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

Jitendra Bajpai, Mattia Cavicchi

TL;DR

The paper develops a geometric realization of predicted extensions in Bloch-Beilinson conjectures for odd-weight algebraic Hecke characters over an imaginary quadratic field by leveraging the cohomology of Picard modular surfaces and Harder’s Eisenstein cohomology framework. It constructs a candidate extension E_φ in Ext^1_MHS(1, H_φ((w+1)/2)) from boundary and interior cohomology data when ε(φ)=−1 and L(φ,−1)=0, and it situates these extensions inside a broader program that uses biextension heights to detect non-triviality and connect to L′(φ,−1). Rogawski’s automorphic results on Picard surfaces provide the necessary cuspidal components in interior cohomology, while the interplay with Eisenstein cohomology supplies the geometric origin of the extensions. The work outlines a concrete program to compute biextension heights and relate them to special L-value derivatives, aiming to connect geometric cycles, Abel–Jacobi maps, and Deligne periods to the Bloch-Beilinson predictions for Hecke characters. Overall, the paper sets a roadmap for proving non-trivial, geometrically realizable extensions that encode central-vanishing data in the L-functions of Hecke characters via Picard-surface geometry and automorphic methods.

Abstract

We consider certain families of Hecke characters $φ$ over a quadratic imaginary field $F$. According to the Bloch-Beilinson conjectures, the order of vanishing of the $L$-function $L(φ,s)$ 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 $φ$. In this article, we construct candidates for the corresponding extensions of Hodge structures, assuming that the sign of the functional equation of $L(φ,s)$ 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.

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

TL;DR

The paper develops a geometric realization of predicted extensions in Bloch-Beilinson conjectures for odd-weight algebraic Hecke characters over an imaginary quadratic field by leveraging the cohomology of Picard modular surfaces and Harder’s Eisenstein cohomology framework. It constructs a candidate extension E_φ in Ext^1_MHS(1, H_φ((w+1)/2)) from boundary and interior cohomology data when ε(φ)=−1 and L(φ,−1)=0, and it situates these extensions inside a broader program that uses biextension heights to detect non-triviality and connect to L′(φ,−1). Rogawski’s automorphic results on Picard surfaces provide the necessary cuspidal components in interior cohomology, while the interplay with Eisenstein cohomology supplies the geometric origin of the extensions. The work outlines a concrete program to compute biextension heights and relate them to special L-value derivatives, aiming to connect geometric cycles, Abel–Jacobi maps, and Deligne periods to the Bloch-Beilinson predictions for Hecke characters. Overall, the paper sets a roadmap for proving non-trivial, geometrically realizable extensions that encode central-vanishing data in the L-functions of Hecke characters via Picard-surface geometry and automorphic methods.

Abstract

We consider certain families of Hecke characters over a quadratic imaginary field . According to the Bloch-Beilinson conjectures, the order of vanishing of the -function at the central point should be equal to the dimension of the space of extensions of the Tate motive by the motive associated with . In this article, we construct candidates for the corresponding extensions of Hodge structures, assuming that the sign of the functional equation of is . This is accomplished through the cohomology of variations of Hodge structures over Picard modular surfaces associated with 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.
Paper Structure (37 sections, 24 theorems, 198 equations)

This paper contains 37 sections, 24 theorems, 198 equations.

Table of Contents

  1. Introduction
  2. Ideas of construction and evidence for non-triviality
  3. Our program
  4. Related work and further future directions
  5. Structure of the article
  6. Preliminaries
  7. Basic notations and conventions
  8. Structure Theory of $\mathrm{G}$
  9. Root System
  10. Irreducible Representations
  11. Standard $\mathbb{Q}$-Parabolic Subgroups
  12. Kostant Representatives
  13. The associated symmetric spaces and Shimura varieties
  14. Local systems on $\mathrm{S}_K({\mathbb{C}})$
  15. Hodge structures with coefficients and variations of Hodge structures on $\mathrm{S}_K({\mathbb{C}})$
  16. ...and 22 more sections

Key Result

Theorem 1

Let $k$ be a positive integer, and let $\phi$ be a Hecke character of $F$ of type $(k,-k-3)$ or $(-k-3,k)$, satisfyingNote the hypothesis on $\phi^{\perp}$ is equivalent to asking that the restriction $\phi_{{\mathbb{Q}}}$ of $\phi$ to ${\mathbb{I}}_{{\mathbb{Q}}}$ verifies \phi_{{\mathbb{Q}}}= \ome If $\epsilon(\phi)=-1$, then there exists an extension of geometric origin.

Theorems & Definitions (69)

  • Conjecture 1: Conjecture \ref{['BBHecke_Hodge']}
  • Theorem 1: Theorem \ref{['sourcext1']}.(\ref{['ext_mainthm']})
  • Remark 2.1
  • Remark 2.2
  • Theorem 2.3
  • Remark 2.4
  • Remark 2.5
  • Remark 2.6
  • Remark 3.1
  • Remark 3.2
  • ...and 59 more