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Arithmetic Geometry over Global Function Fields

  • Textbook
  • © 2014

Overview

Authors:
  1. Gebhard Böckle

    1. Interdisciplinary Center for Scientific, Universität Heidelberg, Heidelberg, Germany

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  2. David Burns

    1. Department of Mathematics, King's College London, London, United Kingdom

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  3. David Goss

    1. Columbus, USA

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  4. Dinesh Thakur

    1. Department of Mathematics, University of Rochester, Rochester, USA

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  5. Fabien Trihan

    1. Department of Information and Communication Sciences, Sophia University, Tokyo, Japan

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  6. Douglas Ulmer

    1. School of Mathematics, Georgia Institute of Technology, Atlanta, USA

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Editors:
  1. Francesc Bars

    1. Departament Matemàtiques, Universitat Autonòma de Barcelona, Bellaterra, Spain

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  2. Ignazio Longhi

    1. Suzhou Industrial Park, Xi'an Jiaotong Liverpool University Dushu Lake Higher Education Town, Suzhou, China

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  3. Fabien Trihan

    1. Department of Information and Communication Sciences, Sophia University, Tokyo, Japan

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  • Includes a thorough introduction to the arithmetic of Jacobians over function fields (including the current status of the BSD conjecture and its geometric analogues, and the construction of Mordell–Weil groups of high rank)
  • Provides an introduction to A-crystals, with applications to some of the central questions in the theory of L-functions in characteristic p
  • Features a discussion of Gamma, Zeta and Multizeta functions in characteristic p, from scratch to the boundary of current research

Part of the book series: Advanced Courses in Mathematics - CRM Barcelona (ACMBIRK)

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Table of contents (5 chapters)

  1. Front Matter

    Pages i-xiv
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  2. Cohomological Theory of Crystals over Function Fields and Applications

    • Gebhard Böckle
    Pages 1-118

  3. On Geometric Iwasawa Theory and Special Values of Zeta Functions

    • David Burns, Fabien Trihan
    Pages 119-181

  4. The Ongoing Binomial Revolution

    • David Goss
    Pages 183-193

  5. Arithmetic of Gamma, Zeta and Multizeta Values for Function Fields

    • Dinesh S. Thakur
    Pages 195-279

  6. Curves and Jacobians over Function Fields

    • Douglas Ulmer
    Pages 281-337

  7. Back Matter

    Pages 338-338
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Keywords

About this book

This volume collects the texts of five courses given in the Arithmetic Geometry Research Programme 2009-2010 at the CRM Barcelona. All of them deal with characteristic p global fields; the common theme around which they are centered is the arithmetic of L-functions (and other special functions), investigated in various aspects. Three courses examine some of the most important recent ideas in the positive characteristic theory discovered by Goss (a field in tumultuous development, which is seeing a number of spectacular advances): they cover respectively crystals over function fields (with a number of applications to L-functions of t-motives), gamma and zeta functions in characteristic p, and the binomial theorem. The other two are focused on topics closer to the classical theory of abelian varieties over number fields: they give respectively a thorough introduction to the arithmetic of Jacobians over function fields (including the current status of the BSD conjecture and its geometric analogues, and the construction of Mordell-Weil groups of high rank) and a state of the art survey of Geometric Iwasawa Theory explaining the recent proofs of various versions of the Main Conjecture, in the commutative and non-commutative settings.

Authors, Editors and Affiliations

  • Departament Matemàtiques, Universitat Autonòma de Barcelona, Bellaterra, Spain

    Francesc Bars

  • Suzhou Industrial Park, Xi'an Jiaotong Liverpool University Dushu Lake Higher Education Town, Suzhou, China

    Ignazio Longhi

  • Department of Information and Communication Sciences, Sophia University, Tokyo, Japan

    Fabien Trihan, Fabien Trihan

  • Interdisciplinary Center for Scientific, Universität Heidelberg, Heidelberg, Germany

    Gebhard Böckle

  • Department of Mathematics, King's College London, London, United Kingdom

    David Burns

  • Columbus, USA

    David Goss

  • Department of Mathematics, University of Rochester, Rochester, USA

    Dinesh Thakur

  • School of Mathematics, Georgia Institute of Technology, Atlanta, USA

    Douglas Ulmer

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