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Gebhard Böckle (Interdisciplinary Center for Scientific Computing, Heidelberg, Germany)
Richard Pink (ETH Zürich, Switzerland)
Cohomological Theory of Crystals over Function Fields
ISBN print 978-3-03719-074-6, ISBN online 978-3-03719-574-1DOI 10.4171/074
October 2009, 195 pages, hardcover, 17 x 24 cm.
48.00 Euro
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.
Keywords: Characteristic $p$ cohomologies, Drinfeld modules, $A$-motives, $L$-functions, Weil conjecture
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