The rank of elliptic surfaces in unramified abelian towers

Abstract

Let E --> C be an elliptic surface defined over a number field K. For each finite covering C' --> C defined over K, let E' --> C' be the pullback. We give a strong upper bound for the rank of E'(C'/K) in the case that C' --> C is an unramified abelian covering and under the assumption that the Tate conjecture is true for the surface E'/K. In the case that C is an elliptic curve and the map C'=C --> C is the multiplication-by-n map, the rank of E'(C'/K) is O(n^e) for every e > 0, which may be compared with the elementary bound of O(n^2).