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Schneider-Teitelbaum duality extends to Banach representations over non-spherically complete fields such as C_p.

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Extends Schneider-Teitelbaum duality to non-spherically complete fields, equates weak irreducibility with algebraic simplicity of the dual, and gives p-adic families of C_p-representations of p-adic Lie groups.

T0 review reviewed 2026-06-26 challenge →

load-bearing objection Extends Schneider-Teitelbaum duality to non-spherically complete fields like C_p with an algebraic take on weak irreducibility, but the abstract leaves the key topological conditions unexamined. the 2 major comments →

arxiv 2606.18999 v1 pith:YTLROWEY submitted 2026-06-17 math.NT math.RT

Schneider--Teitelbaum Duality over a Non-spherically Complete Field

Tomoki Mihara This is my paper
classification math.NT math.RT
keywords Schneider-Teitelbaum dualityBanach representationsprofinite groupsnon-spherically complete fieldsweak irreducibilityO_k[[G]]-modulesp-adic Lie groupsp-adic families
verification ladder T0 review T1 audit T2 compute T3 formal T4 reserved

The pith

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

The reading

The paper formulates Schneider-Teitelbaum duality between wide classes of Banach k-linear representations of a profinite group G and left O_k[[G]]-modules when the scalar field k is not spherically complete. It shows that a topological weak-irreducibility condition on the representations corresponds exactly to an algebraic simplicity condition on the dual modules. Applications include explicit p-adic families of infinite-dimensional Banach C_p-linear representations of p-adic Lie groups that satisfy this weak irreducibility. A sympathetic reader would care because the result removes a common completeness restriction that previously limited such dualities in p-adic representation theory.

Core claim

We formulate Schneider--Teitelbaum duality between wide classes of Banach k-linear representations of G and left O_k[[G]]-modules for a non-spherically complete field k, e.g. C_p, and a profinite group G. We interpret a topological notion of a weak variant of irreducibility of a Banach k-linear representation of G into a purely algebraic notion of a certain simplicity of the dual left O_k[[G]]-module. As applications, we give two p-adic families of infinite dimensional Banach C_p-linear representations of a p-adic Lie group satisfying the weak irreducibility.

What carries the argument

The duality pairing that sends a Banach k-linear representation to its dual left O_k[[G]]-module, converting topological weak irreducibility into algebraic simplicity.

Load-bearing premise

The topological conditions that define the wide classes of representations are assumed to stay compatible with non-spherical completeness of k without extra restrictions that would invalidate the algebraic simplicity statement.

What would settle it

An explicit Banach C_p-linear representation of a p-adic Lie group that meets the topological conditions of the wide class yet whose dual O_k[[G]]-module fails to be simple under the duality pairing.

Watch this falsifier. Get emailed when new claim-graph text bears on it.

If this is right

  • Weak irreducibility of representations becomes verifiable by checking algebraic simplicity of the corresponding modules.
  • Two explicit p-adic families of infinite-dimensional Banach C_p-linear representations of p-adic Lie groups are weakly irreducible.
  • The duality applies directly to profinite groups and non-spherically complete scalar fields without requiring spherical completeness.
  • The correspondence preserves the structure needed for infinite-dimensional examples.

Where Pith is reading between the lines

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

  • The algebraic reformulation may allow module-theoretic tools to classify representations that were previously studied only topologically.
  • Similar dualities could be tested on other non-complete local fields arising in arithmetic geometry.
  • The construction of parametric families suggests a route to deforming representations while preserving the simplicity condition.
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Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

2 major / 0 minor

Summary. The manuscript formulates Schneider--Teitelbaum duality between wide classes of Banach k-linear representations of a profinite group G and left O_k[[G]]-modules, where k is a non-spherically complete field such as C_p. It equates a topological notion of weak irreducibility for the representations with algebraic simplicity of the dual modules and applies the result to two p-adic families of infinite-dimensional Banach C_p-linear representations of a p-adic Lie group.

