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[null,"1.","Record Nr.",null,"UNINA9910151932703321"]
[null,null,"Autore",null,"Böckle Gebhard"]
[null,null,"Titolo",null,"Cohomological Theory of Crystals over Function Fields [[electronic resource] /] / Gebhard Böckle, Richard Pink"]
[null,null,"Pubbl/distr/stampa",null,"Zuerich, Switzerland, : European Mathematical Society Publishing House, 2009"]
[null,null,"ISBN",null,"3-03719-574-6"]
[null,null,"Descrizione fisica",null,"1 online resource (195 pages)"]
[null,null,"Collana",null,"EMS Tracts in Mathematics (ETM) ; 9"]
[null,null,"Classificazione",null,"11-xx14-xx"]
[null,null,"Soggetti",null,"Analytic number theory"]
[{"$nulls":4},"Number theory"]
[{"$nulls":4},"Algebraic geometry"]
[null,null,"Lingua di pubblicazione",null,"Inglese"]
[null,null,"Formato",null,"Materiale a stampa"]
[null,null,null,"Livello bibliografico","Monografia"]
[null,null,null,"Sommario/riassunto","This 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 à 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 étale 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 "]
[{"$nulls":4},"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."]