arxiv: 2604.21605 · v2 · submitted 2026-04-23 · 🧮 math.NT

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Galois representations over convergent de Rham period ring

Hui Gao , Yupeng Wang

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Pith reviewed 2026-05-12 00:51 UTC · model grok-4.3

classification 🧮 math.NT

keywords Galois representationsde Rham period ringconvergent functionsSen weightsGalois cohomologyTate-Sen formalismp-adic non-Liouville conditionalgebraic Sen weights

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The pith

When Sen weights satisfy a p-adic non-Liouville condition, Galois cohomology of representations over the convergent de Rham period ring is finite.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper develops a Tate-Sen formalism that connects Galois representations over the convergent de Rham period ring B_dR^{+,†} to regular connections on convergent functions. This formalism shows that, under the non-Liouville condition on Sen weights of the mod t reduction, the Galois cohomology of such a representation matches that of its base change to the larger ring B_dR^+ and is therefore finite. When the Sen weights are algebraic numbers, the categories of representations over the two rings become equivalent.

Core claim

By establishing a Tate-Sen formalism for Galois representations over B_dR^{+,†}, the authors prove that Galois cohomology compares to the B_dR^+ case and is finite whenever the Sen weights satisfy the p-adic non-Liouville condition, and that the categories of B_dR^{+,†}-representations and B_dR^+-representations are equivalent when restricted to objects with algebraic Sen weights.

What carries the argument

The Tate-Sen formalism relating Galois representations over the convergent de Rham period ring to regular connections over convergent functions.

If this is right

Galois cohomology groups over B_dR^{+,†} are finite under the non-Liouville condition on Sen weights.
The cohomology of a B_dR^{+,†}-representation equals that of its base change to B_dR^+.
Categories of representations over the convergent and completed rings are equivalent when Sen weights are algebraic numbers.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

Properties known for representations over the completed de Rham ring can transfer to the convergent subring under the non-Liouville condition.
This comparison could simplify computations of Galois cohomology by reducing to the convergent case in p-adic arithmetic settings.
Explicit examples with non-Liouville Sen weights could verify the finiteness result by direct calculation.

Load-bearing premise

The Sen weights of the mod t reduction satisfy a p-adic non-Liouville condition.

What would settle it

A specific B_dR^{+,†}-representation with non-Liouville Sen weights whose Galois cohomology group is infinite or differs from the cohomology of its base change to B_dR^+.

read the original abstract

Let $\mathbf{B}_{\mathrm{dR}}^{+, \dagger} \subset \mathbf{B}_{\mathrm{dR}}^{+}$ be the ``convergent" de Rham period ring which is the (un-completed) stalk at the de Rham point of the Fargues--Fontaine curve. We develop a Tate--Sen formalism to relate Galois representations over $\mathbf{B}_{\mathrm{dR}}^{+, \dagger}$ to regular connections over convergent functions. As a consequence, when the Sen weights (of the mod $t$ reduction) satisfy a $p$-adic non-Liouville condition, Galois cohomology of a $\mathbf{B}_{\mathrm{dR}}^{+, \dagger}$-representation compares to that of its $\mathbf{B}_{\mathrm{dR}}^{+}$-base change, and hence is finite. In addition, restricted to objects whose Sen weights are algebraic numbers, the categories of $\mathbf{B}_{\mathrm{dR}}^{+, \dagger}$-representations and $\mathbf{B}_{\mathrm{dR}}^{+}$-representations are equivalent.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

This paper adapts Tate-Sen theory to the convergent de Rham period ring B_dR^{+,†} and obtains cohomology finiteness plus a category equivalence under non-Liouville and algebraic-weight conditions.

read the letter

The core contribution is a Tate-Sen formalism that links Galois representations over the uncompleted convergent stalk B_dR^{+,†} to regular connections, then uses the p-adic non-Liouville condition on Sen weights of the mod-t reduction to compare cohomology with the completed B_dR^+ base change (hence finiteness) and, for algebraic weights, to get an equivalence of categories. This is new in its specific tailoring to the convergent ring and the resulting statements; it does not simply restate earlier results on the completed ring or the Fargues-Fontaine curve. The work sits squarely on established Tate-Sen machinery and prior properties of de Rham rings, which keeps the circularity burden low and makes the claims traceable. The non-Liouville hypothesis is stated explicitly as the safeguard against divergence, which is standard in the area rather than an ad-hoc fix. The algebraic-weight restriction for the equivalence is also clearly delimited, so the scope is honest. The main soft spot is that the non-Liouville condition, while common, narrows the range of representations one can immediately apply the finiteness result to; readers will need to check case-by-case whether their Sen weights satisfy it. The connection to regular connections is sketched at the level of the abstract, and any gaps in handling the mod-t reduction or edge cases would only surface in the full proofs. Overall the central argument looks internally consistent and grounded in the literature. This is for people already working in p-adic Hodge theory or arithmetic geometry who need finer control over period rings for cohomology calculations. A reader who already knows the completed case and wants the convergent variant will find the formalism useful. It deserves a serious referee because the results are concrete, the hypotheses are explicit, and the potential applications to Galois cohomology are clear even if the broader impact stays within the subfield. I would send it to peer review.

