Elliptic Surfaces

Abstract

Elliptic surfaces appear in many guises. They are one-parameter algebraic families of elliptic curves, they are algebraic surfaces containing a pencil of elliptic curves, and they are elliptic curves over one-dimensional function fields. In this chapter we will see elliptic surfaces arising in all of these ways. Since our emphasis in this book is primarily on arithmetic questions, we will concentrate on those properties of elliptic surfaces which resemble the arithmetic properties of elliptic curves defined over number fields. This means we will neglect many of the fascinating geometric questions raised by the study of elliptic surfaces over algebraically closed fields, especially the classical theory of elliptic surfaces defined over ℂ The interested reader will find a nice introduction to this material in Beauville [1], Griffiths-Harris [1, Ch. 4, §5] and Miranda [1].