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Breuil's Lattice Conjecture for GL2(K)

Hymn Chan

TL;DR

This work extends Breuil's lattice conjecture to GL_2(K) with higher, generic Hodge-Tate weights in the unramified p-adic setting, showing that the integral lattice inside the locally algebraic type depends only on the p-adic Galois representation at p. It blends a new structure theorem for mod p GL_2(O_K) representations with an explicit description of higher-weight Galois deformation rings via Kisin modules and local-model techniques, all embedded in a patching framework that links global automorphy to local lattices. The main contributions are a generalization of the injective-envelope framework to m_K1^n-torsion, a complete description of R^{λ,τ}_{r̄} as a normal complete intersection with explicit monodromy data, and a cyclicity result for patched modules, which together establish Breuil's lattice conjecture in this higher-weight regime and pave the way for a mod p Langlands correspondence in the GL_2(K) setting. By constructing minimal patching functors with unramified coefficients and leveraging automorphy lifting, the paper provides a robust global-to-local mechanism that yields lattice-level invariants and a candidate mod TEXT Langlands correspondence through patching and deformation-theoretic control.

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.

Breuil's Lattice Conjecture for GL2(K)

TL;DR

This work extends Breuil's lattice conjecture to GL_2(K) with higher, generic Hodge-Tate weights in the unramified p-adic setting, showing that the integral lattice inside the locally algebraic type depends only on the p-adic Galois representation at p. It blends a new structure theorem for mod p GL_2(O_K) representations with an explicit description of higher-weight Galois deformation rings via Kisin modules and local-model techniques, all embedded in a patching framework that links global automorphy to local lattices. The main contributions are a generalization of the injective-envelope framework to m_K1^n-torsion, a complete description of R^{λ,τ}_{r̄} as a normal complete intersection with explicit monodromy data, and a cyclicity result for patched modules, which together establish Breuil's lattice conjecture in this higher-weight regime and pave the way for a mod p Langlands correspondence in the GL_2(K) setting. By constructing minimal patching functors with unramified coefficients and leveraging automorphy lifting, the paper provides a robust global-to-local mechanism that yields lattice-level invariants and a candidate mod TEXT Langlands correspondence through patching and deformation-theoretic control.

Abstract

We prove Breuil's lattice conjecture for higher Hodge-Tate weights in the case of where is an unramified extension of . More precisely, under some genericity conditions, we show that the lattice inside a locally algebraic type induced by the completed cohomology of a -arithmetic manifold depends only on the Galois representation at places above for arbitrary Hodge-Tate weights, which are small relative to . 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 representations of , 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.

Paper Structure

This paper contains 28 sections, 76 theorems, 214 equations, 3 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Breuil's lattice conjecture
  3. Representation theory result
  4. Galois deformation ring result
  5. Cyclicity of patched modules
  6. Candidate for the mod TEXT Langlands Correspondence
  7. Notation
  8. Acknowledgment
  9. Representation theory results
  10. Notation and background
  11. Main result
  12. Preliminary lemmas
  13. Proof of the main theorem
  14. Galois Deformation rings
  15. Notations
  16. ...and 13 more sections

Key Result

Theorem 1.1.1

(lattice conjecture) Fix $n\geq 1$. If the Hodge-Tate weights of $r_w$ are contained in the set $\{0,\dots, n\}$, $\overline{r}_w$ and $\tau$ are sufficiently generic (depending on $n$), then the lattice $\sigma^\circ(\lambda,\tau)$ depends only on $r_w$.

Figures (3)

  • Figure 1: step 2
  • Figure 2: Step 3
  • Figure 3: Step 4

Theorems & Definitions (159)

  • Theorem 1.1.1
  • Theorem 1.2.1
  • Theorem 1.3.1
  • Theorem 1.4.1
  • Theorem 1.5.1
  • Lemma 2.1.1
  • Definition 2.1.2
  • Definition 2.1.3
  • Lemma 2.1.4
  • Lemma 2.1.5
  • ...and 149 more