03060nam a22003615a 4500001001100000003001200011005001700023006001900040007001500059008004100074020001800115024002100133040001400154072001700168084002200185100003100207245011400238260008200352300003400434336002600468337002600494338003600520347002400556490003900580506006600619520180000685650003502485650002302520650002802543700002802571856003202599856006702631107-091109CH-001817-320091109150325.0a fot ||| 0|cr nn mmmmamaa091109e20091001sz fot ||| 0|eng d a978303719574170a10.4171/0742doi ach0018173 7aPBRH2bicssc a11-xxa14-xx2msc1 aBöckle, Gebhard,eauthor.10aCohomological Theory of Crystals over Function Fieldsh[electronic resource] /cGebhard Böckle, Richard Pink3 aZuerich, Switzerland :bEuropean Mathematical Society Publishing House,c2009 a1 online resource (195 pages) atextbtxt2rdacontent acomputerbc2rdamedia aonline resourcebcr2rdacarrier atext filebPDF2rda0 aEMS Tracts in Mathematics (ETM)v91 aRestricted to subscribers:uhttps://www.ems-ph.org/ebooks.php 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.07aAnalytic number theory2bicssc07aNumber theory2msc07aAlgebraic geometry2msc1 aPink, Richard,eauthor.40uhttps://doi.org/10.4171/074423cover imageuhttps://www.ems-ph.org/img/books/boeckle_mini.jpg