(1)
◽
pp. 198-213
◽
Keyword(s):
Keyword(s):
2018 ◽
Vol 2020
(9)
◽
pp. 2684-2697
Keyword(s):
2014 ◽
Vol 66
(1)
◽
pp. 170-196
◽
Keyword(s):