The scheme parametrizing natural isomorphisms between ( p )-adic étale cohomology and algebraic de Rham cohomology, respecting their Weil cohomology structures, is deeply tied to the motivic Galois group and the formalism of Tannakian categories. Here's a structured answer:

Key Background:

• Weil Cohomology Theories: Both ( p )-adic étale cohomology (( H^{\text{et}} )) and de Rham c...

Short answer

• On the “Galois” sub-Tannakian category generated by the p-adic étale cohomology of smooth proper varieties over Qp, the scheme of tensor isomorphisms between the forgetful fiber functor (étale) and the de Rham fiber functor is, in a precise Tannakian sense, represented by Spec BdR. In particular, there are no points over any smaller Qp–algebra: any R–valued point necessarily factor...

This is a fascinating question that touches on deep aspects of p-adic Hodge theory and the motivic formalism. Let me address several key points:

The Structure of the Problem

You've correctly identified that the scheme of isomorphisms between p-adic étale cohomology and algebraic de Rham cohomology over is much richer than what classical p-adic Hodge theory reveals. The comparis...

Let

• H = Hét(–, ℚp) : Var/ℚp → Vecℚp
• HdR = Hdr(–/ℚp) : Var/ℚp → Vecℚp

be the two Weil co-homology theories on smooth projective ℚp–varieties that you fixed. Put

Perp := Isom⊗(H, HdR)

(the affine ℚp-scheme whose R-valued points are the tensor-compatible, graded, cup–product–, orientation– and cycle–class-preserving natural isomorphisms
H⊗ℚpR ≃ HdR⊗ℚpR).

1. What is already ...

Here is, in my view, what one can and cannot say about “other” points on the p-adic comparison torsor

 Iso⊗(H_et, H_dR) over ℚₚ.

1) What we know: Fontaine’s p-adic Hodge theory produces a single distinguished point

 s_{dR} ∈ Iso⊗(H_et, H_dR)(B_{dR})

namely the usual B_{dR}-comparison isomorphism
H_et^(X)⊗{ℚₚ}B{dR} ≃ H_{dR}^(X)⊗{ℚₚ}B{dR}.
Because this factors through Galois r...

This is an excellent and deeply insightful question that touches upon the foundational differences between archimedean and non-archimedean Hodge theory, and it leads directly to some of the most exciting recent developments in arithmetic geometry.

You have perfectly summarized the situation. Let's formalize it slightly to set the stage.

The Formal Setup

Let be the Tannakian ...

Let

ω ₑₜ : Var / ℚ_p → Vec_{ℚ_p}, X ↦ ⊕i H^{i}{et}(X_{\bar{ℚ}p},ℚ_p)
ω{dR} : Var / ℚ_p → Vec_{ℚ_p}, X ↦ ⊕i H^{i}{dR}(X/ℚ_p)

denote the two standard Weil co-homology functors (with all usual extra structures).
Write

P := Isom^{⊗}(ω ₑₜ , ω_{dR})

for the fpqc sheaf that assigns to a ℚ_p–algebra R the set of invertible tensor-compatible natural transformations

ω ₑₜ ⊗_{ℚ_p...

<think> Okay, the user is asking a highly technical question about Grothendieck's period conjecture and p-adic Hodge theory. They want to know if there are explicit descriptions of points in the scheme of isomorphisms between p-adic étale cohomology and de Rham cohomology beyond the one provided by the period ring.

Hmm, this is deep arithmetic geometry territory. Let me unpack the context first....

Grothendieck's Period Conjecture and p-adic Cohomology Isomorphisms

This is an excellent question about a deep topic in arithmetic geometry. Let me provide some insights on the scheme of isomorphisms between p-adic étale cohomology and de Rham cohomology.

The Current Situation

As you correctly note, p-adic Hodge theory gives us a B_dR-valued point on the scheme of isomorphisms Isom(H_ét, H...