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© 2014

Arithmetic Geometry over Global Function Fields

  • Francesc Bars
  • Ignazio Longhi
  • Fabien Trihan
  • 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

Textbook

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

Table of contents

  1. Front Matter
    Pages i-xiv
  2. David Burns, Fabien Trihan
    Pages 119-181
  3. David Goss
    Pages 183-193
  4. Douglas Ulmer
    Pages 281-337
  5. Back Matter
    Pages 338-338

About this book

Introduction

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.

Keywords

Drinfeld modules Gamma functions L-functions Zeta and Multizeta functions cohomology theory t-motives

Authors and affiliations

  1. 1.Interdisciplinary Center for ScientificUniversität HeidelbergHeidelbergGermany
  2. 2.Department of MathematicsKing's College LondonLondonUnited Kingdom
  3. 3.ColumbusUSA
  4. 4.Department of MathematicsUniversity of RochesterRochesterUSA
  5. 5.Department of Information and Communication SciencesSophia UniversityTokyoJapan
  6. 6.School of MathematicsGeorgia Institute of TechnologyAtlantaUSA

Editors and affiliations

  • Francesc Bars
    • 1
  • Ignazio Longhi
    • 2
  • Fabien Trihan
    • 3
  1. 1.Departament MatemàtiquesUniversitat Autonòma de BarcelonaBellaterraSpain
  2. 2.Suzhou Industrial ParkXi'an Jiaotong Liverpool University Dushu Lake Higher Education TownSuzhouChina
  3. 3.Department of Information and Communication SciencesSophia UniversityTokyoJapan

Bibliographic information

  • Book Title Arithmetic Geometry over Global Function Fields
  • Authors Gebhard Böckle
    David Burns
    David Goss
    Dinesh Thakur
    Fabien Trihan
    Douglas Ulmer
  • Editors Francesc Bars
    Ignazio Longhi
    Fabien Trihan
  • Series Title Advanced Courses in Mathematics - CRM Barcelona
  • Series Abbreviated Title Advanced Courses in Mathematics -CRM Barcelona (Birkhäuser)
  • DOI https://doi.org/10.1007/978-3-0348-0853-8
  • Copyright Information Springer Basel 2014
  • Publisher Name Birkhäuser, Basel
  • eBook Packages Mathematics and Statistics Mathematics and Statistics (R0)
  • Softcover ISBN 978-3-0348-0852-1
  • eBook ISBN 978-3-0348-0853-8
  • Series ISSN 2297-0304
  • Series E-ISSN 2297-0312
  • Edition Number 1
  • Number of Pages XIV, 337
  • Number of Illustrations 0 b/w illustrations, 0 illustrations in colour
  • Topics Number Theory
    General Algebraic Systems
    Algebraic Geometry
  • Buy this book on publisher's site
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