A Bound for the Mordell-weil Rank of an Elliptic Surface after a Cyclic Base Extension

Let E ! P 1 be an elliptic surface deened over a number eld K, or equivalently an elliptic curve deened over K(T). In this note we prove, assuming Tate's conjecture, that the rank of E(K(T 1=n)) is bounded by F (E)d K (n), where F (E) is an explicit constant independent of n and d K (n) is an explicit elementary function. In particular, if K \ Q(d) = Q for all djn, then d K (n) = d(n) is just the number of divisors of n. Let E ! P 1 be an elliptic surface deened over a eld K, or equivalently an elliptic curve over K(T), say given by a Weierstrass equation E : y 2 = x 3 + A(T)x + B(T) with A; B 2 K(T). (We will assume throughout that char(K) = 0.) There has recently been interest in studying the rank of E over the tower of elds K(T 1=n), n = 1; 2; since clearly E(K(T 1=n)) = E n (K(T)). In this note we will combine recent work of Michael Rosen and the author 10] with an estimate of Michel 7] (which is based on work of Deligne 2]) to show that Tate's conjecture 15] implies a very strong bound for the rank of E n (K(T)) when K is a xed number eld and n ! 1. We recall that if K is a number eld,