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arxiv: 2606.28818 · v1 · pith:GO7WVP7Mnew · submitted 2026-06-27 · 🧮 math.NT · math.AG

To be or not to be local

Christophe Breuil , Florian Herzig , Yongquan Hu , Karol Koziol , Stefano Morra , Benjamin Schraen , Sug Woo Shin This is my paper

Pith reviewed 2026-06-30 08:50 UTC · model grok-4.3

classification 🧮 math.NT math.AG
keywords mod p Langlands correspondenceGalois representationsShimura curvesperfectoid geometrysupersingular representationsGL_2 over p-adic fieldslocality questionunramified quadratic extensions
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The pith

For quadratic unramified extensions of Q_p, perfectoid geometry produces from any 2-dimensional mod p Galois representation an infinite-dimensional smooth representation of the upper-triangular subgroup that is expected to be the restrictio

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

The paper investigates whether smooth mod p representations of GL_2(K) that appear in the cohomology of Shimura curves depend only on their attached 2-dimensional Galois representation. This is the locality question in the mod p setting. For the case where K is quadratic over Q_p the authors use perfectoid geometry to build directly from the Galois side an infinite-dimensional mod p representation of the subgroup of matrices with bottom row (0,1). They conjecture that this construction recovers the restriction of the irreducible supersingular subquotient of the automorphic representation.

Core claim

For a smooth representation π of GL_2(K) occurring in a Hecke eigenspace of the mod p cohomology of a Shimura curve, the authors explore strategies to decide whether π depends only on the underlying Galois representation ρ-bar. When [K:Q_p]=2 they use perfectoid geometry to associate to ρ-bar an infinite-dimensional mod p smooth representation of the subgroup (K^x K ; 0 1) that they hope coincides with the restriction of the irreducible supersingular subquotient of π.

What carries the argument

The perfectoid-geometry construction that produces an infinite-dimensional mod p smooth representation of the subgroup of upper-triangular matrices from the Galois representation ρ-bar.

If this is right

  • If the constructed representation matches the supersingular subquotient, then π is determined by ρ-bar alone for quadratic unramified K.
  • The locality question for quadratic extensions can be attacked by direct comparison of the perfectoid output with cohomology representations.
  • Strategies already known for K equal to Q_p can be extended using perfectoid methods when the degree is two.
  • Realization of the hoped-for equality would give an explicit local construction of the supersingular part independent of global Shimura curve data.

Where Pith is reading between the lines

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

  • The same perfectoid technique might produce candidate representations for higher-degree unramified extensions once suitable Shimura varieties are available.
  • Success would imply that supersingular representations are classified by their attached Galois data in the quadratic case.
  • The construction supplies a concrete test object that could be used to study local-global compatibility questions in the mod p Langlands correspondence.

Load-bearing premise

The perfectoid-geometry construction produces a representation whose restriction to the indicated subgroup coincides with the supersingular subquotient of the representation π that occurs in the mod p cohomology.

What would settle it

For a concrete choice of ρ-bar, compute the representation produced by the perfectoid construction and compare it directly with the supersingular subquotient extracted from an actual π appearing in the mod p cohomology of a Shimura curve; mismatch would disprove the hoped-for equality.

read the original abstract

Let $p$ be a prime number and $K$ a finite unramified extension of $\mathbf{Q}_p$. For a smooth representation $\pi$ of $\mathrm{GL}_2(K)$ occurring in some Hecke eigenspace of the mod $p$ cohomology of a Shimura curve, we explore different strategies (inspired by the case $K=\mathbf{Q}_p$) to attack the locality question: does $\pi$ depend only on the underlying $2$-dimensional representation $\overline{\rho}$ of ${\rm Gal}(\overline K/K)$? In particular when $[K:\mathbf{Q}_p]=2$, crucially using perfectoid geometry, we associate to $\overline{\rho}$ an infinite-dimensional mod $p$ smooth representation of $\begin{pmatrix}K^\times&K\\0&1\end{pmatrix}$ which we hope is the restriction to $\begin{pmatrix}K^\times&K\\0&1\end{pmatrix}$ of the (irreducible) supersingular subquotient of $\pi$.

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

1 major / 0 minor

Summary. The manuscript explores the locality question for smooth mod p representations π of GL_2(K) occurring in Hecke eigenspaces of the cohomology of Shimura curves: whether π depends only on the underlying 2-dimensional Galois representation ho-bar. It reviews strategies inspired by the K = Q_p case and, for the quadratic extension case [K : Q_p] = 2, describes (via perfectoid geometry) an association of an infinite-dimensional mod p smooth representation of the subgroup egin{pmatrix} K^ imes & K \ 0 & 1 \end{pmatrix} to ho-bar, with the explicit hope that this representation coincides with the restriction of the irreducible supersingular subquotient of π.

Significance. If the hoped-for identification via perfectoid geometry can be made rigorous and verified, the construction would supply a concrete geometric link between Galois data and the local automorphic representation in the mod p setting, extending known results from the rational case and potentially informing the mod p Langlands correspondence for GL_2 over unramified quadratic extensions.

major comments (1)
  1. Abstract and main text: the central contribution is framed as an exploratory construction whose output is hoped (but not shown) to match the supersingular subquotient; no geometric argument, error controls, or explicit verification that the produced representation restricts correctly to the indicated subgroup is supplied, leaving the load-bearing identification unsupported.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their thoughtful summary and for highlighting the exploratory nature of the work. We address the single major comment below.

read point-by-point responses

  1. Referee: Abstract and main text: the central contribution is framed as an exploratory construction whose output is hoped (but not shown) to match the supersingular subquotient; no geometric argument, error controls, or explicit verification that the produced representation restricts correctly to the indicated subgroup is supplied, leaving the load-bearing identification unsupported.

    Authors: We agree that the manuscript does not supply a proof or verification that the constructed representation coincides with the restriction of the supersingular subquotient; the identification is explicitly presented as a hope rather than a theorem. The central contribution is the geometric construction, via perfectoid spaces, of an infinite-dimensional mod p smooth representation of the indicated subgroup attached to ρ-bar. The text already qualifies the claim with the verb 'hope' and does not assert that the matching has been established. We can revise the abstract and introduction to state even more explicitly that the identification remains conjectural and that the paper offers a candidate rather than a verified correspondence. revision: partial

Circularity Check

0 steps flagged

No circularity; purely exploratory construction stated as a hope

full rationale

The manuscript contains no derivation chain, fitted parameters, or asserted theorems whose supporting steps reduce to self-definition, self-citation, or renaming. The central construction for [K:Q_p]=2 is explicitly framed as exploratory via perfectoid geometry, with the key identification to the supersingular subquotient of π labeled only as a 'hope' rather than a proven equality or prediction. No equations, uniqueness theorems, or ansatzes are invoked in a load-bearing way that could create circularity. The text is self-contained as an open-ended exploration without any completed claim that could be reduced to its inputs by construction.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract-only review supplies no explicit free parameters, axioms, or invented entities; the construction is described at the level of a hoped-for geometric association whose details are not given.

pith-pipeline@v0.9.1-grok · 5723 in / 1016 out tokens · 31889 ms · 2026-06-30T08:50:33.163627+00:00 · methodology

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Reference graph

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