The main goal of the present paper is to give a detailed exposition of the essential logical structure of inter-universal Teichmüller theory from the point of view of the Boolean operators – such as the logical AND and logical OR operators – of propositional calculus.
In this paper k always denotes a finite field of characteristic . A variety over k means a reduced scheme separated and of finite type over k. An étale cover is always assumed to be finite (as in [SGA1, I, Def. 4.9]).
This document is a short user\rqs guide to the theory of motives and homotopy theory in the setting of logarithmic geometry. We review some of the basic ideas and results in relation to other works on motives with modulus, motivic homotopy theory, and reciprocity sheaves.
The present article is based on the four hours mini-courses “Introduction to Mono-anabelian Geometry” which the author gave at the conference “Fundamental Groups in Arithmetic Geometry” (Paris, 2016). The purpose of the present article is to introduce mono-anabelian geometry by focusing on mono-anabelian geometry for mixed-characteristic local fields, which provides elementary but nontrivial examples of typical arguments in the study of mono-anabelian geometry.
In his “Letter to Faltings”, Grothendieck lays the foundation of what will become part of his multi-faceted legacy to arithmetic geometry. This includes the following three branches discussed in the workshop: the arithmetic of Galois covers, the theory of motives and the theory of anabelian Galois representations. Their geometrical paradigms endow similar but complementary arithmetic insights for the study of the absolute Galois group Gq of the field of rational numbers that initially crystallized into a functorially group-theoretic unifying approach. Recent years have seen some new enrichments based on modern geometrical constructions – e.g. simplicial homotopy, Tannaka perversity, automorphic forms – that endow some higher considerations and outline new geometric principles. This workshop brought together an international panel of young and senior experts of arithmetic geometry who sketched the future desire paths of homotopic and geometric Galois theory.
In his Annals paper in 1986, Y.Ihara introduced the universal power series for Jacobi sums and showed deep arithmetic phenomena arising in Galois actions on profinite fundamental groups. In particular, the explicit formula established by Anderson, Coleman, Ihara-Kaneko-Yukinari opened remarkable connection to theory of cyclotomic fields (Iwasawa theory) and shed new lights on circle of ideas surrounding Grothendieck\rqs philosophy on anabelian geometry as well as various geometric approaches in inverse Galois theory. In this article, I will illustrate some of these aspects from a viewpoint of Grothendieck-Teichmüller theory.
The geometric study of the absolute Galois group of the rational numbers has been a highly active research topic since the first milestones: Hilbert\rqs Irreducibility Theorem, Noether\rqs program, Riemann\rqs Existence Theorem. It gained special interest in the last decades with Grothendieck\rqs “Esquisse d\rqun programme”, his “Letter to Faltings” and Fried\rqs introduction of Hurwitz spaces. It grew on and thrived on a wide range of areas, e.g. formal algebraic geometry, Diophantine geometry, group theory. The recent years have seen the development and integration in algebraic geometry and Galois theory of new advanced techniques from algebraic stacks, \ell\ensuremath\ell-adic representations and homotopy theories. It was the goal of this mini-workshop, to bring together an international panel of young and senior experts to draw bridges towards these fields of research and to incorporate new methods, techniques and structures in the development of geometric Galois theory.
In the mid sixties, A. Grothendieck envisioned a vast generalization of Galois theory to systems of polynomials in several variables, motivic Galois theory, and introduced tannakian categories on this occasion. In characteristic zero, various unconditional approaches were later proposed. The most precise one, due to J. Ayoub, relies on Voevodsky theory of mixed motives and on a new tannakian theory. It sheds new light on periods of algebraic varieties, and shows in particular that polynomial relations between periods of a pencil of algebraic varieties always arise from Stokes formula.
The ultimate goal of this work, as we explain in the general introduction, is to prove that the Grothendieck–Teichm¨uller group represents, in the rational setting at least, the group of homotopy automorphism classes of E2-operads. This objective can be taken as a motivation to read this book or as a guiding example of an application of our methods...
This Lecture Notes volume is the fruit of two research-level summer schools jointly organized by the GTEM node at Lille University and the team of Galatasaray University (Istanbul): "Geometry and Arithmetic of Moduli Spaces of Coverings (2008)" and "Geometry and Arithmetic around Galois Theory (2009)". The volume focuses on geometric methods in Galois theory. The choice of the editors is to provide a complete and comprehensive account of modern points of view on Galois theory and related moduli problems, using stacks, gerbes and groupoids. It contains lecture notes on étale fundamental group and fundamental group scheme, and moduli stacks of curves and covers. Research articles complete the collection.
On March 29-April 2, 2010, a meeting was organized at the Luminy CIRM (France) on geometric and differential Galois theories, witnessing the close ties these theories have woven in recent years. The present volume collects the Proceedings of this meeting. Although it may be viewed as a continuation of the one held 6 years earlier on arithmetic and differential Galois groups (see Séminaires & Congrès, vol. 13), several new and promising themes have appeared. The articles gathered here cover the following topics: moduli spaces of connexions, differential equations and coverings in finite characteristic, liftings, monodromy groups in their various guises (tempered fundamental group, motivic groups, generalized difference Galois groups), and arithmetic applications.
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, \ensuremath\ell-adic, p-adic, pro-algebraic and motivic. It explores a wealth of topics that range from anabelian geometry (in particular the section conjecture), the \ensuremath\ell-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.
Field Arithmetic explores Diophantine fields through their absolute Galois groups. This largely self-contained treatment starts with techniques from algebraic geometry, number theory, and profinite groups. Graduate students can effectively learn generalizations of finite field ideas. We use Haar measure on the absolute Galois group to replace counting arguments. New Chebotarev density variants interpret diophantine properties. Here we have the only complete treatment of Galois stratifications, used by Denef and Loeser, et al, to study Chow motives of Diophantine statements. Progress from the first edition starts by characterizing the finite-field like P(seudo)A(lgebraically)C(losed) fields. We once believed PAC fields were rare. Now we know they include valuable Galois extensions of the rationals that present its absolute Galois group through known groups. PAC fields have projective absolute Galois group. Those that are Hilbertian are characterized by this group being pro-free. These last decade results are tools for studying fields by their relation to those with projective absolute group. There are still mysterious problems to guide a new generation: Is the solvable closure of the rationals PAC; and do projective Hilbertian fields have pro-free absolute Galois group (includes Shafarevich’s conjecture)? The third edition improves the second edition in two ways: First it removes many typos and mathematical inaccuracies that occur in the second edition (in particular in the references). Secondly, the third edition reports on five open problems (out of thirtyfour open problems of the second edition) that have been partially or fully solved since that edition appeared in 2005.
This is a survey paper. The contents corresponds roughly to my talk at the Number Theory Camp held at Pohang Unversity of Science and Technology, January, 2004. Most part of the paper is dedicated to an informal introduction to the Fontaine-Mazur Conjectures. Explained also is a relation between the two versions of the Finiteness Conjecture of Fontaine-Mazur.
La théorie des motifs, introduite par A. Grothendieck il y a 40 ans et demeurée longtemps conjecturale, a connu depuis une quinzaine d’années des développements spectaculaires. Ce texte a pour objectif de rendre ces avancées accessibles au non-spécialiste, tout en donnant, au cours de ses deux premières parties, une vision unitaire des fondements géométriques de la théorie (pure et mixte). La troisième partie, consacrée aux périodes des motifs, en propose une illustration concrète ; on y traite en détail les exemples des valeurs de la fonction gamma aux points rationnels, et des nombres polyzêta.