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001			107-091109
003			CH-001817-3
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008			091109e20091001sz fot ||| 0|eng d
020			_a9783037195741
024	7	0	_a10.4171/074 _2doi
040			_ach0018173
072		7	_aPBRH _2bicssc
084			_a11-xx _a14-xx _2msc
100	1		_aBöckle, Gebhard, _eauthor.
245	1	0	_aCohomological Theory of Crystals over Function Fields _h[electronic resource] / _cGebhard Böckle, Richard Pink
260	3		_aZuerich, Switzerland : _bEuropean Mathematical Society Publishing House, _c2009
264		1	_aZuerich, Switzerland : _bEuropean Mathematical Society Publishing House, _c2009
300			_a1 online resource (195 pages)
336			_atext _btxt _2rdacontent
337			_acomputer _bc _2rdamedia
338			_aonline resource _bcr _2rdacarrier
347			_atext file _bPDF _2rda
490	0		_aEMS Tracts in Mathematics (ETM) _v9
506	1		_aRestricted to subscribers: _uhttp://www.ems-ph.org/ebooks.php
520			_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 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.
650	0	7	_aAnalytic number theory _2bicssc
650	0	7	_aNumber theory _2msc
650	0	7	_aAlgebraic geometry _2msc
700	1		_aBöckle, Gebhard, _eauthor.
700	1		_aPink, Richard, _eauthor.
856	4	0	_uhttps://doi.org/10.4171/074
856	4	2	_3cover image _uhttp://www.ems-ph.org/img/books/boeckle_mini.jpg
942			_2EBK13785 _cEBK
999			_c50409 _d50409