The Moduli Stack of Breuil-Kisin Modules with Descent Data for Reductive Groups

Eivind Otto Hjelle

REVIEW 3 major objections 4 minor 6 references

The Moduli Stack of Breuil-Kisin Modules with Descent Data for Reductive Groups

T0 review · 3 major / 4 minor · reviewed 2026-08-07 · deepseek-v4-flash

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Pith's one-line read This paper constructs the moduli stack of Breuil–Kisin modules with reductive-group descent data and proves it is smoothly equivalent to a Pappas–Zhu twisted Schubert variety.

desk verdict A careful, honest thesis-level generalization of the GLn local-model story to tamely ramified reductive groups; the main theorem is real work but conditional on an assumption the intro states more loosely than the proofs do. read the letter →

arxiv 2506.11910 v1 pith:5DNOHWHJ submitted 2025-06-13 math.NT

Eivind Otto Hjelle This is my paper · ORCID

classification math.NT MSC 14D2311S3720G25

keywords Breuil-KisinmodulesdescentdatareductivegroupsaffineGrassmanniantwistedSchubertvarietyp-adicLanglandsprogramBruhat-Titsgroupschemesformalalgebraicstacks

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

This paper introduces the moduli stack $Y^{\leq \mu}$ of Breuil–Kisin modules with $\hat{G}$-structure and descent data, called Breuil–Kisin $(\Gamma,\hat{G})$-torsors, where $\Gamma$ is the Galois group of a tamely ramified extension of $\mathbb{Q}_p$ and $\mu$ is a dominant cocharacter bounding the Hodge–Tate weights. The paper proves that $Y^{\leq \mu}$ is a $p$-adic formal algebraic stack and that it is smoothly equivalent to the $p$-adic completion of the Pappas–Zhu twisted Schubert variety $\mathrm{Gr}^{\leq \mu}_G$ attached to $\mu$. This equivalence gives a concrete loop-group model for a moduli problem defined through $p$-adic Hodge theory, extending to general reductive groups the relation previously known for $\mathrm{GL}_n$ and used in the study of crystalline Emerton–Gee stacks and Serre weight conjectures. If the result is right, it makes the geometry of integral crystalline $L$-parameters for reductive groups accessible through affine Grassmannian techniques.

What carries the argument

The central object is the Bruhat–Tits group scheme $G_x$ over $\mathbb{A}^1_O$, obtained as the invariant pushforward of a $\Gamma$-twisted form of the dual group $\hat{G}$ along the ramified cover $u^e = v$; under the connected-fiber assumption it coincides with the Pappas–Zhu group scheme attached to a point $x$ of the enlarged Bruhat–Tits building, where $x$ encodes the Galois type of the descent data. The stack is then realized as the quotient $Y^{\leq \mu} = [L^{\leq \mu}G_x^{\wedge \varpi} /_{c,\varphi}\, L^+G_x^{\wedge \varpi}]$ by the $c$-twisted Frobenius action $X \star A = A^{-1} X \varphi_c(A)$. The proof's engine is straightening: on $\mathbb{Z}/p^a$-algebras, once the depth inequality $(p-1)\lfloor f\rfloor + d - h_\mu - 2a + 2 > 0$ holds, the operator $A \mapsto X\varphi_c(A)X^{-1}$ is a contraction in the $v$-adic metric, so the $c$-twisted $\varphi$-conjugation orbits coincide with left-translation orbits modulo sufficiently deep congruence subgroups; Banach's fixed point theorem turns one action into the other. The negative loop group $L^{--}G_x$ provides the affine open charts $U(z) = L^{--,\leq \mu}G\cdot z$ in $\mathrm{Gr}^{\leq \mu}$, which are used to split the relevant torsors Zariski locally.

What would settle it

Take the explicit failure of connectedness from Example 3.10: $\hat{G} = \mathrm{PGL}_3$ with trivial $\Gamma$-action and $x$ the barycenter of an alcove, so that $G_x$ has disconnected fibers. For a dominant $\mu$ and a Frobenius-invariant type represented by $x$, compute the map $Z^{\wedge\varpi} \to Y^{\leq\mu}$ constructed in Section 6; if any geometric fiber fails to be formally smooth, or the map is not a covering of $p$-adic formal algebraic stacks, then Theorem A fails without Assumption ConFib.

