Reduction of homomorphisms mod p and algebraicity

Chandrashekhar Khare; Dipendra Prasad

paper

Reduction of homomorphisms mod p and algebraicity

arXiv:math/0211004

Abstract

{Let $K$ be a number field, and $A_{1}, A_{2}$ abelian varieties over $K$ . Let $P$ (resp. $Q$ ) be a non-torsion point in $A_{1} (K)$ (resp. $A_{2} (K)$ ) such that for almost all places $v$ of $K$ , the order of $Q$ mod $v$ divides the order of $P$ mod $v$ . Then we prove (under some conditions on $A_{i}, i = 1, 2$ ) that there is a homomorphism $j$ from $A_{1}$ to $A_{2}$ such that $j (P) = Q$ . We formulate and extend such a result for any subgroups of $A_{i} (K), i = 1, 2$ . These results in particular extend the work of Corrales and Schoof from elliptic curves to abelian varieties.