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Number Fields and Function Fields – Two Parallel Worlds

  • Book
  • © 2005

Overview

Editors:
  1. Gerard Geer

    1. Korteweg-de Vries Instituut, Universiteit van Amsterdam, Amsterdam, The Netherlands

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  2. Ben Moonen

    1. Korteweg-de Vries Instituut, Universiteit van Amsterdam, Amsterdam, The Netherlands

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  3. René Schoof

    1. Dipartimento di Matematica, Università di Roma “Tor Vergata”, Roma, Italy

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  • Invited articles by leading researchers explore various aspects of the parallel worlds of function fields and number fields
  • Topics range from Arakelov geometry, the search for a theory of varieties over the field with one element, via Eisenstein series to Drinfeld modules, and t-motives
  • Graduate students, mathematicians, and researchers interested in geometry and arithmetic and their connections

Part of the book series: Progress in Mathematics (PM, volume 239)

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Table of contents (15 chapters)

  1. Front Matter

    Pages i-xiii
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  2. Arithmetic over Function Fields: A Cohomological Approach

    • Gebhard Böckle
    Pages 1-38

  3. Algebraic Stacks Whose Number of Points over Finite Fields is a Polynomial

    • Theo van den Bogaart, Bas Edixhoven
    Pages 39-49

  4. On a Problem of Miyaoka

    • Holger Brenner
    Pages 51-59

  5. Monodromy Groups Associated to Non-Isotrivial Drinfeld Modules in Generic Characteristic

    • Florian Breuer, Richard Pink
    Pages 61-69

  6. Irreducible Values of Polynomials: A Non-Analogy

    • Keith Conrad
    Pages 71-85

  7. Schemes over \( \mathbb{F}_1 \)

    • Anton Deitmar
    Pages 87-100

  8. Line Bundles and p-Adic Characters

    • Christopher Deninger, Annette Werner
    Pages 101-131

  9. Arithmetic Eisenstein Classes on the Siegel Space: Some Computations

    • Gerd Faltings
    Pages 133-166

  10. Uniformizing the Stacks of Abelian Sheaves

    • Urs Hartl
    Pages 167-222

  11. Faltings’ Delta-Invariant of a Hyperelliptic Riemann Surface

    • Robin de Jong
    Pages 223-236

  12. A Hirzebruch Proportionality Principle in Arakelov Geometry

    • Kai Köhler
    Pages 237-268

  13. On the Height Conjecture for Algebraic Points on Curves Defined over Number Fields

    • Ulf Kühn
    Pages 269-277

  14. A Note on Absolute Derivations and Zeta Functions

    • Jeffrey C. Lagarias
    Pages 279-285

  15. On the Order of Certain Characteristic Classes of the Hodge Bundle of Semi-Abelian Schemes

    • Vincent Maillot, Damian Roessler
    Pages 287-310

  16. A Note on the Manin-Mumford Conjecture

    • Damian Roessler
    Pages 311-318
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Keywords

About this book

Ever since the analogy between number fields and function fields was discovered, it has been a source of inspiration for new ideas, and a long history has not in any way detracted from the appeal of the subject.

As a deeper understanding of this analogy could have tremendous consequences, the search for a unified approach has become a sort of Holy Grail. The arrival of Arakelov's new geometry that tries to put the archimedean places on a par with the finite ones gave a new impetus and led to spectacular success in Faltings' hands. There are numerous further examples where ideas or techniques from the more geometrically-oriented world of function fields have led to new insights in the more arithmetically-oriented world of number fields, or vice versa.

These invited articles by leading researchers in the field explore various aspects of the parallel worlds of function fields and number fields. Topics range from Arakelov geometry, the search for a theory of varieties over the field with one element, via Eisenstein series to Drinfeld modules, and t-motives.

This volume is aimed at a wide audience of graduate students, mathematicians, and researchers interested in geometry and arithmetic and their connections.

Reviews

From the reviews:

“I thoroughly enjoyed the book; referring to it now and then through the various pages has been a wonderful experience. … It is a stimulating and well-researched volume, aimed at a wide audience of gradute students, mathematicians, and researchers interested in geometry and arithmetic and their connections. In short, it places a most engaging progress in mathematics volume in the hands of the target audience who will enjoy, not just profit from, the different aspects of the involved parallelism.” (Current Engineering Practice, Vol. 48, 2005-2006)

Editors and Affiliations

  • Korteweg-de Vries Instituut, Universiteit van Amsterdam, Amsterdam, The Netherlands

    Gerard Geer, Ben Moonen

  • Dipartimento di Matematica, Università di Roma “Tor Vergata”, Roma, Italy

    René Schoof

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