Selmer groups of abelian varieties in extensions of function fields

Let k be a field of characteristic q, a smooth geometrically connected curve defined over k with function field . Let A/K be a non-constant abelian variety defined over K of dimension d. We assume that q = 0 or >  2d + 1. Let p ≠ q be a prime number and a finite geometrically Galois and étale cover defined over k with function field . Let (τ′, B′) be the K′/k-trace of A/K. We give an upper bound for the -corank of the Selmer group Sel _p (A × _K K′), defined in terms of the p-descent map. As a consequence, we get an upper bound for the -rank of the Lang–Néron group A(K′)/τ′B′(k). In the case of a geometric tower of curves whose Galois group is isomorphic to , we give sufficient conditions for the Lang–Néron group of A to be uniformly bounded along the tower. This work was partially supported by CNPq research grant 305731/2006-8.