I will first give an exposition of Dwork’s Theorem about zeta functions on varieties over finite fields, together with the other parts of the Weil conjectures, highlighting the connection to the Riemann zeta function, because we are all analytic number theorists here. I will then introduce non-archemedian valuations and p-adic numbers and outline Dwork’s proof of his theorem, emphasizing the passage from finite fields to p-adic fields. This will be the first part of a two-part talk; the second part will be given in the Geometry and Analysis seminar on March 21, and both parts should be accessible to a general grad student audience independently of each other.

Talk by Alex Song.