Watch

Extended reading notes

Core claim

The paper's central claim is Theorem A (Theorem 6.4): for every dominant cocharacter $\mu : \mathbb{G}_m \to \hat{T}$, the substack $Y^{\leq \mu}$ of Breuil–Kisin $(\Gamma,\hat{G})$-torsors of height at most $\mu$ is a $p$-adic formal algebraic stack over $\mathrm{Spf}\,O$, and there is a $p$-adic formal scheme $Z^{\wedge \varpi}$ over $\mathrm{Spf}\,O$ with smooth covering maps to both $Y^{\leq \mu}$ and $\mathrm{Gr}^{\leq \mu,\wedge\varpi}_G$. In other words, the two stacks are smoothly equivalent. Under the additional hypotheses that $G$ is unramified, the derived group of $\hat{G}$ is simply connected, and the point $x$ is lowest-alcove and $d$-generic with $d > h_\mu$, Theorem B (Theorem 6.6) upgrades this to explicit Zariski-open charts, indexed by the admissible set $\mathrm{Adm}(\mu)$, on which the equivalence is fibered by $\hat{T}$-torsors.

Load-bearing premise

The main theorems assume that the group scheme $G_x$ has connected fibers; if this fails, the identification of $G_x$ with the Pappas–Zhu group scheme, and hence the smooth equivalence, can break, with an explicit failure for $\hat{G} = \mathrm{PGL}_3$ at the barycenter of an alcove.

Editorial extensions

If this is right

For every dominant cocharacter $\mu$, $Y^{\leq \mu}$ is a $p$-adic formal algebraic stack, giving finite-type approximations to the full moduli stack $Y$ of Breuil–Kisin $(\Gamma,\hat{G})$-torsors.
The smooth equivalence $Y^{\leq \mu} \simeq \mathrm{Gr}^{\leq \mu,\wedge\varpi}_G$ transfers geometric information from the explicit Pappas–Zhu local model to the Breuil–Kisin side.
Under unramified hypotheses with simply connected derived group, the equivalence admits explicit Zariski-open charts indexed by $\mathrm{Adm}(\mu)$, making the local model diagram concrete enough for deformation-ring computations.
The straightening inequalities quantify exactly which congruence subgroups of $L^+G_x$ are needed, with the bound $h_\mu = \max_{a\in\Phi}\langle a,\mu\rangle$ shown to be sharp.
The theorem directly generalizes the local model diagrams of earlier $\mathrm{GL}_n$-type and $\mathrm{GSp}_4$ results to arbitrary tamely ramified reductive groups.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

The paper leaves the monodromy condition to future work; if the smooth equivalence survives after imposing a monodromy condition, it would give local models for crystalline Emerton–Gee stacks for general reductive groups, not just the Breuil–Kisin side.
The connected-fiber assumption is probably not essential at the level of formal algebraic stacks: one could work with the neutral connected component of $G_x$ and track the finite component group through the quotient, with Example 3.10 suggesting the failure is a finite étale phenomenon rather than a breakdown of smoothness in the interior.
Because the straightening estimates are purely group-theoretic and often optimal, they could be implemented algorithmically to compute explicit $p$-adic charts, making deformation-ring computations for reductive groups more tractable.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

Short version: this is a solid PhD-thesis-level paper that genuinely extends the GLn/GSp4 results of [Le+23] and [Lee23] to arbitrary tamely ramified reductive groups, introducing Breuil–Kisin (Γ, Ĝ)-torsors as new objects and proving a local model theorem relating their bounded moduli stack Y^{≤μ} to Pappas–Zhu twisted Schubert varieties. It deserves a serious referee. But the advertised main theorem is slightly broader than what the proofs establish, because Assumption ConFib (connected fibers of the Bruhat–Tits group scheme Gx) is load-bearing.

What is actually new: the moduli stack Y and its quotient presentation via the c-twisted Frobenius action, the invariant pushforward formalism in Section 2.6 and Appendix A, the building-theoretic classification of Galois types (Proposition 2.23), and the contraction/straightening machinery in Section 5 with explicit bounds. The proofs are detailed and the assumptions are stated up front. The paper is honest about its limitations: Remark 1.15 and Section 3.1.4 flag ConFib, Example 3.10 shows it can fail (PGL3 at an alcove barycenter), and the main theorems are stated conditionally there.

