The Arithmetic of Fundamental Groups: PIA 2010

ISBN-13: 9783642439421 / Angielski / Miękka / 2014 / 380 str.

Jakob Stix

Opis

In the more than 100 years since the fundamental group was first introduced by Henri Poincare 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 Tamas Szamuely, respectively.

Spis treści

Part I Heidelberg Lecture Notes: 1 Heidelberg lectures on Coleman integration by A.Besser.- 2 Heidelberg lectures on fundamental groups by T. Szamuely.- Part II The Arithmetic of Fundamental Groups: 3 Vector bundles trivialized by proper morphisms and the fundamental group scheme, II by I. Biswas and J.P.P. dos Santos.- 4 Note on the gonality of abstract modular curves by A. Cadoret.- 5 The motivic logarithm for curves by G.Faltings.- 6 On a motivic method in diophantine geometry by M.Hadian.- 7 Descent obstruction and fundamental exact sequence by D. Harari and J. Stix.- 8 On monodromically full points of configuration spaces of hyperbolic Curves by Y.Hoshi.- 9 Tempered fundamental group and graph of the stable reduction by E.Lepage.- 10 ℤ/ℓ abelian-by-central Galois theory of prime divisors by F.Pop.- 11 On ℓ-adic pro-algebraic and relative pro- ℓ fundamental groups by J.P.Pridham.- 12 On 3-nilpotent obstructions to π_1 sections for ℙ^1­_ℚ-{0,1, ∞} by K.Wickelgren.- 13 Une remarque sur les courbes de Reichardt–Lind et de Schinzel by O.Wittenberg.- 14 On ℓ-adic iterated integrals V : linear independence, properties of ℓ-adic polylogarithms, ℓ-adic sheaves by Z. Wojtkowiak.- Workshop Talks

Blurb

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.