Automorphic Forms and the Picard Number of an Elliptic Surface

  • Peter F. Stiller

Part of the Aspects of Mathematics / Aspekte der Mathematik book series (ASMA, volume 5)

Table of contents

  1. Front Matter
    Pages i-vi
  2. Peter F. Stiller
    Pages 1-5
  3. Peter F. Stiller
    Pages 6-22
  4. Peter F. Stiller
    Pages 23-40
  5. Peter F. Stiller
    Pages 41-85
  6. Peter F. Stiller
    Pages 86-102
  7. Peter F. Stiller
    Pages 103-186
  8. Back Matter
    Pages 187-194

About this book

Introduction

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.

Keywords

Abelian group Dimension Divisor algebra cohomology elliptic surface

Authors and affiliations

  • Peter F. Stiller
    • 1
  1. 1.Texas A & M UniversityCollege StationUSA

Bibliographic information

  • DOI https://doi.org/10.1007/978-3-322-90708-0
  • Copyright Information Vieweg+Teubner Verlag | Springer Fachmedien Wiesbaden GmbH, Wiesbaden 1984
  • Publisher Name Vieweg+Teubner Verlag, Wiesbaden
  • eBook Packages Springer Book Archive
  • Print ISBN 978-3-322-90710-3
  • Online ISBN 978-3-322-90708-0
  • Series Print ISSN 0179-2156
  • About this book