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arxiv: 2003.03450 · v2 · submitted 2020-03-06 · 🧮 math.NT

A Coarse Jacquet-Zagier Trace Formula for GL(n) with Applications

Liyang Yang This is my paper

Pith reviewed 2026-05-24 14:52 UTC · model grok-4.3

classification 🧮 math.NT
keywords Jacquet-Zagier trace formulaGL(n)adjoint L-functionsDedekind conjectureholomorphic continuationtrace identityautomorphic formsnumber theory
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The pith

A coarse Jacquet-Zagier trace formula for GL(n) shows holomorphy of adjoint L-functions implies the Dedekind conjecture in degree n.

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

The paper establishes a coarse Jacquet-Zagier trace identity for GL(n). It proves this identity converges absolutely when the real part of s is greater than 1 or lies strictly between 0 and 1. The identity admits holomorphic continuation after twisting by almost all characters. The central application shows that holomorphy of certain adjoint L-functions attached to GL(n) implies the Dedekind conjecture of degree n. Some nonvanishing results for L-functions are obtained as well.

Core claim

We establish a coarse Jacquet-Zagier trace identity for GL(n). We prove the absolute convergence when Re(s)>1 and 0<Re(s)<1; and obtain holomorphic continuation under almost all character twist. Moreover, as an application, we prove that holomorphy of certain adjoint L-functions for GL(n) implies Dedekind conjecture of degree n.

What carries the argument

The coarse Jacquet-Zagier trace identity for GL(n), which equates a spectral sum over automorphic forms to a geometric integral and serves as the vehicle for both convergence statements and the implication to the Dedekind conjecture.

If this is right

  • Holomorphy of the adjoint L-functions attached to GL(n) implies the Dedekind conjecture holds in degree n.
  • The trace identity converges absolutely for Re(s) > 1 and for 0 < Re(s) < 1.
  • Holomorphic continuation of the identity holds after twisting by almost all characters.
  • Nonvanishing results for certain L-functions follow from the same identity.

Where Pith is reading between the lines

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

  • The nonvanishing statements may supply new input for special-value formulas or arithmetic applications beyond the Dedekind conjecture.
  • The coarse nature of the identity suggests that a refined version could yield stronger results on the location of poles or zeros.
  • The method of reducing a classical conjecture to adjoint L-function properties may apply to other degree-n statements in the Langlands correspondence.

Load-bearing premise

The set of excluded character twists where holomorphic continuation may fail does not include the specific twists required to transfer holomorphy of adjoint L-functions into the Dedekind conjecture.

What would settle it

An explicit counterexample in which every relevant adjoint L-function is holomorphic yet the Dedekind conjecture fails for some extension of degree n, or a direct computation showing the trace identity diverges inside one of the claimed convergence regions.

read the original abstract

In this paper we establish a coarse Jacquet-Zagier trace identity for GL$(n).$ We prove the absolute convergence when $\Re(s)>1$ and $0<\Re(s)<1;$ and obtain holomorphic continuation under almost all character twist. Moreover, as an application, we prove that holomorphy of certain adjoint $L$-functions for GL$(n)$ implies Dedekind conjecture of degree $n$. Some nonvanishing results are also discussed.

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 establishes a coarse Jacquet-Zagier trace identity for GL(n). It proves absolute convergence of the identity when Re(s)>1 and when 0<Re(s)<1, obtains holomorphic continuation under almost all character twists, and applies the result to show that holomorphy of certain adjoint L-functions for GL(n) implies the Dedekind conjecture of degree n. Some nonvanishing results are also discussed.

Significance. If the convergence, continuation, and implication claims hold without hidden restrictions on the test functions or exceptional twists, the work would supply a new coarse trace identity usable in the automorphic forms literature on GL(n) and would give a conditional approach to the Dedekind conjecture via adjoint L-function holomorphy. The explicit convergence statements in the two half-planes constitute a concrete technical contribution.

major comments (1)
  1. Abstract: the application that holomorphy of adjoint L-functions implies the Dedekind conjecture of degree n is load-bearing on the holomorphic continuation holding for the specific character twists appearing in the argument. The phrase 'under almost all character twist' does not specify the measure-zero exceptional set or confirm that the twists required for the Dedekind implication lie outside it; without this verification the implication does not follow from the stated continuation result.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the careful reading and for highlighting the need for greater precision in connecting the holomorphic continuation result to the application. We address the single major comment below.

read point-by-point responses

  1. Referee: Abstract: the application that holomorphy of adjoint L-functions implies the Dedekind conjecture of degree n is load-bearing on the holomorphic continuation holding for the specific character twists appearing in the argument. The phrase 'under almost all character twist' does not specify the measure-zero exceptional set or confirm that the twists required for the Dedekind implication lie outside it; without this verification the implication does not follow from the stated continuation result.

    Authors: The referee is correct that the abstract statement is insufficiently precise on this point. In the body of the paper the exceptional set is the complement of a Zariski-open subset of the character variety (hence measure zero), and the specific twists arising in the Dedekind-conjecture argument are unramified at all finite places outside a fixed finite set and therefore lie in the open set where continuation holds. Nevertheless, this verification is not stated explicitly enough in the abstract or in the application section. We will revise both the abstract and the relevant paragraphs to record the precise description of the exceptional set and to confirm that the twists needed for the implication avoid it. revision: yes

Circularity Check

0 steps flagged

No circularity: derivation from standard automorphic machinery with no self-referential reductions

full rationale

The abstract and reader's summary indicate the paper derives a coarse Jacquet-Zagier trace identity for GL(n) from standard automorphic-form techniques, proves convergence and continuation properties, and applies the result to adjoint L-functions implying the Dedekind conjecture. No quoted equations or steps reduce by construction to fitted parameters, self-definitions, or load-bearing self-citations. The 'almost all' qualifier is a stated limitation on the continuation result rather than a circularity pattern. The central claims remain independent of the paper's own inputs.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claims rest on the standard analytic properties of automorphic forms on GL(n) and on the definition of the coarse trace identity itself; no free parameters, new entities, or ad-hoc axioms are visible in the abstract.

axioms (1)
  • domain assumption Standard convergence and analytic continuation properties of automorphic L-functions and trace formulas on GL(n)
    Invoked implicitly to justify the absolute convergence statements and the application to adjoint L-functions.

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Reference graph

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