Dwork's $p$-adic proof of rationality

Abstract: In 1959, ex-electrical engineer Bernard Dwork shocked the mathematical world by proving the first Weil conjecture on the rationality of the zeta function. Dwork's proof introduced striking new $p$-adic methods, and defied the expectation that the Weil conjectures could only be solved by developing a suitable Weil cohomology theory (later found to be $l$-adic etale cohomology). In this talk we will outline Dwork's proof and begin the initial part of the argument, introducing Dwork's general notion of "splitting functions", the Artin-Hasse exponential and Dwork's lemma.

Reference: Koblitz, p-adic numbers, p-adic analysis, and zeta-functions, pp. 92-95 and then Section V.2 to the end of the book, some of which may be covered in a second lecture.

algebraic geometrynumber theory

Audience: advanced learners

( slides )

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STAGE

Series comments: STAGE (Seminar on Topics in Arithmetic, Geometry, Etc.) is a learning seminar in algebraic geometry and number theory, featuring speakers talking about work that is not their own. Talks will be at a level suitable for graduate students. Everyone is welcome.

Spring 2024 topic: The contents of Serre, Lectures on the Mordell-Weil theorem

Some topics might take more or less time than allotted. If a speaker runs out of time on a certain date, that speaker might be allowed to borrow some time on the next date. So the topics below might not line up exactly with the dates below.

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