On a weak variant of the geometric torsion conjecture

Abstract

A consequence of the geometric torsion conjecture for abelian varieties over function fields is the following. Let k be an algebraically closed field of characteristic 0. For any integers d , g ⩾ 0 there exists an integer N : = N ( k , d , g ) ⩾ 1 such that for any function field L / k with transcendence degree 1 and genus ⩽ g and any d-dimensional abelian variety A → L containing no nontrivial k-isotrivial abelian subvariety, A ( L ) tors ⊂ A [ N ] . In this paper, we deal with a weak variant of this statement, where A → L runs only over abelian varieties obtained from a fixed ( d-dimensional) abelian variety by base change. More precisely, let K / k be a function field with transcendence degree 1 and A → K an abelian variety containing no nontrivial k-isotrivial abelian subvariety. Then we show that if K has genus ⩾1 or if A → K has semistable reduction over all but possibly one place, then, for any integer g ⩾ 0 , there exists an integer N : = N ( A , g ) ⩾ 1 such that for any finite extension L / K with genus ⩽ g, A ( L ) tors ⊂ A [ N ] . Previous works of the authors show that this holds—without any restriction on K—for the ℓ-primary torsion (with ℓ a fixed prime). So, it is enough to prove that there exists an integer N : = N ( A , g ) ⩾ 1 such that for any finite extension L / K with genus ⩽ g, the prime divisors of | A ( L ) tors | are all ⩽ N.