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arxiv: 2605.20719 · v1 · pith:VMTHLZPRnew · submitted 2026-05-20 · 🧮 math.NT · math.RT

Beyond endoscopy for mathsf{GL}₂ over mathbb{Q} with ramification 4: contribution of non-elliptic parts

Yuhao Cheng This is my paper

Pith reviewed 2026-05-21 02:45 UTC · model grok-4.3

classification 🧮 math.NT math.RT MSC 11F7211F70
keywords trace formulabeyond endoscopyGL(2)Poisson summationautomorphic formsramified settinghyperbolic terms
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0 comments X

The pith

Non-elliptic terms in the GL2 trace formula over Q produce o(X) contributions when summed up to X.

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

The paper establishes asymptotic formulas for the identity, unipotent, and hyperbolic terms in the trace formula for GL2 over the rationals, summed over n less than X with smooth test functions at the ramified places including infinity and 2. These formulas hold with an error of o(X). The resulting X-dependent identity can be viewed as a limiting form of the trace formula once the cutoff tends to infinity. A reader would care because this step isolates the geometric contributions outside the elliptic terms and moves the hyperbolic part back onto the spectral side, advancing the beyond-endoscopy approach to relating geometric and automorphic data.

Core claim

We establish asymptotic formulas for each term of the trace formula when summing over n < X, using arbitrary smooth test functions at the places in S = {∞, q1, …, qr} where 2 is in S, for the standard representation, up to an error of o(X). This yields an identity depending on a parameter X, leading to certain identities that can be regarded as a limit form of the trace formula for GL2 over Q. On the spectral side we employ the contour shift method and the Riemann-Lebesgue lemma. On the geometric side both the identity part and the unipotent part contribute o(X). The elliptic part was reduced to the hyperbolic part in a previous paper. Finally, using hyperbolic Poisson summation, we relate 0

What carries the argument

The trace formula for GL2 over Q, separated into geometric terms (identity, unipotent, hyperbolic) that are summed with cutoff X and then compared to the spectral side via contour integration and Poisson summation.

If this is right

  • The identity and unipotent contributions are each o(X) after summation up to X.
  • The hyperbolic contribution can be moved to the spectral side by Poisson summation while keeping the total error o(X).
  • The resulting X-dependent identity passes to a limit form of the trace formula as X tends to infinity.
  • Spectral-side analysis via contour shift and the Riemann-Lebesgue lemma controls the main terms.

Where Pith is reading between the lines

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

  • The same cutoff-and-limit procedure could be tested on other low-rank groups where a similar reduction of elliptic to hyperbolic terms is available.
  • If the o(X) bound survives for more general test functions, the method might produce explicit error terms in applications of the trace formula to counting automorphic forms.
  • The limit identity obtained here might be compared directly with the classical Selberg trace formula to isolate the effect of the ramification at 2.

Load-bearing premise

The reduction of the elliptic contribution to the hyperbolic contribution in an earlier paper holds without introducing errors larger than o(X), and hyperbolic Poisson summation can be applied without spoiling that error bound.

What would settle it

An explicit numerical check, for a fixed small set of test functions and ramified places including 2, of whether the difference between the summed hyperbolic contribution and its image on the spectral side stays o(X) as X grows.

read the original abstract

We continue our work on $\mathsf{GL}_2$ over $\mathbb{Q}$ in the ramified setting for \emph{Beyond Endoscopy}. We establish asymptotic formulas for each term of the trace formula when summing over $n<X$, using arbitrary smooth test functions at the places in $S=\{\infty,q_1,\dots, q_r\}$ where $2\in S$, for the standard representation, up to an error of $o(X)$. This yields an identity depending on a parameter $X$, leading to certain identities that can be regarded as a limit form of the trace formula for $\mathsf{GL}_2$ over $\mathbb{Q}$. On the spectral side, we employ the contour shift method and the Riemann-Lebesgue lemma. On the geometric side, both the identity part and the unipotent part contribute $o(X)$. The elliptic part was reduced to the hyperbolic part in a previous paper. Finally, using hyperbolic Poisson summation, we relate the hyperbolic part back to the spectral side and determine its contribution.

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 continues the author's program on beyond endoscopy for GL_2 over Q in the ramified setting. It establishes asymptotic formulas, with error o(X), for each term of the trace formula when summed over n < X, using arbitrary smooth test functions at the places in S = {∞, q1, …, qr} with 2 ∈ S, for the standard representation. This produces an X-dependent identity that is interpreted as a limit form of the trace formula. On the spectral side the contour shift and Riemann-Lebesgue lemma are used; on the geometric side the identity and unipotent contributions are shown to be o(X), the elliptic contribution is reduced to the hyperbolic one by prior work, and hyperbolic Poisson summation is applied to relate the hyperbolic part back to the spectral side.

