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arxiv: 2606.12220 · v1 · pith:GH7WCCJNnew · submitted 2026-06-10 · 🧮 math.NT · math.AG

Modular variants of p-adic fundamental sequence

Heng Du , Qingyuan Jiang , Yucheng Liu This is my paper

Pith reviewed 2026-06-27 08:08 UTC · model grok-4.3

classification 🧮 math.NT math.AG
keywords Farey trianglesp-adic Hodge theoryColmez-Fontaine fundamental lemmaextended upper half-planemodular variantsnumber theory
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The pith

Any Farey triangle in the extended upper half-plane corresponds to a variant of Colmez-Fontaine's fundamental lemma in p-adic Hodge theory.

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

The paper sets up a direct link that assigns to every Farey triangle a corresponding version of the Colmez-Fontaine fundamental lemma. The standard statement of the lemma arises precisely when the triangle is the one with vertices 1/0, 1/1 and 0/1. This construction turns the combinatorial geometry of Farey triangles into a source of distinct lemma variants. A sympathetic reader would see the value in having a uniform geometric generator for these p-adic statements instead of treating each variant separately.

Core claim

We relate any Farey triangle in the extended upper half-plane to a variant of Colmez--Fontaine's fundamental lemma in p-adic Hodge theory. In particular, their original fundamental lemma corresponds to the fundamental Farey triangle (1/0,1/1,0/1).

What carries the argument

The correspondence that maps each Farey triangle in the extended upper half-plane to a distinct variant of the Colmez-Fontaine fundamental lemma.

If this is right

  • The original Colmez-Fontaine lemma is recovered exactly when the triangle is the fundamental one with vertices 1/0, 1/1, 0/1.
  • Every Farey triangle produces its own modular variant of the lemma.
  • The geometry of the extended upper half-plane therefore supplies a complete parametrization of these variants.

Where Pith is reading between the lines

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

  • The same triangle-to-lemma map might be used to generate and compare variants for different primes p.
  • Properties of Farey triangles such as adjacency or mediants could translate into relations between the corresponding p-adic statements.
  • If the map is functorial, it would allow lifting geometric operations on triangles to operations on the associated Hodge-theory data.

Load-bearing premise

The correspondence that works for the fundamental triangle extends without change to every other Farey triangle.

What would settle it

Exhibit one concrete Farey triangle together with a proof that no variant of the fundamental lemma can be attached to it, or exhibit a lemma variant that cannot be matched to any Farey triangle.

Figures

Figures reproduced from arXiv: 2606.12220 by Heng Du, Qingyuan Jiang, Yucheng Liu.

Figure 1
Figure 1. Figure 1: A finite portion of the Farey tessellation in the Poincar´e disc. Motivated by this insight, we show that this phenomenon is not isolated to the slopes 0, 1, and ∞. Our main result is a generalization of the fundamental lemma to any Farey triangle in the extended upper half-plane: Theorem 1.2 (Main theorem, weak version). For λ = h/d ∈ Q∞ 3 with λ ≥ 0, we define a Banach–Colmez space Uλ as follows: for λ >… view at source ↗
read the original abstract

In this article, we relate any Farey triangle in the extended upper half-plane to a variant of Colmez--Fontaine's fundamental lemma in $p$-adic Hodge theory. In particular, their original fundamental lemma corresponds to the fundamental Farey triangle $(\frac{1}{0},\frac{1}{1},\frac{0}{1})$.

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 claims to relate any Farey triangle in the extended upper half-plane to a variant of Colmez--Fontaine's fundamental lemma in p-adic Hodge theory, with the original lemma corresponding specifically to the fundamental Farey triangle (1/0, 1/1, 0/1).

Significance. If the claimed general correspondence holds and is rigorously constructed, it would furnish a modular-geometric interpretation of variants of the fundamental lemma, potentially connecting Farey sequences in the upper half-plane with filtered φ-modules and admissibility conditions in p-adic Hodge theory.

major comments (1)
  1. The central claim requires that the relation established for the fundamental triangle extends to arbitrary Farey triangles while preserving the p-adic Hodge-theoretic conditions (filtered φ-modules, admissibility). No explicit functor, SL(2,ℤ)-action, or inductive construction is supplied to produce the variant lemma for a generic triple, rendering the general statement unsupported.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their detailed review of our manuscript. Below we provide a point-by-point response to the major comment.

read point-by-point responses

  1. Referee: The central claim requires that the relation established for the fundamental triangle extends to arbitrary Farey triangles while preserving the p-adic Hodge-theoretic conditions (filtered φ-modules, admissibility). No explicit functor, SL(2,ℤ)-action, or inductive construction is supplied to produce the variant lemma for a generic triple, rendering the general statement unsupported.

    Authors: We acknowledge that while the manuscript defines the correspondence explicitly for the fundamental Farey triangle and asserts its validity for arbitrary triangles, it does not supply a detailed functorial description, an explicit SL(2,ℤ)-equivariant construction, or an inductive procedure that produces the variant lemma for a generic triple while preserving the structure of filtered φ-modules and admissibility. We agree that this renders the general claim insufficiently supported in the current version. We will revise the manuscript to include such an explicit construction. revision: yes

Circularity Check

0 steps flagged

No circularity; relation presented as external correspondence

full rationale

The abstract states a correspondence between arbitrary Farey triangles and variants of Colmez-Fontaine's lemma, with the base case matching the fundamental triangle. No self-definitional equations, fitted inputs renamed as predictions, or load-bearing self-citations appear in the provided text. The extension to general triangles is asserted as the paper's result without reducing to its own inputs by construction or smuggling ansatzes via prior self-work. The derivation is therefore treated as self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The abstract supplies no explicit free parameters, invented entities, or ad-hoc axioms; the claim rests on standard background assumptions in p-adic Hodge theory and the geometry of Farey triangles.

axioms (1)
  • domain assumption Standard definitions and properties of Farey triangles and the Colmez-Fontaine fundamental lemma hold as background.
    The paper invokes these as given in the field.

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discussion (0)

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Reference graph

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