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On the order of the reduction of a point on an abelian variety. (English) Zbl 1077.11046
If $$A$$ is an abelian variety over a number field $$K$$ and $$a\in A(K)$$ is of infinite order, then the reduction $$a_v$$ of $$a$$ at a place $$v$$ of good reduction is torsion. What is its order? More precisely, fix a rational prime $$\ell$$: if $$v$$ does not divide $$\ell$$, what can be said about the $$\ell$$-part of $$a_v$$? Such questions are approached here by studying the action of $$\text{ Frob}_v$$ on $\ell^{-\infty}({\mathbb Z}a)=\left\{x\in A(\bar K)\mid \exists n\geq 0 \;\ell^n x\in {\mathbb Z}a\right\}$ which, it is shown here, determines the $$\ell$$-part of $$a_v$$. The groups $$\ell^{-\infty}({\mathbb Z}a)$$ is an extension of $${\mathbb Z}[1/\ell]$$ by $$A[\ell^\infty]$$ so it may be studied via the Tate module of the kernel $T_\ell(A)=\text{ Hom}\big({\mathbb Q}_\ell/{\mathbb Z}_\ell, A[\ell^\infty]),$ the Tate module for the extension $T_\ell(A,a)=\text{ Hom}\big({\mathbb Q}_\ell/{\mathbb Z}_\ell, \ell^{-\infty}({\mathbb Z}a)/{\mathbb Z}a\big),$ and their Galois module structures. These structures are described in §§1–2 of the paper, where it is also shown that there is a finite extension $$L\colon K$$ such that $$A_w$$ has no supersingular abelian subvariety for $$w$$ in a set of places of $$L$$ of density $$1$$. In §3 we see the connection between that $$\ell$$-part of $$a_v$$ and the Galois modules. The results of §4 are density results for the $$\ell$$-part of $$a_v$$: for instance, if $${\mathbb Z}a$$ is Zariski dense and $$b\in A[\ell^\infty]$$ then for $$v$$ in a set of positive density, the $$\ell$$-part of $$a_v$$ is equal to the reduction of $$b$$. The general tenor is that any plausible behaviour actually occurs, and for a set of positive density.
The final section draws some conclusions not depending on $$\ell$$. In particular, if $$A_i$$, $$1\leq i\leq d$$, are abelian varieties over $$K$$, $$a_i\in A_i(K)$$, and for $$v$$ in a set of density $$1$$ the reduction of at least one $$a_i$$ is annihilated by a power of the residue characteristic, then one of the $$a_i$$ must be zero; a similar criterion for one of the $$a_i$$ to be torsion is also given. Some of the results of this part could (as the author indicates) alternatively be deduced quickly from results of S. Wong [“Power residues on Abelian varieties”, Manuscr. Math. 102, No. 1, 129–137 (2000; Zbl 1025.11019)].

##### MSC:
 11G10 Abelian varieties of dimension $$> 1$$ 11R45 Density theorems 14K15 Arithmetic ground fields for abelian varieties
##### Keywords:
abelian variety; torsion; Tate module
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