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The Arithmetic of Fundamental Groups

PIA 2010

  • Conference proceedings
  • © 2012

Overview

Editors:
  1. Jakob Stix

    1. MATCH - Mathematics Center Heidelberg, Dept. of Mathematics, Heidelberg University, Heidelberg, Germany

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  • The book contains the lecture notes of the Heidelberg lectures on fundamental groups/on Coleman integration. These are introductory texts aiming to familiarize non-specialists with essential techniques of the subject
  • An important development in Minhyong Kim's non-abelian Chabauty method by introducing the motivic fundamental group into the subject is reported on by Gerd Faltings and Majid Hadian
  • The book contains current research on important questions in the arithmetic of fundamental groups
  • Includes supplementary material: sn.pub/extras

Part of the book series: Contributions in Mathematical and Computational Sciences (CMCS, volume 2)

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Table of contents (14 papers)

  1. Front Matter

    Pages i-xii
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  2. Heidelberg Lecture Notes

    1. Front Matter

      Pages 1-1
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  3. Heidelberg lecture notes

    1. Heidelberg Lectures on Coleman Integration

      • Amnon Besser
      Pages 3-52

    2. Heidelberg Lectures on Fundamental Groups

      • TamĂ¡s Szamuely
      Pages 53-74

  4. The Arithmetic of Fundamental Groups

    1. Front Matter

      Pages 75-75
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  5. The arithmetic of fundamental groups

    1. Vector Bundles Trivialized by Proper Morphisms and the Fundamental Group Scheme, II

      • Indranil Biswas, JoĂ£o Pedro P. dos Santos
      Pages 77-88

    2. Note on the Gonality of Abstract Modular Curves

      • Anna Cadoret
      Pages 89-106

    3. The Motivic Logarithm for Curves

      • Gerd Faltings
      Pages 107-125

    4. On a Motivic Method in Diophantine Geometry

      • Majid Hadian
      Pages 127-146

    5. Descent Obstruction and Fundamental Exact Sequence

      • David Harari, Jakob Stix
      Pages 147-166

    6. On Monodromically Full Points of Configuration Spaces of Hyperbolic Curves

      • Yuichiro Hoshi
      Pages 167-207

    7. Tempered Fundamental Group and Graph of the Stable Reduction

      • Emmanuel Lepage
      Pages 209-224

    8. ℤ∕ℓ Abelian-by-Central Galois Theory of Prime Divisors

      • Florian Pop
      Pages 225-244

    9. On â„“-adic Pro-algebraic and Relative Pro-â„“ Fundamental Groups

      • Jonathan P. Pridham
      Pages 245-279

    10. On 3-Nilpotent Obstructions to π1 Sections for \( \mathbb{P}^{1}_\mathbb{Q}\)−{0,1, \(\infty\)}

      • Kirsten Wickelgren
      Pages 281-328

    11. Une remarque sur les courbes de Reichardt–Lind et de Schinzel

      • Olivier Wittenberg
      Pages 329-337

    12. On â„“-adic Iterated Integrals V: Linear Independence, Properties of â„“-adic Polylogarithms, â„“-adic Sheaves

      • ZdzisÅ‚aw Wojtkowiak
      Pages 339-374

  6. Back Matter

    Pages 375-380
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Keywords

About this book

In the more than 100 years since the fundamental group was first introduced by Henri PoincarĂ© it has evolved to play an important role in different areas of mathematics. Originally conceived as part of algebraic topology, this essential concept and its analogies have found numerous applications in mathematics that are still being investigated today, and which are explored in this volume, the result of a meeting at Heidelberg University that brought together mathematicians who use or study fundamental groups in their work with an eye towards applications in arithmetic. The book acknowledges the varied incarnations of the fundamental group: pro-finite, â„“-adic, p-adic, pro-algebraic and motivic. It explores a wealth of topics that range from anabelian geometry (in particular the section conjecture), the â„“-adic polylogarithm, gonality questions of modular curves, vector bundles in connection with monodromy, and relative pro-algebraic completions, to a motivic version of Minhyong Kim's non-abelian Chabauty method and p-adic integration after Coleman. The editor has also included the abstracts of all the talks given at the Heidelberg meeting, as well as the notes on Coleman integration and on Grothendieck's fundamental group with a view towards anabelian geometry taken from a series of introductory lectures given by Amnon Besser and TamĂ¡s Szamuely, respectively.

Editors and Affiliations

  • MATCH - Mathematics Center Heidelberg, Dept. of Mathematics, Heidelberg University, Heidelberg, Germany

    Jakob Stix

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