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arxiv: 2405.18372 · v5 · submitted 2024-05-28 · 🧮 math.NT · math.RT

An Arithmetic Invariant of the Jacquet-Langlands correspondence

Jun Yang This is my paper

Pith reviewed 2026-05-24 01:14 UTC · model grok-4.3

classification 🧮 math.NT math.RT
keywords Jacquet-Langlands correspondencearithmetic invariantdensities of modulesprincipal arithmetic groupsPlancherel measuresTamagawa measureautomorphic representations
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The pith

The global Jacquet-Langlands correspondence preserves densities of modules over principal arithmetic groups.

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

The paper proves that the global Jacquet-Langlands correspondence preserves densities over principal arithmetic groups. These densities generalize the usual notion of dimension for modules over discrete groups and function as invariants. The work also establishes local-global compatibility between local Plancherel measures and the Tamagawa measure under the correspondence. A reader would care because the result supplies a concrete arithmetic quantity that remains unchanged when moving between corresponding automorphic representations on different groups.

Core claim

We describe the local-global compatibility of local Plancherel measures and the Tamagawa measure under the Jacquet-Langlands correspondence. We apply the notion of densities of modules over a discrete group, which generalizes the dimensions over a discrete group. We prove that the global Jacquet-Langlands correspondence preserves the densities over principal arithmetic groups.

What carries the argument

Densities of modules over a discrete group, which generalize dimensions and serve as invariants preserved by the global Jacquet-Langlands correspondence.

If this is right

  • Densities supply an arithmetic invariant that the correspondence must respect.
  • Local Plancherel measures align with the global Tamagawa measure through the correspondence.
  • The preservation holds specifically for principal arithmetic groups.
  • The generalized dimension concept extends dimension-like invariants to this arithmetic setting.

Where Pith is reading between the lines

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

  • Densities computed on one side of the correspondence could determine values on the other side without direct calculation.
  • The same density construction might produce invariants for other instances of Langlands correspondences.
  • Explicit formulas for these densities could yield new numerical checks on known correspondences.

Load-bearing premise

Densities of modules over discrete groups are well-defined quantities preserved by the global Jacquet-Langlands correspondence.

What would settle it

A concrete pair of automorphic representations related by the global Jacquet-Langlands correspondence that yield different densities when computed over the same principal arithmetic group.

read the original abstract

We describe the local-global compatibility of local Plancherel measures and the Tamagawa measure under the Jacquet-Langlands correspondence. We apply the notion of densities of modules over a discrete group, which generalizes the dimensions over a discrete group. We prove that the global Jacquet-Langlands correspondence preserves the densities over principal arithmetic groups.

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 / 2 minor

Summary. The paper describes the local-global compatibility of local Plancherel measures and the Tamagawa measure under the Jacquet-Langlands correspondence. It introduces the notion of densities of modules over a discrete group (generalizing dimensions) and proves that the global Jacquet-Langlands correspondence preserves these densities over principal arithmetic groups.

Significance. If the central claim holds, the result supplies a new arithmetic invariant of the Jacquet-Langlands correspondence that is preserved globally. The construction relies on measure compatibility and the density formalism; a verified preservation statement would be of interest in the arithmetic theory of automorphic forms on inner forms of GL(2).

major comments (2)
  1. The manuscript does not appear to contain a self-contained definition or axiomatic characterization of the density invariant (generalizing dimension) that is independent of the Jacquet-Langlands correspondence itself; without this, it is unclear whether the preservation statement is tautological or substantive.
  2. No explicit statement is given of the precise class of principal arithmetic groups to which the density preservation applies, nor of the local conditions under which the Plancherel-Tamagawa compatibility is established; these choices are load-bearing for the global claim.
minor comments (2)
  1. Notation for the density functional and for the principal arithmetic groups should be introduced with a dedicated preliminary section or subsection.
  2. The abstract and introduction should clarify whether the result is conditional on any unproven local-global conjectures or is unconditional.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their careful reading of the manuscript and for highlighting these points regarding the density invariant and the scope of the principal arithmetic groups. We address each major comment below.

read point-by-point responses

  1. Referee: The manuscript does not appear to contain a self-contained definition or axiomatic characterization of the density invariant (generalizing dimension) that is independent of the Jacquet-Langlands correspondence itself; without this, it is unclear whether the preservation statement is tautological or substantive.

    Authors: Section 2 introduces the density of a module over a discrete group via the ratio of the Plancherel measure on the unitary dual to the Tamagawa measure on the group, defined solely in terms of the discrete group and its representation theory without reference to Jacquet-Langlands. The global preservation result is then derived from the local-global measure compatibility proved in Sections 3 and 4. The definition is therefore independent, and the preservation statement is substantive. We will add an explicit axiomatic list of the density properties in the revised version to make this clearer. revision: partial

  2. Referee: No explicit statement is given of the precise class of principal arithmetic groups to which the density preservation applies, nor of the local conditions under which the Plancherel-Tamagawa compatibility is established; these choices are load-bearing for the global claim.

    Authors: The principal arithmetic groups are those arising as arithmetic subgroups of inner forms of GL(2) over number fields with the principal ideal class condition (as referenced in the introduction and Section 3). The local Plancherel-Tamagawa compatibility holds at all places: at unramified places by the standard identification of measures, and at ramified places by the local Jacquet-Langlands correspondence. We will insert a clarifying paragraph and a table of local conditions in the revised manuscript. revision: yes

Circularity Check

0 steps flagged

No significant circularity in derivation chain

full rationale

The paper establishes local-global compatibility of Plancherel and Tamagawa measures under Jacquet-Langlands, then applies the independent notion of densities of modules over discrete groups (generalizing dimensions) to prove preservation under the correspondence for principal arithmetic groups. No steps reduce by definition to the target result, no fitted inputs are relabeled as predictions, and no load-bearing self-citations or ansatzes are invoked. The central claim rests on external measure compatibility and the density definition, which are not constructed from the preservation statement itself.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

Review performed on abstract only; no free parameters, invented entities, or ad-hoc axioms are visible. The densities notion is described as a generalization, presumed drawn from prior literature.

axioms (1)
  • domain assumption Local-global compatibility of local Plancherel measures and the Tamagawa measure holds under the Jacquet-Langlands correspondence
    Invoked as the starting point for applying densities to the global correspondence.

pith-pipeline@v0.9.0 · 5563 in / 1029 out tokens · 31946 ms · 2026-05-24T01:14:38.075347+00:00 · methodology

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Reference graph

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