Selmer groups of abelian varieties in extensions of function fields

Let k be a field of characteristic q, C a smooth geometrically connected curve defined over k with function field K := k(C). 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 not equal = q be a prime number and C' -> C a finite geometrically GALOIS and etale cover defined over k with function field K' := k(C'). Let (tau', B') be the K'/k-trace of A/K. We give an upper bound for the Z(p)-corank of the SELMER group Sel(p)(A x(K) K'), defined in terms of the p-descent map. As a consequence, we get an upper bound for the Z-rank of the LANG-NERON group A(K')/tau' B'(k). In the case of a geometric tower of curves whose GALOIS group is isomorphic to Z(p), we give sufficient conditions for the LANG-NERON group of A to be uniformly bounded along the tower.