  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.

Number Theory;Algebraic Geometry