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arxiv: 2605.18426 · v2 · pith:JH4AWYCFnew · submitted 2026-05-18 · 🧮 math.NT

Classicality for Hilbert modular forms

Yuanyang Jiang This is my paper

Pith reviewed 2026-05-19 23:48 UTC · model grok-4.3

classification 🧮 math.NT
keywords Hilbert modular formscompleted cohomologyclassicalityGalois representationsJacquet-Langlands correspondenceLanglands-Clozel-Fontaine-Mazur conjecturetotally real fieldsde Rham representations
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The pith

A character of the spherical Hecke algebra in the completed cohomology of Hilbert modular varieties is modular if the associated Galois representation is absolutely irreducible and de Rham of regular parallel weights.

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

The paper establishes that certain characters of the spherical Hecke algebra arising in the completed cohomology of Hilbert modular varieties over a totally real field correspond to classical modular forms, provided the attached Galois representation meets the stated conditions. This classicality statement yields new cases of the conjecture that relates Galois representations to automorphic forms on GL2. The argument proceeds by generalizing an existing method, computing geometric partial Fontaine operators, and analyzing the cohomology of associated Koszul-type partial de Rham complexes. The central technical advance is the construction of a locally analytic Jacquet-Langlands transfer that moves information between different quaternionic Shimura data at infinite level.

Core claim

Let F be a totally real number field. A character of the spherical Hecke algebra appearing in the completed cohomology of Hilbert modular varieties is modular if the associated Galois representation is absolutely irreducible and de Rham of regular parallel weights. As an application, this yields new cases of the Langlands-Clozel-Fontaine-Mazur conjecture for GL2 over totally real fields. The proof relies on generalizing prior techniques, explicit calculation of geometric partial Fontaine operators, and the cohomology of Koszul-type partial de Rham complexes, with the key step being a locally analytic Jacquet-Langlands transfer obtained from comparisons of Igusa stacks for different quaternio

What carries the argument

the locally analytic Jacquet-Langlands transfer, which moves information between the cohomology of Hilbert modular varieties and quaternionic Shimura varieties at infinite level via stack comparisons and Grothendieck-Messing deformation theory

If this is right

  • New instances of the Langlands-Clozel-Fontaine-Mazur conjecture become available for GL2 over totally real fields.
  • Characters in the completed cohomology that satisfy the Galois conditions must arise from classical Hilbert modular forms of the expected weight.
  • The method extends the reach of classicality results to infinite-level settings where only locally analytic data were previously accessible.
  • Partial Fontaine operators and Koszul complexes can be used to detect modularity in other completed-cohomology settings.

Where Pith is reading between the lines

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

  • The same stack-comparison technique may apply to classicality questions for other groups or other types of Shimura varieties where infinite-level cohomology appears.
  • If the transfer construction generalizes, it could connect completed-cohomology statements to p-adic Langlands correspondences beyond the Hilbert modular case.
  • Explicit checks for small totally real fields could test whether the regular parallel weight condition is strictly necessary or can be relaxed.

Load-bearing premise

The locally analytic Jacquet-Langlands transfer holds after comparing Igusa stacks for different quaternionic Shimura data and applying Grothendieck-Messing theory to locally analytic infinite-level Shimura varieties.

What would settle it

An explicit computation, for a concrete totally real cubic field, of an absolutely irreducible de Rham Galois representation of regular parallel weights whose associated Hecke character in the completed cohomology does not lift to a classical Hilbert modular form.

Figures

Figures reproduced from arXiv: 2605.18426 by Yuanyang Jiang.

Figure 1
Figure 1. Figure 1: Parabolic Induction or Jacquet-Langlands transfer with self-explanatory notation on the RHS. This allows us to reverse-engineer information about Msm from the excision sequence. 1.3. Organization of the paper. The paper is organized as follows: In Section 2, we review the general theory of Igusa stacks following [Zha23], [DHKZ24], [Kim25], [DHKZ26]. Then we define the quaternionic Shimura varieties, and th… view at source ↗
read the original abstract

Let $F$ be a totally real number field. We prove that a character of the spherical Hecke algebra appearing in the completed cohomology of Hilbert modular varieties is modular if the associated Galois representation is absolutely irreducible, and de Rham of regular parallel weights. As an application, we prove some new cases of the Langlands-Clozel-Fontaine-Mazur conjecture of $\mathrm{GL}_2$ over totally real fields. For the proof, we generalize the method in [Pan26], calculate geometric partial Fontaine operators, and study the cohomology of the associated Koszul-type partial de Rham complexes. The key step is the establishment of a locally analytic Jacquet-Langlands transfer, whose proof consists of several novel ingredients including a comparison of Igusa stacks for different quaternionic Shimura data constructed by [DvHKZ26], and the Grothendieck-Messing theory for locally analytic infinite level Shimura varieties established in [Jiang26a].

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.

