2016;
397 pp;
Hardcover

MSC: Primary 05; 11; 13; 14;
**Print ISBN: 978-3-03719-157-6
Product Code: EMSAAFG**

List Price: $88.00

AMS Member Price: $70.40

# Absolute Arithmetic and \(\mathbb{F}_{1}\)-Geometry

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*Koen Thas*

A publication of the European Mathematical Society

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.