Singular cohomology and algebraic de Rham cohomology are both functors from the category of smooth projective algebraic varieties over $\mathbb Q$ to $\mathbb Q$-vectors spaces. They come with the extra structure of Weil cohomology theories - grading, cup product, orientation map, and cycle class map. We can consider the space of all invertible natural transformations from singular cohomology to algebraic de Rham cohomology that respect this structure. For purely formal reasons, this is an affine scheme. The coordinates are the matrix coefficients of the natural transformation for each algebraic variety, and the relations are given by the various commutative diagrams that must hold. Conjecturally, this scheme is a torsor for the motivic Galois group of $\mathbb Q$.

de Rham's theorem, together with GAGA, gives us an isomorphism between these two functors when tensored with $\mathbb C$. In other words, it gives us a $\mathbb C$-point of this affine scheme.

Grothendieck's period conjecture says that this point is a generic point - so the ring of periods in $\mathbb C$ is equal to the whole ring of functions on this scheme.

On the other hand, consider the $p$-adic analogue. $p$-adic etale cohomology and algebraic De Rham cohomology are both functors from the category of smooth projective algebraic varieties over $\mathbb Q_p$ to $\mathbb Q_p$-vector spaces. They are Weil cohomology theories.

We can consider the space of all isomorphisms between these two functors as an affine scheme over $\mathbb Q_p$.

$p$-adic Hodge theory gives a $B_{dR}$-valued point of this scheme.

However, it is not the generic point. The reason is that the isomorphism described by $p$-adic Hodge theory naturally factors through the category of $\operatorname{Gal}(\overline{\mathbb Q}_p|\mathbb Q_p)$-representations. Hence it lies in a closed subscheme that is a torsor under, not the full motivic Galois group, but the Zariski closure of $\operatorname{Gal}(\overline{\mathbb Q}_p|\mathbb Q_p)$ inside it. This group is much smaller because the Tate conjecture fails over $\mathbb Q_p$, and a map of Tannakian categories that is not a full functor corresponds to a map of Tannakian groups that is not surjective.

So the analogue of the period conjecture fails badly over $\mathbb Q_p$. This leads me to my question: Grothendieck's period conjecture essentially gives us a description of the scheme of isomorphisms betweeen singular and de Rham cohomology - it's the spectrum of the period ring. However, we do not have a similar picture of the scheme of isomorphisms between $p$-adic etale cohomology and de Rham cohomology. The period ring only gives us one point.

Can we explicitly describe any other points of this scheme of isomorphisms?

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