On a weak variant of the geometric torsion conjecture

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.