The compatibility of absolute Hodge classes with the étale–de Rham comparison

Let $X$ be a smooth projective variety over a number field $F$. Let $(s_{\mathbb A_f},s_{\mathrm{dR}})$ be an absolute Hodge class in

$$\bigl(H^w_{\mathrm{\acute et}}(X_{\overline F},\mathbb A_f)\times H^w_{\mathrm{dR}}(X/F')\bigr)^{\otimes}.$$

Let $s_p\in H^w_{\mathrm{\acute et}}(X_{\overline F},\mathbb Q_p)$ be the corresponding $p$-adic étale cohomology class. Compatibility conjecture. Under Faltings' étale–de Rham comparison isomorphism, $s_p$ maps to $s_{\mathrm{dR}}$. The conjecture extends the known compatibility for abelian varieties to arbitrary smooth projective varieties over number fields. The source presents the abelian-variety case as a theorem of Blasius; the general case remains open.