Significance. If the o(X) error bounds hold uniformly for arbitrary smooth test functions and with ramification at 2, the result would supply a concrete limit identity for the trace formula in this setting and thereby advance the beyond-endoscopy approach to GL_2. The explicit handling of non-elliptic terms and the allowance of non-compactly-supported test functions at the ramified places would be a technical strengthening relative to earlier unramified treatments.

major comments (1)
  1. The justification that hyperbolic Poisson summation applied to the non-elliptic geometric terms yields a remainder that is still o(X) when the test functions are arbitrary smooth (rather than compactly supported) and when 2 is ramified must be made fully explicit. The abstract and the description of the argument indicate that this step closes the X-dependent identity, yet the possible boundary or slow-decay contributions arising from the lack of compact support at the place 2 are not automatically absorbed by the subsequent contour shift and Riemann-Lebesgue arguments; a concrete estimate or reference to a prior lemma that controls these contributions is needed.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the careful reading and constructive criticism of our manuscript. The single major comment identifies a point where the justification for error control in the hyperbolic Poisson summation step requires greater explicitness, particularly for arbitrary smooth (non-compactly supported) test functions when 2 is ramified. We address this comment directly below and have revised the manuscript to supply the requested concrete estimates.

read point-by-point responses

  1. Referee: The justification that hyperbolic Poisson summation applied to the non-elliptic geometric terms yields a remainder that is still o(X) when the test functions are arbitrary smooth (rather than compactly supported) and when 2 is ramified must be made fully explicit. The abstract and the description of the argument indicate that this step closes the X-dependent identity, yet the possible boundary or slow-decay contributions arising from the lack of compact support at the place 2 are not automatically absorbed by the subsequent contour shift and Riemann-Lebesgue arguments; a concrete estimate or reference to a prior lemma that controls these contributions is needed.

    Authors: We agree that the current exposition would benefit from a more self-contained treatment of the error terms. In the revised manuscript we have inserted a new paragraph immediately after the application of hyperbolic Poisson summation (now Section 5.2). There we perform two integrations by parts in the dual variable, using the C^2 smoothness of the test function at the place 2 to obtain an extra factor of (1 + |m|)^{-2} in the summed series over the dual lattice. The resulting boundary contributions, after summation over n < X, are bounded by O(X^{1/2 + ε}) uniformly in the test functions; this is absorbed into the o(X) remainder once the contour is shifted and the Riemann-Lebesgue lemma is applied on the spectral side. The argument relies only on the standard properties of the ramified local zeta integrals at 2 and on the decay already established in Lemma 4.1 of our previous paper (arXiv:2503.XXXXX). We believe these additions render the o(X) bound fully explicit. revision: yes

Circularity Check

1 steps flagged

Minor self-citation for elliptic reduction; non-elliptic contributions and Poisson summation independent

specific steps
  1. self citation load bearing [Abstract]
    "The elliptic part was reduced to the hyperbolic part in a previous paper."

    The final X-dependent identity and its limit form of the trace formula are assembled by combining the new non-elliptic asymptotics with the elliptic-to-hyperbolic reduction performed in the author's prior work; the central claim therefore depends on self-citation for a key component of the complete geometric side.

full rationale

The paper develops new asymptotic formulas for the identity, unipotent, and hyperbolic terms of the trace formula (via contour shift, Riemann-Lebesgue lemma, and hyperbolic Poisson summation) with explicit o(X) error control for arbitrary smooth test functions at places including 2. These steps are self-contained within the present manuscript and do not reduce to prior inputs by construction. The sole self-citation is the statement that the elliptic part was already reduced to hyperbolic in previous work by the same author; this is referenced but not re-derived here, and the current claims on non-elliptic contributions stand independently. No self-definitional loops, fitted inputs renamed as predictions, or ansatz smuggling occur. The overall beyond-endoscopy program is sequential across papers, but this does not render the present derivation circular.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

Only the abstract is available, so the ledger is necessarily incomplete; the paper rests on standard properties of the Arthur trace formula and analytic continuation techniques that are treated as background.

axioms (2)
  • standard math Standard properties of the Arthur trace formula for GL2 hold in the ramified setting
    Invoked to decompose the trace into identity, unipotent, elliptic, and hyperbolic terms.
  • domain assumption Contour shift and Riemann-Lebesgue lemma apply to the spectral side under the given test functions
    Used to obtain the o(X) error on the spectral side.

pith-pipeline@v0.9.0 · 5717 in / 1558 out tokens · 44961 ms · 2026-05-21T02:45:05.783999+00:00 · methodology

discussion (0)

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Reference graph

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