On the order of the reduction of a point on an abelian variety.

*(English)*Zbl 1077.11046If \(A\) 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)].

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)].

Reviewer: G. K. Sankaran (Bath)