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 $\mathbb{F}_p$. 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 a 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 etale 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.
Author(s): Gebhard Bockle and Richard Pink
Publisher: European Mathematical Society
Year: 2009
Language: English
Commentary: no
Pages: 197
Cover......Page 1
EMS Tracts in Mathematics 9......Page 2
Editors......Page 3
Title......Page 4
Copyright......Page 5
Preface......Page 6
Contents......Page 8
1 Introduction......Page 10
Categories......Page 22
Localization......Page 26
Abelian categories......Page 29
Grothendieck categories......Page 32
Triangulated categories......Page 35
Derived categories......Page 36
Derived functors......Page 39
Construction of derived functors......Page 41
Comparison of derived categories......Page 45
Conventions......Page 48
-sheaves......Page 49
Nilpotence......Page 51
A-crystals......Page 56
Examples......Page 58
Inverse image......Page 61
Tensor product......Page 67
Change of coefficients......Page 69
Direct image......Page 70
Extension by zero......Page 73
Constructibility......Page 78
The affine case: ind-acyclic T[]-modules......Page 80
Ind-acyclic -sheaves......Page 85
Derived categories of -sheaves and quasi-crystals......Page 90
Cech resolution......Page 93
Inverse image......Page 97
Tensor product......Page 99
Direct image I......Page 102
Direct image II......Page 106
Extension by zero......Page 108
Direct image with compact support......Page 110
Flatness of modules......Page 114
Basic properties......Page 116
Flatness of the canonical representative......Page 118
Functoriality and constructibility......Page 121
Representability......Page 122
Complexes of finite Tor-dimension......Page 126
Regular coefficient rings......Page 129
Basic properties......Page 130
Duality......Page 133
Anderson's trace formula......Page 135
A cohomological trace formula......Page 138
An extended example......Page 143
9 Crystalline L-functions......Page 147
Characteristic polynomials......Page 148
A primary decomposition for rational functions......Page 152
The local L-factor......Page 154
The global L-function......Page 157
The L-function of a complex......Page 159
Functoriality......Page 160
Arbitrary coefficients......Page 166
Change of coefficients......Page 169
Basic Definitions......Page 172
Functors......Page 176
Equivalence of categories......Page 178
Flatness......Page 182
L-functions......Page 183
Bibliography......Page 186
List of notation......Page 190
Index......Page 194
B......Page 197
