Sheaves and Functions Modulo p

The Woods Hole trace formula is a Lefschetz fixed-point theorem for coherent cohomology on algebraic varieties. It leads to a version of the sheaves-functions dictionary of Deligne, relating characteristic-p-valued functions on the rational points of varieties over finite fields to coherent modules equipped with a Frobenius structure. This book begins with a short introduction to the homological theory of crystals of Bockle and Pink with the aim of introducing the sheaves-functions dictionary as quickly as possible, illustrated with elementary examples and classical applications. Subsequently, the theory and results are expanded to include infinite coefficients, L-functions, and applications to special values of Goss L-functions and zeta functions. Based on lectures given at the Morningside Center in Beijing in 2013, this book serves as both an introduction to the Woods Hole trace formula and the sheaves-functions dictionary, and to some advanced applications on characteristic p zeta values.Introduction; 1. tau-sheaves, crystals, and their trace functions; 2. Functors between categories of crystals; 3. The Woods Hole trace formula; 4. Elementary applications; 5. Crystals with coefficients; 6. Cohomology of symmetric powers of curves; 7. Trace formula for L-functions; 8. Special values of L-functions; Appendix A. The trace formula for a transversal endomorphism.