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Let k be a finitely generated field of characteristic 0 and ℓ a prime. Let S be a smooth, separated and geometrically connected scheme over k and let ρ:π1(S)→GLr(Zℓ) be an ℓ-adic representation of the étale fundamental group of S. Let G and G¯ denote the images of π1(S) and π1(Sk¯) respectively. Given a closed point s∈S, we write Gs for the image of the absolute Galois group Γk(s) of the residue field...
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,...
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