The soft spot is packaging, not correctness. Theorem A in the abstract and introduction reads unconditional, but Proposition 3.11's identification of Gx with the Pappas–Zhu group scheme requires connected fibers, and Sections 4–6 run under ConFib throughout. When ConFib fails, the central smooth equivalence is simply not proved. Since the author explicitly flags this, it is not a hidden flaw—but a referee should ask that the main theorems carry the hypothesis in their statements, or that the disconnected-fiber case be addressed.

I found the standing assumptions clear and the contraction bounds plausible; I did not verify every estimate, but nothing looks circular. The results are conditional on ConFib (and Dil/genericity for Theorem B), which limits scope but is honestly disclosed. For people working on Emerton–Gee stacks or categorical p-adic Langlands, this is a useful and citable step.

Recommendation: send it to peer review. The referee should check whether ConFib can be weakened and push for a statement of Theorem A that matches the proofs.

Referee Report

3 major / 4 minor

Summary. The paper introduces a moduli stack ℤ of Breuil–Kisin modules with descent data and Ġ-structure, called Breuil–Kisin (Γ,Ġ)-torsors, for a tamely ramified reductive group G over Q_p. For a dominant cocharacter μ it defines the bounded substack ℤ^{≤μ} by a group-theoretic height condition and proves (Theorem A, Theorem 6.4) that ℤ^{≤μ} is a p-adic formal algebraic stack admitting a smooth covering diagram connecting it to the p-adic completion of the Pappas–Zhu twisted Schubert variety Gr^{≤μ}_G. Under unramifiedness, simple-connectivity of the derived group, and genericity hypotheses, Theorem B (Theorem 6.6) gives explicit Zariski charts. The proof uses invariant pushforward to construct Bruhat–Tits group schemes G_x over A^1_O, identifies them with Pappas–Zhu models under the connected-fibers assumption ConFib, and proves contraction/straightening estimates in Section 5. The paper explicitly states in Remark 1.15 and Section 2.7 that the main theorems are proven under Assumption ConFib and, for results using the negative loop group, Assumption Dil.

Significance. If the results are correct, this is a substantial generalization of the local model theorem for moduli of Breuil–Kisin modules from GL_n and GSp_4 to general tamely ramified reductive groups, with a group-theoretic height condition that avoids choosing an embedding into GL_n. The paper provides detailed proofs of the formal algebraic stack property, explicit contraction bounds in Section 5, Zariski chart constructions via negative loop groups, and an extension of Pappas–Zhu group schemes to concave functions (Proposition 3.19). The main theorems are honestly conditional on clearly stated assumptions, and the failure of the connected-fibers condition is explicitly exhibited in Example 3.10. The main concern is that the abstract and introductory theorem statements do not prominently carry this standing assumption, so the advertised scope is broader than the proof establishes.

major comments (3)

[§1.5, §2.7, §3.1.4, Example 3.10] The abstract and the introductory statement of Theorem A present the smooth equivalence as unconditional, but the proof depends on Assumption ConFib, which is imposed from Section 3.1.5 onward. Proposition 3.11 identifies G_x with the Pappas–Zhu group scheme only when G_x has connected fibers, and Example 3.10(2) shows that ConFib fails for Ġ = PGL_3 at the barycenter of an alcove. No argument in Sections 4–6 covers the disconnected-fiber case. Since the paper itself flags this in Remark 1.15, this is not a hidden inconsistency, but the abstract and the main theorem statements should carry ConFib explicitly; otherwise the claimed scope is broader than what is proved.
[§1.5 Theorem B, §2.7, §3.1.4] Theorem B as stated omits Assumption Dil and the condition F ⊇ F_q (or E ⊇ L) that Proposition 3.16 requires for G_x to be a dilation, even though Theorem B uses the negative loop group of Section 3.3. Please state these hypotheses in the theorem, or explicitly say that they are part of the standing assumptions in force.
[Proposition 3.19, §3.1.8] The proof of Proposition 3.19 is carried out in detail only under the assumption that I acts trivially on Ġ; the general case is dispatched by saying that it follows by arguing with ~G_{x,f} and passing to neutral connected components. Since the group schemes G_{x,f} underlie the congruence subgroups used in the straightening argument and in Corollary 4.14, the general case should either be written out or the results that rely on it should be restricted to the unramified case.

