The Arithmetic of Fundamental Groups

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

      • Tamás 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, João 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 ℓ-adic Pro-algebraic and Relative Pro-ℓ 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 ℓ-adic Iterated Integrals V: Linear Independence, Properties of ℓ-adic Polylogarithms, ℓ-adic Sheaves

       • Zdzisław Wojtkowiak

       Pages 339-374

6. Back Matter

   Pages 375-380

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In the more than 100 years since the fundamental group was first introduced by Henri Poincaré 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, ℓ-adic, p-adic, pro-algebraic and motivic. It explores a wealth of topics that range from anabelian geometry (in particular the section conjecture), the ℓ-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 Tamás Szamuely, respectively.

• anabelian geometry
• etale fundamental group
• fundamental group
• motivic non-abelian Chabauty method

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

  Jakob Stix

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