LEADER 03126nam 22004215a 450 001 9910151932703321 005 20091109150325.0 010 $a3-03719-574-6 024 70$a10.4171/074 035 $a(CKB)3710000000953848 035 $a(CH-001817-3)107-091109 035 $a(PPN)178155691 035 $a(EXLCZ)993710000000953848 100 $a20091109j20091001 fy 0 101 0 $aeng 135 $aurnn|mmmmamaa 181 $ctxt$2rdacontent 182 $cc$2rdamedia 183 $acr$2rdacarrier 200 10$aCohomological Theory of Crystals over Function Fields$b[electronic resource] /$fGebhard Bo?ckle, Richard Pink 210 3 $aZuerich, Switzerland $cEuropean Mathematical Society Publishing House$d2009 215 $a1 online resource (195 pages) 225 0 $aEMS Tracts in Mathematics (ETM)$v9 330 $aThis book develops a new cohomological theory for schemes in positive characteristic p and it applies this theory to give a purely algebraic proof of a conjecture of Goss on the rationality of certain L-functions arising in the arithmetic of function fields. These L-functions are power series over a certain ring A, associated to any family of Drinfeld A-modules or, more generally, of A-motives on a variety of finite type over the finite field Fp. By analogy to the Weil conjecture, Goss conjectured that these L-functions are in fact rational functions. In 1996 Taguchi and Wan gave a first proof of Goss's conjecture by analytic methods a? la Dwork. The present text introduces A-crystals, which can be viewed as generalizations of families of A-motives, and studies their cohomology. While A-crystals are defined in terms of coherent sheaves together with a Frobenius map, in many ways they actually behave like constructible e?tale sheaves. A central result is a Lefschetz trace formula for L-functions of A-crystals, from which the rationality of these L-functions is immediate. Beyond its application to Goss's L-functions, the theory of A-crystals is closely related to the work of Emerton and Kisin on unit root F-crystals, and it is essential in an Eichler-Shimura type isomorphism for Drinfeld modular forms as constructed by the first author. The book is intended for researchers and advanced graduate students interested in the arithmetic of function fields and/or cohomology theories for varieties in positive characteristic. It assumes a good working knowledge in algebraic geometry as well as familiarity with homological algebra and derived categories, as provided by standard textbooks. Beyond that the presentation is largely self-contained. 606 $aAnalytic number theory$2bicssc 606 $aNumber theory$2msc 606 $aAlgebraic geometry$2msc 615 07$aAnalytic number theory 615 07$aNumber theory 615 07$aAlgebraic geometry 686 $a11-xx$a14-xx$2msc 700 $aBo?ckle$b Gebhard$01065141 702 $aPink$b Richard 801 0$bch0018173 906 $aBOOK 912 $a9910151932703321 996 $aCohomological Theory of Crystals over Function Fields$92564456 997 $aUNINA