minor comments (4)

[Definition 3.12] The displayed characterization of “lowest alcove” contains a stray brace: the condition should read “−1 < ⟨a,β⟩ ≤ 0” with no closing brace after the final 0.
[Lemma 3.35] The description of G_x(R[1/(v+p)]) uses the notation “A mod v/(v+p)” without explicitly indicating the ideal (namely v/(v+p) · R[1/(v+p)]); spelling this out would improve readability.
[Section 2.7.1] The fixed choices list n, τ, x, and c, but the dependence of x on the choice of w and λ, and the role of the unramified extension L′ in Lemma 2.37(2), could be stated more explicitly so that the reader can track which choices are canonical and which are auxiliary.
[Remark 2.47] The phrase “it is also possible to be d-deep” appears to conflate the paper’s notion of d-genericity with the notion of depth used in [Le+23]; please align the terminology.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the local model theorem is proved from external building-theoretic input and internal contraction/straightening estimates, not by fitting or self-citation.

full rationale

The central derivation is self-contained in the relevant sense. The stack Y≤μ is defined as a quotient of L≤μG_x by the c-twisted φ-conjugation action, while Gr≤μ is the Pappas–Zhu local model; even though L≤μG is defined as the preimage of Gr≤μ under LG→Gr, the claimed smooth equivalence of the two quotient stacks is not true by construction. It is established by a detailed contraction analysis (Lemmas 5.10–5.12) showing that, after passing to sufficiently deep congruence subgroups, the c-twisted φ-conjugation orbits coincide with left-translation orbits. No parameter is fitted to the target: the c-twisted action is forced by Frobenius invariance of the Galois type (Section 2.5.3 and Proposition 4.11), and the height bound μ is an input rather than an output. Identifications with Pappas–Zhu group schemes cite [PZ13], an external benchmark, and are explicitly conditional on Assumption ConFib; the paper flags when this assumption fails (Example 3.10 and Remark 1.15). The only self-citation, to the author's thesis [Hje24], occurs in the acknowledgements and is not load-bearing. Thus there is no self-definitional step, no fitted input called a prediction, and no load-bearing self-citation chain.

Assumptions & free parameters 0 free parameters · 7 assumptions · 3 invented entities

The construction is parameter-free in the sense of no data fitting; the inputs are the chosen Galois type tau, the point x in the building, and the cocharacter mu, which are part of the theorem statements. The key axioms are the domain assumptions (tame ramification, large coefficient field) and the paper-specific assumptions ConFib and Dil, which limit the scope of the main theorems. Standard theorems from Bruhat-Tits theory and Lang's theorem are used without proof.

assumptions (7)

domain assumption L is a tamely ramified Galois extension of Qp with #mu_e(F_q) = e
Fixed throughout (Section 2.1); guarantees e is prime to p and allows a uniformizer with minimal polynomial u^e + p, which underlies the variable u = v^{1/e}.
domain assumption E is a large finite extension of Qp containing L, with ring of integers O
Coefficient field for the moduli stacks (Section 2.1.1); enlarging E resolves rationality issues, and E containing L is used for the type classification.
ad hoc to paper Assumption ConFib: Gx has connected fibers
Standing assumption for all main theorems (Section 3.1.4); not always true (Example 3.10); used to identify Gx with Pappas-Zhu group schemes (Proposition 3.11) and to prove the smooth equivalence of Theorem A.
ad hoc to paper Assumption Dil: Gx is a dilation of G-hat in a parabolic subgroup
Assumed for the negative loop group and the explicit charts in Theorem B (Section 3.3); needed to prove that L+G times L--G is formally etale in LG (Proposition 3.38).
ad hoc to paper The Galois type tau is Frobenius invariant: phi([tau]) = [tau]
Assumed in Section 2.7.1; guarantees existence of c with c.phi(x) = x and yields the quotient presentation of Y as [LG^{∧ varpi}/_{c,phi} L+G^{∧ varpi}]. This holds for types arising from Breuil-Kisin modules in characteristic zero when I acts trivially (Proposition 2.35).
standard math Steinberg's theorem: H^1(I, G-hat(F((u)))) = 0
Used in the proof of Proposition 2.23 to obtain the building-theoretic classification of Galois types.
standard math Lang's theorem for reductive groups over finite fields
Used in the proof of Lemma 2.37 to find c in G-hat(F_J) with b = c^{-1} phi(c), enabling strict Frobenius invariance.