Significance. If the construction is valid, the result would extend classical duality theorems to fields where spherical completeness fails, which is relevant for p-adic representation theory and Hodge theory over C_p. The explicit families of representations constitute a concrete application that could be checked independently.

major comments (2)
  1. [§2] §2 (definition of wide classes): The topological conditions used to define the admissible Banach representations and the duality pairing must be shown not to invoke properties (such as existence of orthonormal bases, Hahn-Banach extensions, or strictness of dual maps) that require spherical completeness; otherwise the classes become empty or the weak-irreducibility correspondence fails for k = C_p, undermining the central claim.
  2. [§3] §3 (duality statement and simplicity equivalence): The proof that the duality pairing induces an equivalence between weak topological irreducibility and algebraic simplicity of the O_k[[G]]-module must be checked for any implicit use of spherical-completeness-dependent functional analysis; the abstract provides no derivation details, so this step is load-bearing for both the duality and the applications.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and for highlighting the need to make explicit the independence from spherical completeness. We address each major comment below.

read point-by-point responses

  1. Referee: [§2] §2 (definition of wide classes): The topological conditions used to define the admissible Banach representations and the duality pairing must be shown not to invoke properties (such as existence of orthonormal bases, Hahn-Banach extensions, or strictness of dual maps) that require spherical completeness; otherwise the classes become empty or the weak-irreducibility correspondence fails for k = C_p, undermining the central claim.

    Authors: In §2 the admissible classes are defined using only the Banach norm and the topology of continuous linear maps over a complete valued field; no orthonormal bases, Hahn-Banach extensions, or strictness of dual maps are invoked. The duality pairing is constructed via the completed projective tensor product, which remains well-defined without spherical completeness. The concrete families constructed in §4 over C_p already demonstrate that the classes are non-empty. We will add a short verification paragraph in §2 listing the functional-analytic tools that are deliberately avoided. revision: yes

  2. Referee: [§3] §3 (duality statement and simplicity equivalence): The proof that the duality pairing induces an equivalence between weak topological irreducibility and algebraic simplicity of the O_k[[G]]-module must be checked for any implicit use of spherical-completeness-dependent functional analysis; the abstract provides no derivation details, so this step is load-bearing for both the duality and the applications.

    Authors: The equivalence proof in §3 translates the topological definition of weak irreducibility (absence of proper closed invariant subspaces) directly into the algebraic statement that the dual module has no proper submodules, using only the adjointness of the pairing and the definition of the O_k[[G]]-action. No further functional-analytic results are required after the pairing is established. We will insert a brief remark after the main theorem in §3 that records the absence of spherical-completeness-dependent steps. revision: yes

Circularity Check

0 steps flagged

No circularity: duality formulation and weak-irreducibility correspondence are presented as new constructions without reduction to fitted inputs or self-citations.

full rationale

The abstract and available description present a direct formulation of Schneider--Teitelbaum duality for non-spherically complete k (e.g., C_p) together with a topological-to-algebraic correspondence for weak irreducibility. No equations, parameter fits, or load-bearing self-citations are visible that would make any claimed prediction or simplicity statement equivalent to its own inputs by construction. The topological conditions defining the 'wide classes' are stated as assumptions compatible with the setting, and the applications to p-adic families are derived from the new duality rather than presupposing it. This satisfies the default expectation of a self-contained derivation against external benchmarks in p-adic representation theory.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The work rests on standard domain assumptions of p-adic analysis and profinite group representations; no free parameters or invented entities are visible in the abstract.

axioms (1)
  • domain assumption k is a complete non-spherically complete valued field (e.g., C_p) and G is profinite.
    Explicitly stated as the setting in the abstract.

reviewed 2026-06-26 · how reviews work

0 comments
Cite this review

Pith. "Pith review of Schneider--Teitelbaum Duality over a Non-spherically Complete Field." pith.science (2026). https://pith.science/paper/YTLROWEY

@misc{pith2026260618999,
  author       = {Pith},
  title        = {Pith review of: Schneider--Teitelbaum Duality over a Non-spherically Complete Field},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/YTLROWEY}},
  note         = {Machine review of arXiv:2606.18999}
}
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abstract

We formulate Schneider--Teitelbaum duality between wide classes of Banach $k$-linear representations of $G$ and left $O_k[[G]]$-modules for a non-spherically complete field $k$, e.g.\ $\mathbb{C}_p$, and a profinite group $G$. We interpret a topological notion of a weak variant of irreducibility of a Banach $k$-linear representation of $G$ into a purely algebraic notion of a certain simplicity of the dual left $O_k[[G]]$-module. As applications, we give two $p$-adic families of infinite dimensional Banach $\mathbb{C}_p$-linear representations of a $p$-adic Lie group satisfying the weak irreducibility.

discussion (0)

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Forward citations

Cited by 1 Pith paper

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score.

  1. A Mackey criterion for locally analytic representations

    math.NT 2026-07 accept novelty 6.5

    Compact induction of a uniformly simple admissible locally analytic representation from an open compact-mod-centre subgroup is topologically irreducible when there are no nontrivial intertwiners with its conjugates.

Reference graph

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This paper was first reviewed by grok-4.3 on June 26, 2026.