Selmer Groups of Abelian Varieties in Extensions of Function Fields

Let k be a field of characteristic q, C a smooth 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 6= q be a prime number and C → C a finite geometrically Galois and étale cover defined over k with function field K ′ := k(C). Let (τ , B) be the K /k-trace of A/K. We give an upper bound for the Zp-corank of the Selmer group Selp(A ×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-Néron group A(K )/τ B(k). In the case of a geometric tower of curves whose Galois group is isomorphic to Zp, we give conditions for the Lang-Néron group of A to be uniformly bounded along the tower. We produce an example where these conditions are fulfilled.