Algebraicity of critical Hecke $L$-values

TL;DR

The work addresses Deligne's conjecture on the algebraicity of critical values $L( ext{χ},0)$ for algebraic Hecke characters, extending the scope from CM fields to arbitrary totally imaginary fields containing a CM subfield. It introduces an equivariant cohomological framework built from Eisenstein–Kronecker classes on CM abelian varieties, enabling the expression of critical $L$-values in terms of CM-periods $Ω$ and $Ω^ abla$, and connecting these values to algebraic data of the motive. It then develops $p$-adic interpolation by constructing $p$-adic measures on relevant Galois groups, with different treatments for totally real and totally imaginary settings, notably the ordinary-CM-type case where a CM-type $p$-ordinary condition yields a genuine measure. The results generalize prior work by Shimura–Katz and Blasius, and provide a unified approach to Deligne’s conjecture in a broad CM-amenable context, including a pathway to reflex-motive considerations and $p$-adic refinements via Mahler-based $p$-adic theta functions.

Abstract

In this survey, we review the known results on the algebraicity of critical values of Hecke $L$-functions and explain the new developments in \cite{Kings-Sprang}.

Theorems & Definitions (6)

• Proposition 1.1
• Theorem 2.1: Siegel-Klingen, Siegel,Klingen
• Theorem 2.2: Kings-Sprang
• Theorem 2.3: Kufner
• Theorem 3.1
• Theorem 3.2: Kings-Sprang