Bounds for torsion on abelian varieties with integral moduli

Details Bounds for torsion on abelian varieties with integral moduli Bounds for torsion on abelian varieties with integral moduli

2004-07-15

arxiv.org/abs/math/0407264v2

Mathematics

Number Theory

An appeal to a bound on the p-part of the component group has been corrected, with a corresponding change to the function "F".

Scientific paper

We give a function F(d,n,p) such that if K/Q_p is a degree n field extension and A/K is a d-dimensional abelian variety with potentially good reduction, then #A(K)[tors] is at most F(d,n,p). Separate attention is given to the prime-to-p torsion and to the case of purely additive reduction. These latter bounds are applied to classify rational torsion on CM elliptic curves over number fields of degree at most 3, on elliptic curves over Q with integral j (recovering a theorem of Frey), and on abelian surfaces over Q with integral moduli. In the last case, our efforts leave us with 11 numbers which may, or may not, arise as the order of the full torsion subgroup. The largest such number is 72.

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