Automorphic Forms and the Picard Number of an Elliptic Surface

In studying an algebraic surface E, which we assume is non-singular and projective over the field of complex numbers t, it is natural to study the curves on this surface. In order to do this one introduces various equivalence relations on the group of divisors (cycles of codimension one). One such relation is algebraic equivalence and we denote by NS(E) the group of divisors modulo algebraic equivalence which is called the N~ron-Severi group of the surface E. This is known to be a finitely generated abelian group which can be regarded naturally as a subgroup of 2 H (E,Z). The rank of NS(E) will be denoted p and is known as the Picard number of E. 2 Every divisor determines a cohomology class in H(E,E) which is of I type (1,1), that is to say a class in H(E,9!) which can be viewed as a 2 subspace of H(E,E) via the Hodge decomposition. The Hodge Conjecture asserts in general that every rational cohomology class of type (p,p) is algebraic. In our case this is the Lefschetz Theorem on (I,l)-classes: Every cohomology class 2 2 is the class associated to some divisor. Here we are writing H (E,Z) for 2 its image under the natural mapping into H (E,t). Thus NS(E) modulo 2 torsion is Hl(E,n!) n H(E,Z) and th 1 b i f h -~ p measures e a ge ra c part 0 t e cohomology.

Inhalt
I. Differential Equations.- §1. Generalities.- §2. Inhomogeneous equations.- §3. Automorphic forms.- §4. Periods.- II. K-Equations.- §1. Definitions.- §2. Local properties.- §3. Automorphic forms associated to K-equations and parabolic cohomology.- III. Elliptic Surfaces.- §1. Introduction.- §2. A bound on the rank r of Egen (K(X)).- §3. Automorphic forms and a result of Hoyt's.- §4. Periods and the rank of Egen (K(X)).- §5. A generalization.- IV. Hodge Theory.- §1. The filtrations.- §2. Differentials of the second kind.- V. The Picard Number.- §1. Periods and period integrals.- §2. Periods and differential equations satisfied by normal functions.- §3. A formula, a method, and a remark on special values of Dirichlet series.- §4. Examples.- Appendix I. Third Order Differential Equations.