The Arithmetic and Geometry of Elliptic Surfaces

We survey some aspects of the theory of elliptic surfaces and give some results aimed at determining the Picard number of such a surface. For the surfaces considered, this will be equivalent to determining the Mordell-Weil rank of an elliptic curve defined over a function field in one variable. An interesting conjecture concerning Galois actions on the relative de Rham cohomology of these surfaces is discussed. This paper focuses on an important class of algebraic surfaces called elliptic surfaces. The results while geometric in character are arithmetic at heart, and for that reason we devote a fair portion of our discussion to those definitions and facts that make the arithmetic clear. Later in the paper, we will explain some recent results and conjectures. This is a preliminary version, the detailed version will appear elsewhere. There are a number of natural routes leading to the definition of the class of elliptic surfaces. Let E denote a compact connected complex manifold with diml CE = 2. Theorem 1: (Siegel) The field of meromorphic functions on E has transcendence degree ≤ 2 over l C, i.e. the field of meromorphic functions is: 1) l C constant functions 2) a finite separable extension of l C(x) 3) a finite separable extension of l C(x, y). Case 3) is precisely the set of algebraic surfaces, i.e. those admitting an embedding into lPNl C . Case 2) was studied by Kodaira, leading to a series of three papers: Kodaira, K., “On complex analytic surfaces I, II, III,” Annals of Math. 77 and 78, 1963, which expound on elliptic surfaces. Kodaira makes the following definition: AMS subject classification: 14D05, 14J27. Partially supported by ARO grant DAAL 03-88-K-0019. The detailed version of this paper will be submitted for publication elsewhere.