Rational Points on Elliptic Surfaces

x.1. Elliptic Surfaces Deenition. An elliptic surface consists of a smooth (projective) surface E, a smooth (projective) curve C, and a morphism : E ?! C such that almost all bers E t = ?1 (t) are (smooth projective) curves of genus 1. In addition, we will generally assume that our elliptic surfaces come equipped with an identity section 0 : C ?! E which serves as the identity element of the group of sections E(C). (We will refer to families of curves of genus 1 if we do not want to assume the existence of a section. An elliptic surface is split if E = E C for some elliptic curve E. Similarly, the elliptic surface is constant if its j-invariant j(E) is constant, and non-constant otherwise. We will always assume that our elliptic surfaces are non-split. In order to do number theory, we x K=Q a number eld, and we assume that E, C, , and 0 are all deened over K. We will also write E(C=K) for the group of sections deened over K. Our object of study is E(K). Clearly E(K) = t2C(K) E t (K): A key to studying rational points on elliptic surfaces is to exploit the fact that the bers are elliptic curves, hence have a group structure. Note that if g(C) 2, then E(K) lies on a nite union of bers, so the problem of describing E(K) reduces to the two problems of describing the nitely many points in C(K), and the group of rational points in E t (K) for each t 2 C(K). Thus the problem is, in some sense, one-dimensional (although by no means easy, as will be explained in Poonen's talk). We will generally assume that #C(K) = 1.