Eisenstein-Kronecker classes, integrality of critical values of Hecke $L$-functions and $p$-adic interpolation

TL;DR

This paper proves the algebraicity and $p$-adic interpolation of critical values of Hecke $L$-functions for totally complex fields, using new cohomological constructions and Eisenstein-Kronecker series, extending previous results to all cases.

Contribution

It introduces a novel cohomology class construction and explicit calculations that establish algebraicity and $p$-adic interpolation of critical $L$-values for arbitrary totally complex fields.

Findings

Critical $L$-values divided by periods are algebraic integers.

Constructed a $p$-adic measure interpolating critical $L$-values.

Extended algebraicity and interpolation results to all cases of Hecke $L$-functions.

Abstract

We show that for an arbitrary totally complex number field $L$ the (regularized) critical $L$-values of algebraic Hecke characters of $L$ divided by certain periods are algebraic integers. This relies on a new construction of an equivariant coherent cohomology class with values in the completion of the Poincar\'e bundle on an abelian scheme $\cal{A}$. From this we obtain a cohomology class for the automorphism group of a CM abelian scheme $\cal{A}$ with values in some canonical bundles, which can be explicitly calculated in terms of Eisenstein-Kronecker series. As a further consequence, using an infinitesimal trivialization of the Poincar\'e bundle, we construct a $p$-adic measure interpolating the critical $L$-values in the ordinary case. This generalizes previous results for CM fields by Damerell, Shimura and Katz and settles the algebraicity and $p$-adic interpolation in the remaining open cases of critical values of Hecke $L$-functions.