Referee Report

2 major / 2 minor

Summary. The paper introduces the convergent de Rham period ring B_dR^{+,†} as the uncompleted stalk at the de Rham point of the Fargues-Fontaine curve. It develops a Tate-Sen formalism relating Galois representations over B_dR^{+,†} to regular connections over convergent functions. As consequences, under a p-adic non-Liouville condition on the Sen weights of the mod t reduction, the Galois cohomology of a B_dR^{+,†}-representation is finite via comparison to its B_dR^+ base change; moreover, when Sen weights are algebraic numbers the categories of B_dR^{+,†}-representations and B_dR^+-representations are equivalent.

Significance. If the central claims hold, the work supplies a useful extension of Tate-Sen theory into the convergent setting of the Fargues-Fontaine curve. The cohomology finiteness result and the category equivalence under algebraic weights would furnish new tools for p-adic Hodge theory, particularly for controlling Galois cohomology of representations over non-complete period rings and for comparing categories of representations with different convergence conditions.

major comments (2)

[Section 4] The non-Liouville hypothesis on Sen weights of the mod-t reduction is load-bearing for both the cohomology comparison and the finiteness statement. The manuscript should include an explicit verification that this condition prevents divergence in the Sen operator and ensures the comparison isomorphism between H^*(G_K, V) and H^*(G_K, V ⊗ B_dR^+) is well-defined and bijective (see the statement following the development of the Tate-Sen formalism).
[Section 5] For the category equivalence when Sen weights are algebraic, the proof relies on the Tate-Sen formalism producing an equivalence of categories between B_dR^{+,†}-representations and regular connections. It is not immediately clear from the setup whether the algebraic-weight restriction is used only to guarantee that the connection is regular or whether additional convergence arguments are needed; a precise statement of the equivalence functor and its inverse would strengthen the claim.

minor comments (2)

[Introduction] The notation B_dR^{+,†} is introduced without an explicit comparison to the standard completed ring B_dR^+ in the introductory paragraphs; a short diagram or inclusion statement would clarify the relationship for readers unfamiliar with the Fargues-Fontaine stalk construction.
[Section 2] Several references to prior Tate-Sen results (e.g., the classical case over B_dR) are invoked without page or theorem numbers; adding precise citations would improve traceability.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and constructive comments on our manuscript. We address the major comments point by point below and will make the suggested clarifications in the revised version.

read point-by-point responses

Referee: [Section 4] The non-Liouville hypothesis on Sen weights of the mod-t reduction is load-bearing for both the cohomology comparison and the finiteness statement. The manuscript should include an explicit verification that this condition prevents divergence in the Sen operator and ensures the comparison isomorphism between H^*(G_K, V) and H^*(G_K, V ⊗ B_dR^+) is well-defined and bijective (see the statement following the development of the Tate-Sen formalism).

Authors: We agree that an explicit verification would improve the exposition. In Section 4, the non-Liouville condition on the Sen weights of the mod-t reduction is used to ensure that the Sen operator has no eigenvalues in a certain p-adic disk, which prevents divergence of the relevant series in the Tate-Sen formalism and allows the comparison map to be defined on cohomology. This in turn yields bijectivity with the base change to B_dR^+. We will add a short dedicated paragraph immediately after the statement of the comparison isomorphism, recalling the relevant estimates from the formalism and verifying that non-Liouville precisely rules out the divergent cases. revision: yes
Referee: [Section 5] For the category equivalence when Sen weights are algebraic, the proof relies on the Tate-Sen formalism producing an equivalence of categories between B_dR^{+,†}-representations and regular connections. It is not immediately clear from the setup whether the algebraic-weight restriction is used only to guarantee that the connection is regular or whether additional convergence arguments are needed; a precise statement of the equivalence functor and its inverse would strengthen the claim.

Authors: The algebraic-weight hypothesis is used exactly to guarantee regularity of the connection produced by the Tate-Sen formalism; once regularity holds, the equivalence follows directly from the general Tate-Sen correspondence already developed, without further convergence restrictions. The functor sends a B_dR^{+,†}-representation to its associated Sen connection (a regular connection on the convergent functions), while the inverse reconstructs the representation by integrating the Sen differential equation. We will insert a precise statement of both directions of the equivalence, together with a short remark confirming that no extra convergence arguments are required beyond regularity, at the beginning of Section 5. revision: yes

Circularity Check

0 steps flagged

Minor self-citation of prior theory; central derivation remains independent

full rationale

The derivation develops a Tate-Sen formalism relating B_dR^{+,†}-representations to regular connections, then invokes the explicit p-adic non-Liouville condition on Sen weights of the mod-t reduction to obtain the cohomology comparison and category equivalence. This builds on established Tate-Sen theory and Fargues-Fontaine/de Rham ring properties from prior literature without any reduction of the central claims to self-referential definitions, fitted inputs renamed as predictions, or load-bearing self-citation chains. The non-Liouville hypothesis is stated as an external safeguard, and the algebraic-weights equivalence is a direct consequence under that hypothesis rather than a circular renaming or ansatz smuggling.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The results rest on standard background from p-adic Hodge theory and the geometry of the Fargues-Fontaine curve; no free parameters or new invented entities are introduced in the abstract.

axioms (2)

domain assumption Standard properties of the de Rham period rings B_dR^+ and the Fargues-Fontaine curve, including the definition of the convergent stalk B_dR^{+,†}.
Invoked throughout the development of the Tate-Sen formalism and comparison of cohomologies.
domain assumption Existence and basic properties of Sen weights for mod t reductions of representations.
Central to stating the non-Liouville condition and the algebraic weight equivalence.

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