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.

• First book to give a complete treatment of the theory of rigid cohomology, from full proofs for all the basics, to discussion of the most recent developments • Essential for specialists: contains proofs and results not available elsewhere • Accessible for non-specialists: written from a practical point of view, with many worked examples

### Contents

Introduction; 1. Prologue; 2. Tubes; 3. Strict neighborhoods; 4. Calculus; 5. Overconvergent sheaves; 6. Overconvergent calculus; 7. Overconvergent isocrystals; 8. Rigid cohomology; 9. Epilogue; Index; Bibliography.

### Reviews

'…written in a student friendly style.' Zentralblatt MATH

'… well-written, with a mixture of concrete examples and abstract theory. It is accessible to readers familiar with basic concepts of abstract algebraic geometry and p-adic analytic geometry.' European Mathematical Society Newsletter

'… a very nice book, poised to play a role of considerable importance in the literature. Its style is a bit terse but this should be no problem for the reader coming to this field, whose very interest in this fascinating subject bespeaks a commensurate maturity. This brand new book obviously fills an important niche.' MAA Reviews