$p$-adic properties of Eisenstein-Kronecker cocycles over imaginary quadratic fields and $p$-adic interpolation

TL;DR

This paper studies the $p$-adic properties of Eisenstein-Kronecker cocycles over imaginary quadratic fields, establishing integrality, congruences, and constructing a measure for $p$-adic interpolation of critical L-values.

Contribution

It proves integrality and congruence properties of Eisenstein-Kronecker cocycles and constructs a $p$-adic measure for interpolating critical Hecke L-values over imaginary quadratic fields.

Findings

Proved integrality and congruence properties of Eisenstein-Kronecker cocycles.

Established the integrality of critical Hecke L-values in the split case.

Constructed a $p$-adic measure for interpolating these critical values.

Abstract

We establish integrality and congruence properties for the Eisenstein-Kronecker cocycle of Bergeron, Charollois and Garc\'ia introduced in [arXiv:2107.01992v2 [math.NT]]. As a consequence, we recover the integrality of the critical values of Hecke $L$-functions over imaginary quadratic fields in the split case. Additionally, we construct a $p$-adic measure that interpolates these critical values.