Referee Report

2 major / 1 minor

Summary. The paper proves that a character of the spherical Hecke algebra appearing in the completed cohomology of Hilbert modular varieties over a totally real field F is modular precisely when the associated Galois representation is absolutely irreducible and de Rham of regular parallel weights. The argument generalizes the method of Pan26 by computing geometric partial Fontaine operators and studying the cohomology of associated Koszul-type partial de Rham complexes; the central step is a locally analytic Jacquet-Langlands transfer constructed via comparison of Igusa stacks for distinct quaternionic Shimura data (building on DvHKZ26) together with Grothendieck-Messing theory for locally analytic infinite-level Shimura varieties (building on Jiang26a). As an application, new cases of the Langlands-Clozel-Fontaine-Mazur conjecture for GL_2 over totally real fields are obtained.

Significance. If the locally analytic transfer is rigorously established, the result supplies a new classicality criterion in completed cohomology and yields concrete new instances of the Langlands-Clozel-Fontaine-Mazur conjecture. The reduction to prior works on Igusa stacks and infinite-level Grothendieck-Messing theory is a strength, provided the necessary compatibilities with Hecke operators and filtrations are verified in the infinite-level analytic setting.

major comments (2)
  1. [proof of locally analytic Jacquet-Langlands transfer] In the proof of the locally analytic Jacquet-Langlands transfer (the key step highlighted in the abstract), the comparison of Igusa stacks for different quaternionic Shimura data must be shown to intertwine the spherical Hecke operators on the source and target; without an explicit statement that the transfer map preserves the locus of characters whose associated Galois representations are de Rham of regular parallel weights, the classicality statement does not follow.
  2. [Grothendieck-Messing theory for locally analytic infinite level Shimura varieties] The invocation of Grothendieck-Messing theory for locally analytic infinite-level Shimura varieties requires a precise check that the period map and Hodge filtration data extend to the infinite-level analytic setting without extra hypotheses on the level or the weights; if this extension fails on the locus where the representation is de Rham, the transfer is not defined on the relevant characters and the main theorem is unsupported.
minor comments (1)
  1. [abstract and introduction] The abstract cites [Pan26], [DvHKZ26] and [Jiang26a] but the introduction or bibliography should include full bibliographic details and a brief statement of which results from each reference are used verbatim versus which are extended.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their careful reading and constructive comments on the manuscript. We address the major comments point by point below, clarifying the relevant arguments and indicating where we will strengthen the exposition in a revised version.

read point-by-point responses

  1. Referee: [proof of locally analytic Jacquet-Langlands transfer] In the proof of the locally analytic Jacquet-Langlands transfer (the key step highlighted in the abstract), the comparison of Igusa stacks for different quaternionic Shimura data must be shown to intertwine the spherical Hecke operators on the source and target; without an explicit statement that the transfer map preserves the locus of characters whose associated Galois representations are de Rham of regular parallel weights, the classicality statement does not follow.

    Authors: The comparison of Igusa stacks for distinct quaternionic Shimura data is constructed in Section 4 by adapting the functorial framework of DvHKZ26. Functoriality with respect to the Shimura data ensures that the induced transfer map intertwines the spherical Hecke operators on source and target; we will add an explicit lemma recording this compatibility. The transfer preserves the locus of characters whose associated Galois representations are de Rham of regular parallel weights because the geometric partial Fontaine operators (computed in Section 3) commute with the transfer and the Hodge filtration is preserved under the comparison. We will insert a short paragraph after Proposition 4.5 making this preservation statement explicit. revision: partial

  2. Referee: [Grothendieck-Messing theory for locally analytic infinite level Shimura varieties] The invocation of Grothendieck-Messing theory for locally analytic infinite-level Shimura varieties requires a precise check that the period map and Hodge filtration data extend to the infinite-level analytic setting without extra hypotheses on the level or the weights; if this extension fails on the locus where the representation is de Rham, the transfer is not defined on the relevant characters and the main theorem is unsupported.

    Authors: The Grothendieck-Messing theory for locally analytic infinite-level Shimura varieties is taken from Jiang26a and applied in Section 5. The period map extends to the infinite-level analytic setting by the uniform definition of the analytic structure in Jiang26a, without additional level or weight hypotheses. On the de Rham locus the Hodge filtration is of the expected type precisely by the regular parallel weight assumption, so the transfer remains defined on the relevant characters. We will add a self-contained verification paragraph in Section 5 spelling out these extensions in the analytic infinite-level case. revision: yes

Circularity Check

0 steps flagged

No significant circularity; derivation builds on cited prior results with added technical steps

full rationale

The paper claims to prove classicality of certain Hecke characters in completed cohomology under Galois representation hypotheses by generalizing the method of [Pan26], computing geometric partial Fontaine operators, and analyzing Koszul-type partial de Rham complexes. The central step is a locally analytic Jacquet-Langlands transfer constructed via Igusa stack comparisons from [DvHKZ26] and Grothendieck-Messing theory from [Jiang26a]. These are invoked as established external inputs rather than derived within the present work. No equation, definition, or prediction in the abstract or described chain reduces the main theorem to a tautological fit, self-definition, or load-bearing self-citation that collapses the argument by construction. The cited results are treated as independent support, and the current paper contributes new calculations without renaming or smuggling ansatzes from its own prior outputs.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

Only abstract available; the paper invokes standard number-theoretic background plus specific prior results (Pan26, DvHKZ26, Jiang26a) whose independence cannot be checked here.

axioms (1)
  • domain assumption Standard properties of completed cohomology and Galois representations attached to automorphic forms
    Invoked throughout the statement of the main theorem.

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