Construction of Katz $p$-adic $L$-functions

Abstract: We will describe Katz's construction of a $p$-adic measure on the $p^{\infty}$ ray class group of CM fields, whose Mellin transform is a $p$-adic $L$-function interpolating critical values of Hecke $L$-functions. First, we will recall some basics of measures and the construction of the $p$-adic modular form-valued Eisenstein measure. Next, we will obtain Katz's measure by evaluating the Eisenstein measure at CM points. Finally, we will recover the aforementioned interpolation via Katz's insight that the values of the $p$-adic and complex differential operators at CM points coincide, which follows from the moduli-theoretic definitions of these operators.

algebraic geometrynumber theory

Audience: advanced learners

( paper | 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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