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Cohomological Theory of Crystals ove...

Böckle, Gebhard,

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Cohomological Theory of Crystals over Function Fields

紀錄類型:	書目-電子資源 : Monograph/item
正題名/作者:	Cohomological Theory of Crystals over Function Fields/ Gebhard Böckle, Richard Pink
作者:	Böckle, Gebhard,
其他作者:	Pink, Richard,
出版者:	Zuerich, Switzerland :European Mathematical Society Publishing House, : 2009,
面頁冊數:	1 online resource (195 pages)
標題:	Analytic number theory -
電子資源:	https://doi.org/10.4171/074
電子資源:	https://www.ems-ph.org/img/books/boeckle_mini.jpg
ISBN:	9783037195741

Cohomological Theory of Crystals over Function Fields
Böckle, Gebhard,

Cohomological Theory of Crystals over Function Fields[electronic resource] /Gebhard Böckle, Richard Pink - Zuerich, Switzerland :European Mathematical Society Publishing House,2009 - 1 online resource (195 pages) - EMS Tracts in Mathematics (ETM)9.

Restricted to subscribers:https://www.ems-ph.org/ebooks.php

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 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.

ISBN: 9783037195741

Standard No.: 10.4171/074doiSubjects--Topical Terms:

3480868
Analytic number theory

Cohomological Theory of Crystals over Function Fields
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