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arxiv: 1907.03915 · v1 · pith:KPXJSEGUnew · submitted 2019-07-09 · 🧮 math.NT · math.RT

The automorphic discrete spectrum of Mp₄

Wee Teck Gan , Atsushi Ichino This is my paper

Pith reviewed 2026-05-25 00:40 UTC · model grok-4.3

classification 🧮 math.NT math.RT
keywords automorphic formsmetaplectic groupdiscrete spectrummultiplicity formulaMp4endoscopic classificationLanglands correspondence
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The pith

The multiplicity formula for the automorphic discrete spectrum of Mp4 is proven.

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

This paper establishes the multiplicity formula that determines how often each irreducible representation occurs in the discrete part of the automorphic spectrum on the metaplectic group Mp4. Mp4 is the rank-2 metaplectic cover of the symplectic group Sp4. The formula provides an explicit count for these multiplicities using endoscopic data. A reader would care because it completes the classification of the discrete automorphic forms on this group once the local correspondences are in place.

Core claim

We prove the multiplicity formula for the automorphic discrete spectrum of the metaplectic group Mp4 of rank 2.

What carries the argument

The multiplicity formula, which gives the multiplicity of each irreducible admissible representation in the discrete spectrum in terms of its endoscopic transfers and local Arthur parameters.

If this is right

  • The discrete automorphic spectrum of Mp4 is completely determined once the local parameters are fixed.
  • Multiplicities are computed from the number of compatible endoscopic transfers from smaller groups.
  • The result aligns the global multiplicity with predictions from the local Langlands correspondence.
  • The discrete spectrum decomposes into packets whose sizes are controlled by the formula.

Where Pith is reading between the lines

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

  • The same approach may yield multiplicity formulas for Mp2n when the corresponding endoscopic classifications become available.
  • This supplies a concrete test case for conjectural multiplicity formulas on covering groups.
  • The formula could be used to compute dimensions of spaces of automorphic forms on Mp4 for specific congruence subgroups.

Load-bearing premise

The endoscopic classification or local Langlands correspondence for Mp4 and related groups holds.

What would settle it

An explicit irreducible admissible representation whose multiplicity in the discrete automorphic spectrum of Mp4 differs from the value given by the formula.

read the original abstract

We prove the multiplicity formula for the automorphic discrete spectrum of the metaplectic group $\mathrm{Mp}_4$ of rank $2$.

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 proves the multiplicity formula for the automorphic discrete spectrum of the metaplectic group Mp_4 of rank 2.

Significance. If the result is unconditional, it would advance the endoscopic classification of automorphic representations for low-rank metaplectic groups, supplying explicit multiplicity data that is otherwise unavailable and that feeds into trace formula comparisons and global Langlands correspondences.

major comments (1)
  1. [Abstract / Introduction] The central claim of an unconditional multiplicity formula is load-bearing on the endoscopic classification and local Langlands correspondence for Mp_4 and its endoscopic groups. The abstract states the result without indicating whether these inputs are supplied in the paper or imported from prior work (e.g., Gan–Savin or Arthur’s endoscopic classification). If the latter, the formula holds only conditionally; this must be clarified with explicit citations to the precise statements used.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their careful reading and constructive comments. We address the single major comment below.

read point-by-point responses

  1. Referee: [Abstract / Introduction] The central claim of an unconditional multiplicity formula is load-bearing on the endoscopic classification and local Langlands correspondence for Mp_4 and its endoscopic groups. The abstract states the result without indicating whether these inputs are supplied in the paper or imported from prior work (e.g., Gan–Savin or Arthur’s endoscopic classification). If the latter, the formula holds only conditionally; this must be clarified with explicit citations to the precise statements used.

    Authors: We agree that the abstract and introduction should explicitly indicate the foundational inputs. The multiplicity formula proved in the manuscript relies on the endoscopic classification and local Langlands correspondence for Mp_4 and its endoscopic groups, as established in prior work (in particular Gan–Savin and Arthur’s endoscopic classification). These are imported rather than reproved here. We will revise the abstract and introduction to state this dependence clearly and to include precise citations to the relevant theorems, thereby making the conditional nature of the result explicit. revision: yes

Circularity Check

0 steps flagged

No circularity: proof relies on external assumptions without self-referential reduction

full rationale

The provided abstract and context state only that the paper proves the multiplicity formula for the automorphic discrete spectrum of Mp4, depending on the validity of endoscopic classification or LLC for Mp4 and related groups (assumed or cited from prior work). No equations, self-citations, or derivation steps are quoted that reduce a claimed prediction or result to its own inputs by construction. Dependence on prior results (even by the same authors) does not trigger circularity under the rules unless the load-bearing step explicitly reduces via self-definition, fitted input renamed as prediction, or an unverified self-citation chain that replaces independent evidence. The central claim remains a proof under stated assumptions, with no exhibited internal loop. This is the normal case of a paper building on external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

No information on free parameters, axioms, or invented entities is available from the abstract alone.

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discussion (0)

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Reference graph

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