Dwork's $p$-adic proof of rationality, continued

Abstract: We will go over the main steps of Dwork's argument in detail. First, we will construct a splitting function for the standard additive character and show it has good convergence properties using Dwork's lemma. Next we will establish the "analytic Lefschetz fixed point formula" by studying the trace of this splitting function acting on $p$-adic Banach spaces of power series. Finally, we will show this analytic fixed point formula implies the zeta-function is the ratio of two entire functions, and conclude with a general rationality criterion for $p$-adic power series that implies the zeta-function is rational.

Reference: Koblitz, p-adic numbers, p-adic analysis, and zeta-functions, whatever remains of Chapter V after the first 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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