p-adic cohomology: from theory to practice

These notes (somewhat revised from the version presented at the 2007 AWS) present a few facets of the relationship between p-adic analysis, algebraic de Rham cohomology, and zeta functions of algebraic varieties. A key theme is the explicit, computable nature of these constructions, which makes them suitable for numerical calculations. For instance, if you ask the computer algebra system Magma for the order of the Jacobian of a hyperelliptic curve over a field of small characteristic, this order is computed using p-adic cohomology. The same is true if you ask the system Sage for the p-adic regulator of an elliptic curve over Q, for p a good ordinary prime.