The motivic action conjecture for Hilbert modular forms

Let $F$ be a real quadratic field and let $f$ be a normalized weight-one Hilbert modular eigenform. Let $X$ be the associated Hilbert modular surface, let $\omega$ be its Hodge bundle, and let $\omega_f^1,\omega_f^2\in H^1(X,\omega)\otimes_\mathbb{C}$ be the associated coherent cohomology classes. Let $u_{ij}$ be the Stark units and set

$$R_f:=\begin{pmatrix}\log|u_{11}|&\log|u_{12}|\\log|u_{21}|&\log|u_{22}|\end{pmatrix}.$$

The motivic action conjecture. The classes

$$\frac{\log|u_{22}|\omega_f^1-\log|u_{12}|\omega_f^2}{\det R_f},\qquad \frac{-\log|u_{21}|\omega_f^1+\log|u_{11}|\omega_f^2}{\det R_f}$$

lie in $H^1(X,\omega)\otimes_\mathbb{Q}\mathbb{Q}(f)$. Equivalently, they are rational coherent cohomology classes. This conjecture relates rationality of higher coherent cohomology classes to motivic cohomology and Stark regulators; the supplied text records a result of Stark for rational traces but does not resolve the motivic action conjecture in general.