invented entities (3)

Breuil-Kisin (Gamma, G-hat)-torsor (also called Breuil-Kisin module with G-hat-structure and descent data)
purpose: Models integral crystalline L-parameters for a connected reductive group G over Qp with tame ramification, generalizing Breuil-Kisin modules with descent data for GLn.
Introduced in Definition 1.9; it is the central object of study, defined purely by the paper, with no independent evidence beyond the definition.
Moduli stack Y^{≤ mu} of bounded height
purpose: Moduli of Breuil-Kisin (Gamma, G-hat)-torsors with Hodge-Tate weights bounded by mu; the target of the local model theorem.
Defined as a quotient stack in Section 4.2.1; its smooth equivalence to Gr^{≤ mu} is the main theorem, so it has no independent evidence.
Group scheme Gx (and variants Gx,f)
purpose: Descents (Gamma, G-hat)-torsors to torsors over the affine line, connecting the construction to Pappas-Zhu Bruhat-Tits group schemes; used to construct Y.
Constructed via invariant pushforward in Section 3.1.1 and identified with Pappas-Zhu group schemes under ConFib; a mathematical tool with no independent falsifiable handle.

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Pith. "Pith review of The Moduli Stack of Breuil-Kisin Modules with Descent Data for Reductive Groups." pith.science (2026). https://pith.science/paper/5DNOHWHJ

@misc{pith2026250611910,
  author       = {Pith},
  title        = {Pith review of: The Moduli Stack of Breuil-Kisin Modules with Descent Data for Reductive Groups},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/5DNOHWHJ}},
  note         = {Machine review of arXiv:2506.11910}
}

abstract

We introduce and study the moduli stack $\mathcal{Y}$ of Breuil-Kisin modules with $\hat{G}$-structure and descent data, or Breuil-Kisin $(\Gamma,\hat{G})$-torsors for short. Specifically, for a dominant cocharacter $\mu$, we define the moduli stack $\mathcal{Y}^{\leq \mu}$ of Breuil-Kisin $(\Gamma,\hat{G})$-torsors with Hodge-Tate weights bounded by $\mu$. We prove that $\mathcal{Y}^{\leq \mu}$ is a $p$-adic formal algebraic stack, and show that it is smoothly equivalent to (the $p$-adic completion of) a twisted Schubert variety $\operatorname{Gr}^{\leq \mu}_{\mathcal{G}}$ in the sense of Pappas-Zhu. This is a reformatted and lightly edited version of the author's PhD thesis, submitted to Northwestern University in August 2024.

Figures

Figures reproduced from arXiv: 2506.11910 by the authors.

**Figure 1.** A snapshot of the apartment Ae T , b F((v)) = Ae T , b F((u))I , when Tb ⊂ Gb = GL3 is the diagonal torus and e = 6. The small triangles bounded by think gray lines are the alcoves of Ae T , b F((u)) , whereas the large triangles bounded by thick black lines are the alcoves of Ae T , b F((v)) . 2.3.5. The building. The enlarged building Be G, b F J ((u)) is the quotient of Gb(F J ((u)))×Ae T , b F J… view at source ↗

**Figure 2.** The base alcove of A T , b F((v)) where Tb ⊂ SL2 is the diagonal torus, for p = 7 and e = 24. The large nodes are vertices of A T , b F((v)) and the small nodes are vertices of A T , b F((u)) . The white nodes are vertices of A T , b F((u)) which are not in the orbit of o. The green nodes classify types which are Frobenius invariant, and the red nodes classify types which are not Frobenius invariant. Sin… view at source ↗

**Figure 3.** A snapshot of the apartment Ae T , b F((v)) , where Tb ⊂ GL3 is the diagonal torus (the same one that we saw in [PITH_FULL_IMAGE:figures/full_fig_p037_3.png] view at source ↗

Figures from the paper (3 more)

**Figure 4.** Figure 4: An alcove in the apartment A (S, F((v))), where S ⊂ GL3 is the diagonal torus, when p = 19. The large triangles bounded by the thick black lines are the walls of the apartment A (S, F((v))). The smaller triangles bounded by gray lines represent the walls of the apartme…

