To an algebraic variety over the complex numbers, we can associate a complex analytic space. When the result is a smooth complex manifold, we can compute its de Rham cohomology. I would like to discuss some ways to compute this cohomology directly from our algebraic variety, and how these methods can be adapted to more general varieties. None of the material I will present is original. The results are due to many people, especially Grothendieck.

In joint work with Barry Mazur, we show that over every number field there are many elliptic curves of rank zero, and (assuming the finiteness of Shafarevich-Tate groups) many elliptic curves of rank one.

Combining our results about ranks of twists with ideas of Poonen and Shlapentokh, we show that if one assumes the finiteness of Shafarevich-Tate groups of elliptic curves, then Hilbert's Tenth Problem is undecidable (i.e., has a negative answer) over the ring of integers of every number field.

The aim of the talk is to describe the overconvergent de Rham-Witt complex. It is a subcomplex of the de Rham-Witt complex and it can be used to compute Monsky-Washnitzer cohomology for affine varieties, and rigid cohomology in general. (All our varieties are over a perfect field of characteristic $p$.)

We will begin by reviewing Monsky-Washnitzer cohomology and the de Rham-Witt complex. Next we will define overconvergent Witt vectors and then the overconvergent de Rham-Witt complex. As time permits, we will say something about the proof of the comparison theorem between Monsky-Washnitzer cohomology and overconvergent de Rham-Witt ohomology.

This is joint work with Andreas Langer and Thomas Zink.