Deligne's conjecture on the critical values of Hecke $L$-functions

TL;DR

This paper proves Deligne's conjecture on the critical values of Hecke L-functions for all algebraic Hecke characters, extending previous results from CM fields to arbitrary totally imaginary fields using cohomological methods.

Contribution

It introduces a new approach linking Eisenstein-Kronecker classes to de Rham classes of reflex motives, generalizing prior work to broader number fields.

Findings

Proves Deligne's conjecture for arbitrary algebraic Hecke characters.

Establishes a cohomological interpretation of L-values using Eisenstein-Kronecker classes.

Extends results from CM fields to all totally imaginary fields.

Abstract

In this paper we give a proof of Deligne's conjecture on the critical values of $L$-functions for arbitrary algebraic Hecke characters. This extends a result of Blasius, which only works in the case of CM fields. The key new insight is that the Eisenstein-Kronecker classes of Kings-Sprang, which allow for a cohomological interpretation of the value $L(\chi,0)$ for Hecke characters $\chi$ of arbitrary totally imaginary fields, can be regarded as de Rham classes of Blasius' reflex motive.