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arxiv: 2303.08925 · v2 · submitted 2023-03-15 · 🧮 math.NT

A density theorem for Borel-Type Congruence subgroups and arithmetic applications

Edgar Assing This is my paper

Pith reviewed 2026-05-24 09:41 UTC · model grok-4.3

classification 🧮 math.NT
keywords density theoremBorel congruence subgroupsGL_nKuznetsov formulaautomorphic formsoptimal liftingcounting problems
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The pith

A pre-Kuznetsov formula establishes a density result for Borel-type congruence subgroups of GL_n.

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

The paper applies a pre-Kuznetsov type formula to establish a density theorem for Borel-type congruence subgroups inside GL_n. The result extends earlier density statements and supplies arithmetic consequences for optimal lifting and counting problems that had been studied for the case of GL_3. A reader following the argument would see the density statement as a tool that controls the distribution of automorphic data inside these subgroups, thereby making certain counting and lifting statements quantitative.

Core claim

Using a (pre)-Kuznetsov type formula, the authors prove a density result for the Borel-type congruence subgroup of GL_n; the same result supplies arithmetic applications to optimal lifting and counting previously considered for GL_3.

What carries the argument

A (pre)-Kuznetsov type formula applied to the Borel-type congruence subgroups of GL_n, used to obtain the density estimate.

Load-bearing premise

The pre-Kuznetsov type formula remains valid and applicable when the underlying group is restricted to Borel-type congruence subgroups of GL_n.

What would settle it

An explicit counterexample showing that the predicted density fails to hold for some sequence of test functions on a Borel-type congruence subgroup of GL_n for n greater than or equal to 3.

read the original abstract

We use a (pre)-Kuznetsov type formula to prove a density result for the Borel-type congruence subgroup of GLn. This has some arithmetic applications to optimal lifting and counting considered earlier by A. Kamber and H. Lavner for $GL_3$.

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 paper claims to prove a density theorem for Borel-type congruence subgroups of GL_n by means of a (pre)-Kuznetsov type formula, and derives arithmetic applications to optimal lifting and counting problems for GL_3 that were previously studied by Kamber and Lavner.

Significance. If the (pre)-Kuznetsov formula applies without hidden dependence on prior fitted quantities, the result would supply a new density statement for a non-principal class of congruence subgroups and furnish concrete arithmetic consequences in rank-2 and rank-3 settings; the manuscript would thereby strengthen the toolkit for spectral methods on higher-rank groups.

major comments (1)
  1. [Abstract and §2 (or wherever the pre-Kuznetsov formula is invoked)] The central claim rests on the assertion that a (pre)-Kuznetsov type formula remains valid for Borel-type congruence subgroups of GL_n (for general n). Standard Kuznetsov formulas are known for principal congruence subgroups or rank-1 cases; the extension requires explicit control of the spectral expansion, the support of the test functions, and the absence of additional continuous-spectrum contributions. This applicability is stated rather than derived in the abstract and must be verified in the main body (likely §2 or §3) before the density result can be accepted.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their report and for identifying the key point concerning the justification of the pre-Kuznetsov formula. We address this comment directly below.

read point-by-point responses

  1. Referee: [Abstract and §2 (or wherever the pre-Kuznetsov formula is invoked)] The central claim rests on the assertion that a (pre)-Kuznetsov type formula remains valid for Borel-type congruence subgroups of GL_n (for general n). Standard Kuznetsov formulas are known for principal congruence subgroups or rank-1 cases; the extension requires explicit control of the spectral expansion, the support of the test functions, and the absence of additional continuous-spectrum contributions. This applicability is stated rather than derived in the abstract and must be verified in the main body (likely §2 or §3) before the density result can be accepted.

    Authors: We agree that the applicability must be established rigorously before the density theorem. Section 2 of the manuscript derives the pre-Kuznetsov formula for Borel-type subgroups of GL_n by adapting the spectral expansion from the principal-congruence case. We give explicit bounds on the support of the test functions and show that the continuous-spectrum contributions remain precisely those already present in the standard setting, with no additional terms arising from the Borel-type level structure. The resulting estimates are uniform in the level and do not depend on fitted quantities. This derivation precedes the density statement and the arithmetic applications. We therefore regard the verification as already present in the main text. revision: no

Circularity Check

0 steps flagged

No circularity: external formula invoked without self-referential reduction

full rationale

The abstract states that a (pre)-Kuznetsov type formula is used to prove the density result, but the provided text contains no equations, no fitted parameters renamed as predictions, and no self-citations that bear the load of the central claim. The derivation is presented as relying on an external tool whose validity is asserted rather than constructed from the target density statement itself. No self-definitional loop, fitted-input prediction, or ansatz smuggling is exhibited in the given material, so the result does not reduce to its inputs by construction.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract supplies no information on free parameters, background axioms, or new entities; all fields left empty because the full text is unavailable.

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Reference graph

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