The rank of elliptic surfaces in unramified abelian towers

Abstract

Let ℰ → C be an elliptic surface defined over a number field K. For a finite covering C′ → C defined over K, let ℰ′ = ℰ ×CC′ be the corresponding elliptic surface over C′. In this paper we give a strong upper bound for the rank of ℰ′ (C′/K) in the case of geometrically abelian unramified coverings C′ → C and under the assumption that the Tate conjecture is true for ℰ′/K. In the case that C is an elliptic curve and the map C′ = C → C is the multiplication-by-n map, the bound for rank(ℰ′(C′/K)) takes the form O(nk/log logn), which may be compared with the elementary bound of O(n2).