**Figure 5.** Figure 5: A snapshot of A (S, F((v))), where S ⊂ GL3 is the diagonal torus. Shaded alcoves correspond to elements of Adm(µ ′ ). The simple reflections se1, se2 and se3 correspond to the reflections across the walls of C of the same colors, and we can factorize any we in the affi…

**Figure 6.** Figure 6: A snapshot of the apartment Ae(U3, F((v))) = Ae(GL3, F((u)))I (the solid black line) viewed inside Ae(GL3, F((u))) (the faint blue lines), for e = 6. A point classifying a Galois type corresponds to one of the black dots, for example x = o + (1/6, 0, −1/6). The three l…

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Works this paper leans on

6 extracted references · 3 canonical work pages

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Pith tools

Reviewed August 7, 2026 · model on record in the stance chip above.

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Conformal Blocks Attached to Twisted Groups

doi: 10.24033/asens.2354. [Con14] Brian Conrad. Reductive Group Schemes . 2014. url: https://math.stanford.edu/ ˜conrad/papers/luminysga3.pdf. [Dam17] Chiara Damiolini. “Conformal Blocks Attached to Twisted Groups”. PhD thesis. Uni- versit¨ at Duisburg-Essen, 2017. [DG80] Michel Demazure and Peter Gabriel. Introduction to Algebraic Geometry and Algebraic ...

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N´ eron Models and Tame Ramification

doi: 10.1112/plms.12544. [Edi92] Bas Edixhoven. “N´ eron Models and Tame Ramification”. In: Compositio Mathematicae 81.3 (1992), pp. 291–306. [EG23] Matthew Emerton and Toby Gee. Moduli stacks of ´ etale(φ, Γ)-modules and the existence of crystalline lifts . Annals of mathematics studies no. 215. Princeton University Press,

work page doi:10.1112/plms.12544 1992

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doi: 10.1090/S0894-0347-06-00542-X . [Wil95] Andrew Wiles. “Modular Elliptic Curves and Fermat’s Last Theorem”. In: The Annals of Mathematics 141.3 (1995), p. 443. doi: 10.2307/2118559. [Yu02] Jiu Kang Yu. “Smooth Models Associated to Concave Functions in Bruhat–Tits The- ory”. In: 2002. url: https://api.semanticscholar.org/CorpusID:16192779. [Zhu14] Xinw...

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Phi-modules and coefficient spaces

doi: 10.1007/s40062-014-0083-6. [NSW08] J¨ urgen Neukirch, Alexander Schmidt, and Kay Wingberg.Cohomology of Number Fields. Grundlehren Der Mathematischen Wissenschaften. Springer Berlin Heidelberg, 2008. doi: 10.1007/978-3-540-37889-1 . 90 REFERENCES [PR08] Georgios Pappas and Michael Rapoport. “Twisted Loop Groups and Their Affine Flag Varieties”. In: A...

work page Pith review arXiv doi:10.48550/arxiv.0811.1170 2008

[5] [2001]

Emerton--Gee stacks, Serre weights, and Breuil--M\'ezard conjectures for $\mathrm{GSp}_4$

276 pp. doi: 10.1515/9781400837205. [Kis06] Mark Kisin. “Crystalline Representations and F-crystals”. In: Algebraic Geometry and Number Theory: In Honor of Vladimir Drinfeld’s 50th Birthday . Birkh¨ auser Boston, 2006, pp. 459–496. doi: 10.1007/978-0-8176-4532-8_7 . [Kis09] Mark Kisin. “Moduli of Finite Flat Group Schemes, and Modularity”. In: Annals of M...

work page Pith review arXiv 2009

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Zero Divisors and Nilpotent Elements in Power Series Rings

doi: 10.2307/j.ctv2nv8q48. [EGH23] Matthew Emerton, Toby Gee, and Eugen Hellmann. An Introduction to the Categorical P-Adic Langlands Program. 2023. arXiv: 2210.01404 [math]. Pre-published. [Fie71] David E. Fields. “Zero Divisors and Nilpotent Elements in Power Series Rings”. In: Proceedings of the American Mathematical Society 27.3 (1971), pp. 427–433. d...

arXiv 1971