{"id":"902a4a3e-c872-4959-9bfc-1edb808920e7","arxiv_id":"2606.18999","paper_version":1,"verdict":"UNVERDICTED","confidence":"LOW","novelty_score":6.0,"correctness_risk":"unknown","formal_verification":"none","parameter_count":0,"one_line_summary":"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.","lead":"The paper formulates Schneider-Teitelbaum duality for Banach k-linear representations of profinite groups over non-spherically complete fields such as C_p. It translates topological weak irreducibility into algebraic simplicity of the dual module and constructs two p-adic families of infinite-dimensional examples.","discovery_kind":"unclear","skeptic_critique":{"model":"grok-4.3","headline":"Topological conditions for 'wide classes' may implicitly require spherical completeness, risking invalidation of weak-irreducibility correspondence over C_p","rationale":"Reader's weakest assumption correctly isolates the compatibility issue. The claim is a formulation plus interpretation, so the load-bearing point is whether the chosen topological conditions avoid the usual dependence on spherical completeness; if they do not, the algebraic simplicity statement does not transfer. Full text would allow checking the definitions directly; absent that verification the result remains conditional on the conditions being genuinely weaker.","tokens_in":1639,"tokens_out":367,"duration_ms":15834,"concrete_test":"Extract the precise definition of the wide classes and the duality functor (e.g., the continuous dual or the module M(V)); recompute the pairing on the standard example V = C(Z_p, C_p) with the sup norm; check whether the dual module is simple precisely when V is weakly irreducible, and whether any step uses a supremum over a countable set of norms that may not exist in C_p.","verdict_should_be":"CONDITIONAL","load_bearing_attack":"The formulation defines wide classes of Banach k-representations via topological conditions (e.g., admissibility, continuity of action, or dual module properties) that must yield a perfect duality pairing and map weak topological irreducibility to algebraic simplicity of the O_k[[G]]-module. Standard p-adic functional analysis (e.g., existence of orthonormal bases, Hahn-Banach extensions, or strictness of dual maps) fails without spherical completeness; if any step in the construction (likely §§2-3) invokes such a property for the pairing or the simplicity equivalence, the classes become empty or the correspondence fails for k=C_p, undermining the two families in the applications.","agreement_with_reader":"agree"},"referee_report":{"model":"grok-4.3","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.","tokens_in":1769,"tokens_out":379,"duration_ms":11250,"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":[{"comment":"§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.","section":"§2"},{"comment":"§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.","section":"§3"}],"minor_comments":[],"recommendation":"major_revision","confidential_remarks":null},"author_rebuttal":{"model":"grok-4.3","summary":"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.","responses":[{"response":"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_made":"yes","referee_comment":"[§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."},{"response":"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_made":"yes","referee_comment":"[§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."}],"tokens_in":1286,"tokens_out":461,"duration_ms":21136,"standing_objections":[]},"desk_editor":{"model":"grok-4.3","letter":"The paper sets up Schneider-Teitelbaum duality between certain Banach k-representations and O_k[[G]]-modules when k is not spherically complete, such as C_p, and maps a weak topological irreducibility condition on the representation side to algebraic simplicity on the module side. It then produces two explicit p-adic families of infinite-dimensional C_p-representations that meet the weak irreducibility condition.\n\nThe extension itself is the concrete advance. Most earlier statements of this duality relied on spherical completeness for the underlying functional analysis, so removing that hypothesis opens the tool to fields that actually appear in p-adic arithmetic. The algebraic reinterpretation of irreducibility is a clean way to turn a topological statement into something checkable on the dual module.\n\nThe applications to families are the part that could matter beyond the subfield. If those families really satisfy the conditions, they give examples that prior work could not reach.\n\nThe main uncertainty is whether the “wide classes” of representations stay non-empty once spherical completeness is dropped. The stress-test note is reasonable: many standard facts about Banach spaces over C_p (orthonormal bases, strictness of dual maps, Hahn-Banach extensions) fail, and the abstract gives no sign of how the duality pairing or the simplicity equivalence is proved without them. If any step in §§2-3 uses one of those facts, the correspondence could collapse or the classes could turn out empty. That is the load-bearing point that needs checking.\n\nThis is for people already working with p-adic representations and profinite group algebras who need duality statements that match the fields they use. A specialist in the area will see the value quickly; outsiders will not.\n\nIt deserves peer review. The claim is narrow and technical enough that a referee can verify whether the extension holds or where it breaks.","headline":"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.","tokens_in":2221,"tokens_out":460,"would_cite":false,"duration_ms":11839,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":[],"pacs":[],"model":"grok-4.3","headline":"Schneider-Teitelbaum duality extends to Banach representations over non-spherically complete fields such as C_p.","keywords":["Schneider-Teitelbaum duality","Banach representations","profinite groups","non-spherically complete fields","weak irreducibility","O_k[[G]]-modules","p-adic Lie groups","p-adic families"],"falsifier":"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.","tokens_in":2535,"feed_emoji":"","tokens_out":717,"duration_ms":17106,"temperature":0.7,"pith_summary":"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.","feed_headline":"Duality between Banach reps and modules holds over C_p","feed_subtitle":"Weak irreducibility of representations translates to algebraic simplicity of dual modules for non-spherically complete fields.","key_machinery":"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.","core_discovery":"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.","pith_inferences":["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."],"forward_implications":["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."],"fun_headline_variants":["Schneider-Teitelbaum duality over non-spherically complete fields","Duality for Banach k-reps and modules over C_p","Weak irreducibility maps to algebraic module simplicity","p-adic families of infinite dim C_p reps are weakly irreducible"],"cache_read_input_tokens":2112,"weakest_assumption_plain":"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.","fun_headline_variants_meta":{"raw":{"variants":["Schneider-Teitelbaum duality over non-spherically complete fields","Duality for Banach k-reps and modules over C_p","Weak irreducibility maps to algebraic module simplicity","p-adic families of infinite dim C_p reps are weakly irreducible"]},"model":"grok-4.3","cost_usd":0.00775,"raw_usage":{"total_tokens":3517,"prompt_tokens":619,"num_sources_used":0,"completion_tokens":60,"cost_in_usd_ticks":77499500,"prompt_tokens_details":{"text_tokens":619,"audio_tokens":0,"image_tokens":0,"cached_tokens":256},"completion_tokens_details":{"audio_tokens":0,"reasoning_tokens":2838,"accepted_prediction_tokens":0,"rejected_prediction_tokens":0}},"tokens_in":619,"tokens_out":60,"duration_ms":20468,"temperature":1.0,"reasoning_tokens":2838,"cache_read_input_tokens":256,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-06-26T19:25:38.180165+00:00","model_set":{"reader":"grok-4.3"},"falsifier":"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.","supporting_citations":[],"review_version":1}