Bloch-Beilinson conjectures for Hecke characters and Eisenstein cohomology of Picard surfaces

TL;DR

This paper explores the Bloch-Beilinson conjectures for Hecke characters over imaginary quadratic fields, constructing Hodge structure extensions via Eisenstein cohomology on Picard surfaces, and proposes a method to verify their non-triviality.

Contribution

It introduces a new approach to realize and analyze extensions of Hodge structures associated with Hecke characters using Eisenstein cohomology and biextensions on Picard modular surfaces.

Findings

Constructed candidates for Hodge structure extensions assuming negative functional equation sign.

Demonstrated these extensions are realized within certain biextensions.

Outlined a program to compute biextension heights and verify non-triviality.

Abstract

We consider certain families of Hecke characters $\phi$ over a quadratic imaginary field $F$. According to the Bloch-Beilinson conjectures, the order of vanishing of the $L$-function $L(\phi,s)$ at the central point $s=-1$ should be equal to the dimension of the space of extensions of the Tate motive $\mathbb{Q}(1)$ by the motive associated with $\phi$. In this article, we construct candidates for the corresponding extensions of Hodge structures, assuming that the sign of the functional equation of $L(\phi,s)$ is $-1$. This is accomplished through the cohomology of variations of Hodge structures over Picard modular surfaces associated with $F$ and Harder's theory of Eisenstein cohomology. Furthermore, we demonstrate that these extensions are naturally realized within certain biextensions. We outline a program to compute the biextension height and utilize it to establish the non-triviality of these extensions.