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Absolute Arithmetic and $\mathbb{F}{_1}$-Geometry
 
Edited by: Koen Thas Ghent University, Ghent, Belgium
A publication of European Mathematical Society
Absolute Arithmetic and $\mathbb{F}{_1}$-Geometry
Hardcover ISBN:  978-3-03719-157-6
Product Code:  EMSAAFG
List Price: $88.00
AMS Member Price: $70.40
Please note AMS points can not be used for this product
Absolute Arithmetic and $\mathbb{F}{_1}$-Geometry
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Absolute Arithmetic and $\mathbb{F}{_1}$-Geometry
Edited by: Koen Thas Ghent University, Ghent, Belgium
A publication of European Mathematical Society
Hardcover ISBN:  978-3-03719-157-6
Product Code:  EMSAAFG
List Price: $88.00
AMS Member Price: $70.40
Please note AMS points can not be used for this product
  • Book Details
     
     
    2016; 397 pp
    MSC: Primary 05; 11; 13; 14

    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, \(\mathbb{F}_1\), 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 \(\mathbb{F}_1\)-world, covers several areas in \(\mathbb{F}_1\)-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 \(\mathbb{F}_1\), categorical foundations, the blend of different scheme theories over \(\mathbb{F}_1\) 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 an expert. In addition to elaborating on known results, the authors introduce brand-new results, open problems and conjectures. The diversity of the contents, together with the mystery surrounding the field with one element, should attract any mathematician, regardless of speciality.

    Readership

    Any mathematician interested in absolute arithmetic and \(\mathbb{F}_1\)-geometry.

  • Additional Material
     
     
  • Requests
     
     
    Review Copy – for publishers of book reviews
    Accessibility – to request an alternate format of an AMS title
2016; 397 pp
MSC: Primary 05; 11; 13; 14

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, \(\mathbb{F}_1\), 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 \(\mathbb{F}_1\)-world, covers several areas in \(\mathbb{F}_1\)-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 \(\mathbb{F}_1\), categorical foundations, the blend of different scheme theories over \(\mathbb{F}_1\) 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 an expert. In addition to elaborating on known results, the authors introduce brand-new results, open problems and conjectures. The diversity of the contents, together with the mystery surrounding the field with one element, should attract any mathematician, regardless of speciality.

Readership

Any mathematician interested in absolute arithmetic and \(\mathbb{F}_1\)-geometry.

Review Copy – for publishers of book reviews
Accessibility – to request an alternate format of an AMS title
Please select which format for which you are requesting permissions.