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arxiv: 2512.00178 · v2 · pith:UU5WCS3Onew · submitted 2025-11-28 · 🧮 math.NT

Breuil's Lattice Conjecture for GL2(K)

Hymn Chan This is my paper

Pith reviewed 2026-05-25 07:28 UTC · model grok-4.3

classification 🧮 math.NT
keywords Breuil lattice conjectureGL2 over p-adic fieldslocally algebraic representationscompleted cohomologyGalois deformation ringspatched modulesHodge-Tate weightsmod p representations
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The pith

Under genericity conditions the lattice inside a locally algebraic type for GL2(K) is fixed by the Galois representation at places above p.

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

The paper proves Breuil's lattice conjecture for GL2 over an unramified extension of Qp, allowing arbitrary Hodge-Tate weights that remain small relative to p. It shows that the lattice induced by completed cohomology of a U(2)-arithmetic manifold is determined solely by the associated Galois representation at p, once genericity conditions hold. The argument also establishes that the corresponding patched modules for lattices with irreducible cosocle are always cyclic. These statements connect the automorphic and Galois sides of the p-adic Langlands correspondence without requiring extra lattice data.

Core claim

Under genericity conditions, the lattice inside a locally algebraic type induced by the completed cohomology of a U(2)-arithmetic manifold depends only on the Galois representation at places above p, for arbitrary Hodge-Tate weights small relative to p. The patched modules of all lattices inside the locally algebraic types with irreducible cosocle are cyclic. The proof uses a structure theorem for mod p representations of GL2(OK) that are residually multiplicity free and of finite length, together with explicit computations of universal framed Galois deformation rings that parameterize potentially crystalline lifts with fixed tame inertial types and higher Hodge-Tate weights.

What carries the argument

The structure theorem for mod p representations of GL2(OK) that are residually multiplicity free and finite length, together with explicit framed deformation ring computations for potentially crystalline lifts.

If this is right

  • The lattice choice is independent of any further data from the arithmetic manifold once the Galois representation at p is fixed.
  • Patched modules remain cyclic whenever the lattice has irreducible cosocle.
  • The result extends the range of Breuil's conjecture from small to higher Hodge-Tate weights that are still small relative to p.
  • The structure theorem for the mod p representations applies uniformly to all such residually multiplicity free finite-length cases.

Where Pith is reading between the lines

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

  • The cyclic property of the patched modules may allow explicit computation of the corresponding automorphic representations from the Galois side alone.
  • The same deformation-ring techniques could be tested on other groups or at places where the genericity conditions fail.
  • If the small-weight hypothesis can be relaxed, the independence statement would apply to a wider class of crystalline lifts.

Load-bearing premise

The representations satisfy the stated genericity conditions and the Hodge-Tate weights remain small relative to p.

What would settle it

A concrete example, under the genericity and weight hypotheses, of two distinct lattices inside the same locally algebraic type that arise from the same Galois representation at p, or of a non-cyclic patched module with irreducible cosocle.

Figures

Figures reproduced from arXiv: 2512.00178 by Hymn Chan.

Figure 1
Figure 1. Figure 1: step 2 [PITH_FULL_IMAGE:figures/full_fig_p013_1.png] view at source ↗
read the original abstract

We prove Breuil's lattice conjecture for higher Hodge-Tate weights in the case of $\mathrm{GL}_2(K)$ where $K$ is an unramified extension of $\mathbb{Q}_p$. More precisely, under some genericity conditions, we show that the lattice inside a locally algebraic type induced by the completed cohomology of a $U(2)$-arithmetic manifold depends only on the Galois representation at places above $p$ for arbitrary Hodge-Tate weights, which are small relative to $p$. We further prove that the patched modules of all lattices inside the locally algebraic types with irreducible cosocle are cyclic. One key input of the paper is a structure theorem for mod $p$ representations of $\mathrm{GL}_2(\mathcal{O}_K)$, which are residually multiplicity free and of finite length. Another input is an explicit computation of universal framed Galois deformation rings, which parameterize potentially crystalline lifts with fixed tame inertial types and higher Hodge-Tate weights.

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

0 major / 2 minor

Summary. The manuscript proves Breuil's lattice conjecture for GL_2(K) where K is an unramified extension of Q_p. Under genericity conditions on the representations and with Hodge-Tate weights small relative to p, it shows that the lattice inside a locally algebraic type induced by the completed cohomology of a U(2)-arithmetic manifold depends only on the Galois representation at places above p. It further establishes that the patched modules of all lattices inside the locally algebraic types with irreducible cosocle are cyclic. The proof takes as key inputs a structure theorem for residually multiplicity-free finite-length mod p representations of GL_2(O_K) and explicit computations of universal framed Galois deformation rings parameterizing potentially crystalline lifts with fixed tame inertial types and higher Hodge-Tate weights.

Significance. If the result holds, it extends Breuil's lattice conjecture to higher Hodge-Tate weights in the GL_2 setting over unramified extensions, confirming the local Galois dependence of the relevant lattices and establishing cyclicity for the associated patched modules. This provides concrete evidence linking completed cohomology to Galois deformation data and supplies explicit framed deformation ring computations that may be reusable in related modularity lifting problems.

minor comments (2)
  1. The abstract states that the structure theorem and framed deformation ring computations are 'key inputs'; the introduction should explicitly indicate whether these are proved in the manuscript or cited from prior work, with precise references.
  2. The genericity conditions and the bound on Hodge-Tate weights relative to p are invoked repeatedly; a dedicated subsection collecting all such hypotheses (with cross-references to where they are used) would improve readability.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive summary of the manuscript, recognition of its significance, and recommendation for minor revision. The report correctly captures the main results on Breuil's lattice conjecture for GL_2(K) under genericity conditions with Hodge-Tate weights small relative to p, as well as the cyclicity of the relevant patched modules.

Circularity Check

0 steps flagged

No significant circularity; derivation relies on independent external inputs

full rationale

The paper states its key inputs explicitly as a structure theorem for residually multiplicity-free finite-length mod p representations of GL2(OK) and explicit computations of universal framed Galois deformation rings for potentially crystalline lifts. These are invoked under stated genericity conditions and small Hodge-Tate weights relative to p, with the claims scoped precisely to that regime. No equations or steps in the provided text reduce a derived quantity to a fitted parameter or self-citation by construction; the central results (lattice dependence only on Galois data at p, cyclicity of patched modules) are presented as consequences of applying those independent tools rather than re-deriving or renaming them.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract-only; no explicit free parameters, axioms, or invented entities can be extracted. The proof relies on standard background in p-adic Hodge theory and Galois deformation theory whose precise invocation cannot be audited.

pith-pipeline@v0.9.0 · 5692 in / 1169 out tokens · 32557 ms · 2026-05-25T07:28:49.077673+00:00 · methodology

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Reference graph

Works this paper leans on

6 extracted references · 6 canonical work pages

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