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On higher regulators of Picard modular surfaces

Linli Shi

TL;DR

The paper advances Beilinson’s conjectures in the setting of Picard modular surfaces by constructing nontrivial motivic classes in $H^3_M(S,V(2))$ via Eisenstein symbols and showing their Beilinson regulators land in a well-behaved subspace tied to interior cohomology. It then establishes a precise link between these motivic classes and non-critical values of motivic L-functions attached to cuspidal automorphic representations of GU(2,1), encoding this relation in explicit period comparisons and L-value formulas. A core technical achievement is the simultaneous vanishing-on-the-boundary analysis in both absolute Hodge and motivic cohomology, implemented through boundary triangles, Burgos–Wildeshaus/Kostant degeneration, and motivic weight structures, yielding a motivic lifting of boundary-vanishing phenomena. The results provide evidence for Beilinson-type conjectures in a higher-rank unitary setting, and offer a framework for relating Eisenstein-derived motivic classes to automorphic L-values via explicit regulators and period relations, with potential implications for constructing Euler systems in families.

Abstract

We prove the motivic classes in the motivic cohomology groups of Picard modular surfaces with non-trivial coefficients constructed in a paper of Loeffler\textendash Skinner\textendash Zerbes are in the motivic cohomology groups of the interior motives. Then we establish a relation between the motivic classes and non-critical values of the motivic $L$-functions associated to cuspidal automorphic representations of $\mathrm{GU}(2,1)$, thus deducing non-triviality of the motivic classes and providing evidence for Beilinson's conjectures.

On higher regulators of Picard modular surfaces

TL;DR

The paper advances Beilinson’s conjectures in the setting of Picard modular surfaces by constructing nontrivial motivic classes in via Eisenstein symbols and showing their Beilinson regulators land in a well-behaved subspace tied to interior cohomology. It then establishes a precise link between these motivic classes and non-critical values of motivic L-functions attached to cuspidal automorphic representations of GU(2,1), encoding this relation in explicit period comparisons and L-value formulas. A core technical achievement is the simultaneous vanishing-on-the-boundary analysis in both absolute Hodge and motivic cohomology, implemented through boundary triangles, Burgos–Wildeshaus/Kostant degeneration, and motivic weight structures, yielding a motivic lifting of boundary-vanishing phenomena. The results provide evidence for Beilinson-type conjectures in a higher-rank unitary setting, and offer a framework for relating Eisenstein-derived motivic classes to automorphic L-values via explicit regulators and period relations, with potential implications for constructing Euler systems in families.

Abstract

We prove the motivic classes in the motivic cohomology groups of Picard modular surfaces with non-trivial coefficients constructed in a paper of Loeffler\textendash Skinner\textendash Zerbes are in the motivic cohomology groups of the interior motives. Then we establish a relation between the motivic classes and non-critical values of the motivic -functions associated to cuspidal automorphic representations of , thus deducing non-triviality of the motivic classes and providing evidence for Beilinson's conjectures.

Paper Structure

This paper contains 85 sections, 111 theorems, 484 equations.

Table of Contents

  1. Introduction
  2. Motivation
  3. Statement of the main results
  4. Geometric results
  5. Results on special values of $L$-functions
  6. An overview of the proof
  7. An outline of the proof of the geometric results
  8. An outline of the proof of the $L$-value results
  9. Structure of the paper
  10. Notations and Conventions
  11. Preliminaries
  12. Algebraic groups, algebraic representations and branching laws
  13. Algebraic groups
  14. Algebraic representations
  15. Branching law
  16. ...and 70 more sections

Key Result

Theorem 1.1

For $V = V^{a, b}\{r, s\}$ as in Definition def:rep_G, under certain conditions (see Theorem Thm: Hdg vanish on the boudary), the map $\mathcal{E}is_{H}^n\colon \mathcal{B}_{n,\mathbb{R}} \rightarrow \mathrm{H}^{3}_{H}(S, V(2))$ factors through the inclusion where $\mathrm{MHS}_{\mathbb{R}}^{+}$ is the abelian category of mixed $\mathbb{R}$-Hodge structures, ${\mathbf{1}}$ denotes the trivial Hod

Theorems & Definitions (289)

  • Theorem 1.1: Theorem \ref{['Thm: Hdg vanish on the boudary']}
  • Remark 1.2
  • Theorem 1.3: Theorem \ref{['Thm: mot vanish on the boudary']}
  • Remark 1.4
  • Theorem 1.5: Theorem \ref{['Them_mot_L_function']}
  • Remark 1.6
  • Remark 1.7
  • Definition 3.2
  • proof
  • Remark 3.4
  • ...and 279 more