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Automorphic Forms and the Picard Number of an Elliptic Surface

  • Book
  • © 1984

Overview

Authors:
  1. Peter F. Stiller

    1. Texas A & M University, College Station, USA

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Part of the book series: Aspects of Mathematics (ASMA, volume 5)

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Table of contents (6 chapters)

  1. Front Matter

    Pages i-vi
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  2. Introduction

    • Peter F. Stiller
    Pages 1-5

  3. Differential Equations

    • Peter F. Stiller
    Pages 6-22

  4. K-Equations

    • Peter F. Stiller
    Pages 23-40

  5. Elliptic Surfaces

    • Peter F. Stiller
    Pages 41-85

  6. Hodge Theory

    • Peter F. Stiller
    Pages 86-102

  7. The Picard Number

    • Peter F. Stiller
    Pages 103-186

  8. Back Matter

    Pages 187-194
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Keywords

About this book

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.

Authors and Affiliations

  • Texas A & M University, College Station, USA

    Peter F. Stiller

Bibliographic Information

  • Book Title: Automorphic Forms and the Picard Number of an Elliptic Surface

  • Authors: Peter F. Stiller

  • Series Title: Aspects of Mathematics

  • DOI: https://doi.org/10.1007/978-3-322-90708-0

  • Publisher: Vieweg+Teubner Verlag Wiesbaden

  • eBook Packages: Springer Book Archive

  • Copyright Information: Springer Fachmedien Wiesbaden 1984

  • Softcover ISBN: 978-3-322-90710-3Published: 10 November 2012

  • eBook ISBN: 978-3-322-90708-0Published: 17 April 2013

  • Series ISSN: 0179-2156

  • Edition Number: 1

  • Number of Pages: VI, 194

  • Topics: Engineering, general

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