03156nam a22003735a 4500001001100000003001200011005001700023006001900040007001500059008004100074020001800115024002100133040001400154072001600168084003600184245008700220260008200307300003400389336002600423337002600449338003600475347002400511505052000535506006601055520138001121650004102501650002302542650002302565650004002588650002802628700002502656856003202681856006902713204-160630CH-001817-320160630234501.0a fot ||| 0|cr nn mmmmamaa160630e20160725sz fot ||| 0|eng d a978303719657170a10.4171/1572doi ach0018173 7aPBV2bicssc a05-xxa11-xxa13-xxa14-xx2msc10aAbsolute Arithmetic and $\mathbb F_1$-Geometryh[electronic resource] /cKoen Thas3 aZuerich, Switzerland :bEuropean Mathematical Society Publishing House,c2016 a1 online resource (397 pages) atextbtxt2rdacontent acomputerbc2rdamedia aonline resourcebcr2rdacarrier atext filebPDF2rda00tThe Weyl functor. Introduction to Absolute Arithmetic /rKoen Thas --tBelian categories /rAnton Deitmar --tThe combinatorial-motivic nature of $\mathbb F_1$-schemes /rKoen Thas --tA blueprinted view on $\mathbb F_1$-geometry /rOliver Lorscheid --tAbsolute geometry and the Habiro topology /rLieven Le Bruyn --tWitt vectors, semirings, and total positivity /rJames Borger --tModuli operad over $\mathbb F_1$ /rYuri I. Manin, Matilde Marcolli --tA taste of Weil theory in characteristic one /rKoen Thas.1 aRestricted to subscribers:uhttps://www.ems-ph.org/ebooks.php aIt 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 Deningerâ€“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 â€“ 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 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.07aCombinatorics & graph theory2bicssc07aCombinatorics2msc07aNumber theory2msc07aCommutative rings and algebras2msc07aAlgebraic geometry2msc1 aThas, Koen,eeditor.40uhttps://doi.org/10.4171/157423cover imageuhttps://www.ems-ph.org/img/books/thas_mini_2016.jpg