Author	Stiller, Peter F. author
Title	Automorphic Forms and the Picard Number of an Elliptic Surface [electronic resource] / by Peter F. Stiller
Imprint	Wiesbaden : Vieweg+Teubner Verlag : Imprint: Vieweg+Teubner Verlag, 1984
Connect to	http://dx.doi.org/10.1007/978-3-322-90708-0
Descript	VI, 194 p. online resource

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 Ñron-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

I. Differential Equations -- ยง1. Generalities -- ยง2. Inhomogeneous equations -- ยง3. Automorphic forms -- ยง4. Periods -- II. K-Equations -- ยง1. Definitions -- ยง2. Local properties -- ยง3. Automorphic forms associated to K-equations and parabolic cohomology -- III. Elliptic Surfaces -- ยง1. Introduction -- ยง2. A bound on the rank r of Egen (K(X)) -- ยง3. Automorphic forms and a result of Hoytโs -- ยง4. Periods and the rank of Egen (K(X)) -- ยง5. A generalization -- IV. Hodge Theory -- ยง1. The filtrations -- ยง2. Differentials of the second kind -- V. The Picard Number -- ยง1. Periods and period integrals -- ยง2. Periods and differential equations satisfied by normal functions -- ยง3. A formula, a method, and a remark on special values of Dirichlet series -- ยง4. Examples -- Appendix I. Third Order Differential Equations

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