Motivic action for Siegel modular forms

TL;DR

The paper develops a conjectural motivic action of Beilinson regulators on the coherent cohomology of Siegel modular varieties, tying the action to the adjoint motive of Siegel forms and to Beilinson’s conjecture for the adjoint L-function. It provides a precise framework for normalizing contributions from L-packets via Whittaker models, derives explicit regulator formulas, and proves the main rationality result under Beilinson and Deligne conjectures, with unconditional advances in special cases such as Yoshida lifts from real and imaginary quadratic fields. The Hilbert and Bianchi specializations reveal deep compatibilities: in the real-quadratic case via Ramakrishnan asai-factors, and in the imaginary-quadratic case via PV conjectures and theta-lift constructions, suggesting a broader link between motivic actions for hermitian and non-hermitian locally symmetric spaces. The work situates the motivic-action program within a network of period relations (Deligne, Beilinson, Chen–Ichino) and Langlands functoriality, providing both conceptual insights and concrete, computable instances that illuminate the arithmetic of coherent cohomology on Shimura varieties.

Abstract

We study the coherent cohomology of automorphic sheaves corresponding to Siegel modular forms $f$ of low weight on ${\rm GSp}(4)$ Shimura varieties. Inspired by the work of Prasanna--Venkatesh on singular cohomology of locally symmetric spaces, we propose a conjecture that explains all the contributions of a Hecke eigensystem to coherent cohomology in terms of the action of a motivic cohomology group. Under some technical conditions, we prove that our conjecture is equivalent to Beilinson's conjecture for the adjoint $L$-function of $f$. We also prove some unconditional results in special cases. For a lift $f$ of a Hilbert modular form $f_0$ to ${\rm GSp}(4)$, we produce elements in the motivic cohomology group for which the conjecture holds, using the results of Ramakrishnan on the Asai $L$-function of $f_0$. For a lift $f$ of a Bianchi modular form $f_0$ to ${\rm GSp}(4)$, we show that our conjecture for $f$ is equivalent to the conjecture of Prasanna-Venkatesh for $f_0$, thus establishing a connection between the motivic action conjectures for locally symmetric spaces of non-hermitian type and those for coherent cohomology of Shimura varieties.

Figures (4)

• Figure 1: Contributions to coherent cohomology according to the Harish--Chandra parameter $\lambda$. In this example, $\lambda = (3, 2)$. The Harish--Chandra $\lambda$ parameters are labeled by red dots, the Blattner parameters $\Lambda$, given by Table \ref{['table:min_K-types']}, are labeled by green dots, and the $K$-types for the vector bundles $\mathcal{E}_i$ are labeled by blue circles. Serre duality corresponds to reflection about the line $\lambda_1 = -\lambda_2$.
• Figure 2: This diagram indicates the proof of Lemma \ref{['lemma:PK']} for the representation $V_0$ on the left hand side and $V_1$ on the right hand side when $\lambda = (4,1)$. The shaded regions represent the $K^\circ$-types occurring in $X_\lambda^1$ and $X_{\overline \lambda}^2$ and their minimal $K^\circ$-types are indicated by $\Lambda$ and $\overline \Lambda$. We also label the representations $\bigwedge^j \mathfrak P \otimes V_i^\vee$ for $j = 0, 1, 2, 3$ and $i = 0, 1$. One obtains similar diagrams for $i = 2,3$ by reflecting over the $\lambda_2 = -\lambda_1$ axis.
• Figure 3: A graph of $E \cong \mathbb{R}^2 \supseteq \mathbb{Z}^2$ and the root system for $\operatorname{Sp}_4( \mathbb{R})$.
• Figure 4: This shaded region shows the $K$-types occurring in the parabolic induction $\sigma_{\lambda_2} \rtimes D_{\lambda_1}^+$ from the Klingen parabolic $Q$, according to Muic. The central character determines the parity of the occurring $K$-types and we do not indicate this here. We also do not indicate the multiplicities. The subrepresentations $X_\lambda^1$ and $X_{\overline \lambda}^2$ are shown in blue and green, respectively, and the Langlands quotient is shown in pink.

Theorems & Definitions (118)

• Conjecture 1: Conjecture \ref{['conj:motivic_action']}
• Theorem 2: Theorem \ref{['thm:main']}
• Theorem 3: Theorem \ref{['thm:real_quadratic']}
• Theorem 4: Theorem \ref{['thm:HP_implies_PV']}
• Remark 1.2
• Theorem 2.1: Faltings_Chai
• Remark 2.2
• Definition 2.3
• Remark 2.4
• Definition 2.6