Rigid Cohomology

Bernard Le Stum0521875242, 9780521875240, 9780511342554

Dating back to work of Berthelot, rigid cohomology appeared as a common generalization of Monsky-Washnitzer cohomology and crystalline cohomology. It is a p-adic Weil cohomology suitable for computing Zeta and L-functions for algebraic varieties on finite fields. Moreover, it is effective, in the sense that it gives algorithms to compute the number of rational points of such varieties. This is the first book to give a complete treatment of the theory, from full discussion of all the basics to descriptions of the very latest developments. Results and proofs are included that are not available elsewhere, local computations are explained, and many worked examples are given. This accessible tract will be of interest to researchers working in arithmetic geometry, p-adic cohomology theory, and related cryptographic areas.

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Table of contents :
Cover……Page 1
Half-title……Page 3
Title……Page 5
Copyright……Page 6
Dedication……Page 7
Contents……Page 9
Preface……Page 11
Outline……Page 13
Conventions and notations……Page 16
1.1 Alice and Bob……Page 19
1.2 Complexity……Page 20
1.3 Weil conjectures……Page 21
1.4 Zeta functions……Page 22
1.5 Arithmetic cohomology……Page 23
1.6 Bloch–Ogus cohomology……Page 24
1.7 Frobenius on rigid cohomology……Page 25
1.8 Slopes of Frobenius……Page 26
1.10 F-isocrystals……Page 27
2.1.1 Ultrametric fields……Page 30
2.1.2 Fibers of formal schemes……Page 31
2.1.3 Base change……Page 33
2.2 Tubes of radius one……Page 34
2.3 Tubes of smaller radius……Page 41
3.1 Frames……Page 53
3.2 Frames and tubes……Page 61
3.3 Strict neighborhoods and tubes……Page 72
3.4 Standard neighborhoods……Page 83
4.1.1 Crystals and stratifications……Page 92
4.1.2 Connections and differential modules……Page 94
4.1.3 The smooth case……Page 96
4.1.4 Cohomology……Page 98
4.2.1 The Dwork module L……Page 101
4.2.2 The Kummer module K……Page 105
4.2.3 Playing with Dwork and Kummer modules……Page 108
4.2.4 The Legendre family……Page 111
4.2.5 Hypergeometric differential equations……Page 113
4.3 Calculus on strict neighborhoods……Page 115
4.4 Radius of convergence……Page 125
5.1 Overconvergent sections……Page 143
5.2 Overconvergence and abelian sheaves……Page 155
5.3 Dagger modules……Page 171
5.4 Coherent dagger modules……Page 178
6.1 Stratifications and overconvergence……Page 195
6.2 Cohomology……Page 202
6.3 Cohomology with support in a closed subset……Page 210
6.4 Cohomology with compact support……Page 216
6.5 Comparison theorems……Page 229
7.1 Overconvergent isocrystals on a frame……Page 248
7.2 Overconvergence and calculus……Page 254
7.3 Virtual frames……Page 263
7.4 Cohomology of virtual frames……Page 269
8.1 Overconvergent isocrystal on an algebraic variety……Page 282
8.2 Cohomology……Page 289
8.3 Frobenius action……Page 304
9.1 A brief history……Page 317
9.2 Crystalline cohomology……Page 318
9.3 Alterations and applications……Page 320
9.4 The Crew conjecture……Page 321
9.5 Kedlaya’s methods……Page 322
9.6 Arithmetic D-modules……Page 324
9.7 Log poles……Page 325
References……Page 328
Index……Page 333