Absolute arithmetic and F₁-geometry
By: Koen Thas, editor
Publisher: European Mathematical Society
Published: 2016
Place of publication: Zürich, Switzerland
Language: eng
Number of pages: 386
ISBN-13: 9783037196571; 9783037191576
Subjects: Algebraic Geometry; Géométrie algébrique; Combinatorics & graph theory; MATHEMATICS; Geometry; Algebraic; Combinatorics; Number theory; Commutative rings and algebras; Algebraic geometry
Description: It has been known for some time that geometries over finite fields, their automorphism groups and certain counting formulae involving these geometries have interesting guises when one lets the size of the field go to 1. On the other hand, the nonexistent field with one element, F₁, presents itself as a ghost candidate for an absolute basis in Algebraic Geometry to perform the Deninger-Manin program, which aims at solving the classical Riemann Hypothesis. This book, which is the first of its kind in the F₁-world, covers several areas in F₁-theory, and is divided into four main parts - Combinatorial Theory, Homological Algebra, Algebraic Geometry and Absolute Arithmetic. Topics treated include the combinatorial theory and geometry behind F₁, categorical foundations, the blend of different scheme theories over F₁ which are presently available, motives and zeta functions, the Habiro topology, Witt vectors and total positivity, moduli operads, and at the end, even some arithmetic. Each chapter is carefully written by experts, and besides elaborating on known results, brand new results, open problems and conjectures are also met along the way. The diversity of the contents, together with the mystery surrounding the field with one element, should attract any mathematician, regardless of speciality.
Notes: Includes bibliographical references and index.
Table of contents: title: The Weyl functor : introduction to absolute arithmetic  -- Koen Thas; title: Belian categories  -- Anton Deitmar; title: The combinatorial-motivic nature of F₁-schemes  -- Koen Thas; title: A blueprinted view on F₁-geometry  -- Oliver Lorscheid; title: Absolute geometry and the Habiro topology  -- Lieven Le Bruyn; title: Witt vectors, semirings, and total positivity  -- James Borger; title: Moduli operad over F₁  -- Yuri I. Manin and Matilde Marcolli; title: A taste of Weil theory in characteristic one -- Koen Thas.