Motivic action for Siegel modular forms

TL;DR

This paper investigates the relationship between motivic cohomology and the coherent cohomology of Siegel modular forms on GSp(4), proposing a conjecture linking Hecke eigensystems to motivic actions and relating it to Beilinson's conjecture.

Contribution

It formulates a new conjecture connecting motivic cohomology to coherent cohomology of Siegel modular forms and proves its equivalence to Beilinson's conjecture under certain conditions.

Findings

Proves the conjecture is equivalent to Beilinson's conjecture for the adjoint L-function.

Constructs motivic cohomology elements for lifts of Hilbert modular forms.

Establishes the conjecture for lifts of Bianchi modular forms, linking to Prasanna-Venkatesh conjecture.

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.