Beyond endoscopy for $\mathsf{GL}_2$ over $\mathbb{Q}$ with ramification 3: contribution of the elliptic part

Yuhao Cheng

arxiv: 2508.07167 · v3 · submitted 2025-08-10 · 🧮 math.NT · math.RT

Beyond endoscopy for mathsf{GL}₂ over mathbb{Q} with ramification 3: contribution of the elliptic part

Yuhao Cheng This is my paper

Pith reviewed 2026-05-19 00:04 UTC · model grok-4.3

classification 🧮 math.NT math.RT

keywords beyond endoscopytrace formulaGL_2elliptic contributionPoisson summationHitchin-Steinberg baseramified representations

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The pith

An explicit asymptotic formula for the elliptic part is derived for the trace formula on GL_2 over Q with ramification.

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

This paper derives an explicit asymptotic formula for the elliptic contribution when summing over integers n less than X, using arbitrary smooth test functions at the ramified places in S for the standard representation of GL_2. A key consequence is that the simple trace formula attains its desired limit, a feature that appears only in the ramified setting. The method applies a second Poisson summation formula with respect to the determinant to arrive at an expression on the Hitchin-Steinberg base, then changes variables from (T, N) to (T, Delta) to carry out the asymptotic analysis.

Core claim

By applying a second Poisson summation with respect to the determinant and changing variables from (T, N) to (T, Delta) on the Hitchin-Steinberg base g//G, an explicit asymptotic formula is obtained for the elliptic part of the sum over n < X with smooth test functions at places in S, which in turn yields the limit of the simple trace formula in the ramified case. An asymptotic formula for the traces of Hecke operators on cusp forms with arbitrary level and weight greater than 2 is also proved.

What carries the argument

The second Poisson summation with respect to the determinant, which produces a formula on the Hitchin-Steinberg base g//G, followed by the change of variables from (T, N) to (T, Delta) for the asymptotic analysis.

If this is right

The simple trace formula reaches its expected limit in the presence of ramification at places including 2 and infinity.
An asymptotic formula holds for traces of Hecke operators on spaces of cusp forms of arbitrary level and weight exceeding 2.
The results extend previous work on the unramified case to the ramified setting using coordinate changes on the quotient space.

Where Pith is reading between the lines

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

This coordinate change technique on the base might allow similar asymptotic analyses in higher-dimensional or higher-rank trace formulas.
Improved understanding of the elliptic contribution could facilitate applications to counting problems involving automorphic forms with ramification.

Load-bearing premise

The second Poisson summation with respect to the determinant is valid and the change of variables from (T, N) to (T, Delta) on the Hitchin-Steinberg base preserves the necessary analytic properties without introducing uncontrolled errors.

What would settle it

Numerical computation of the elliptic contribution for a concrete small set S with a specific smooth test function and comparison against the predicted asymptotic formula for large X.

read the original abstract

We continue to work on \emph{Beyond Endoscopy} for $\mathsf{GL}_2$ over $\mathbb{Q}$ with ramification at $S = \{\infty, q_1, \dots, q_r\}$ (where $2 \in S$), generalizing the final step of Altu\u{g}'s work in the unramified setting. We derive an explicit asymptotic formula for the elliptic part when summing over $n<X$ with arbitrary smooth test functions at places in $S$ for the standard representation. As a consequence, we obtain the desired limit for the simple trace formula which only occurs in the ramified case. Moreover, we prove an asymptotic formula for the traces of Hecke operators on cusp forms with arbitrary level and weight $>2$, directly generalizing Altu\u{g}'s final result. Our approach differs entirely from Altu\u{g}'s: We apply a second Poisson summation with respect to the determinant, obtaining a formula on the Hitchin-Steinberg base $\mathfrak{g}/\!/ \mathsf{G}$. By changing variables from $(T, N)$ to $(T, \Delta)$ on $\mathfrak{g}/\!/ \mathsf{G}$, we perform analysis in the new coordinates.

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.

Desk Editor's Note private letter to a colleague

Cheng gets an explicit elliptic asymptotic in the ramified GL2 beyond endoscopy case by a second Poisson summation on the determinant followed by a coordinate change on the Hitchin-Steinberg base, but the analytic control after that change is the part that needs the most checking.

read the letter

This paper gives an explicit asymptotic for the elliptic part when summing over n less than X with arbitrary smooth test functions at the ramified places in S for the standard representation on GL2. As a result it obtains the limit for the simple trace formula, which only shows up in the ramified setting. It also produces an asymptotic for the traces of Hecke operators on cusp forms of arbitrary level and weight greater than 2.

Referee Report

2 major / 2 minor

Summary. The paper continues work on Beyond Endoscopy for GL_2 over Q with ramification at S = {∞, q1, …, qr} (including 2). It derives an explicit asymptotic formula for the elliptic part when summing over n < X with arbitrary smooth test functions at places in S for the standard representation. The derivation proceeds by a second Poisson summation with respect to the determinant, yielding a formula on the Hitchin-Steinberg base g//G, followed by a change of variables from (T, N) to (T, Δ). As a consequence the paper obtains the desired limit for the simple trace formula (which occurs only in the ramified case) and proves an asymptotic formula for traces of Hecke operators on cusp forms of arbitrary level and weight > 2, generalizing Altuǧ’s unramified result. The approach is stated to differ entirely from prior work by relying on standard trace-formula machinery and the indicated coordinate change.

Significance. If the central derivation is valid, the result advances the beyond-endoscopy program into the ramified setting at 3 by supplying an explicit asymptotic for the elliptic contribution and the associated limit of the simple trace formula. The generalization to arbitrary smooth test functions at places in S and the direct proof of the Hecke-trace asymptotic are concrete strengths. The use of a second Poisson summation to reach the Hitchin-Steinberg base and the subsequent coordinate change constitute a technically distinct route from Altuǧ’s work.

major comments (2)

[Derivation of the elliptic asymptotic (second Poisson summation and coordinate change)] The central asymptotic for the elliptic part rests on the second Poisson summation with respect to the determinant and the subsequent change of variables from (T, N) to (T, Δ) on g//G. The manuscript must supply explicit bounds showing that the Jacobian of this coordinate change, together with the support and decay of the pulled-back test functions, does not produce singularities or slower decay that would render the error terms non-uniform when the level or ramification at 3 grows. Without such estimates the main-term extraction cannot be guaranteed to proceed with the claimed precision.
[Error-term estimates following the (T, N) → (T, Δ) change] The error-term analysis after the variable change must be checked for dependence on the ramification at 3. If the new coordinates introduce factors that grow with the conductor or with the weight, the claimed uniformity of the asymptotic (and therefore the limit of the simple trace formula) would require additional justification.

minor comments (2)

Notation for the Hitchin-Steinberg base and the coordinates (T, N), (T, Δ) should be introduced with a brief reminder of the identification g//G ≅ Spec k[T, Δ] to aid readers unfamiliar with the ramified setting.
The statement of the Hecke-trace asymptotic (final theorem) should explicitly record the dependence on the weight and level to make the generalization of Altuǧ’s result immediately visible.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and constructive comments on our manuscript. We address the two major comments point by point below, clarifying the uniformity of our estimates and indicating the revisions we will make to strengthen the exposition.

read point-by-point responses

Referee: The central asymptotic for the elliptic part rests on the second Poisson summation with respect to the determinant and the subsequent change of variables from (T, N) to (T, Δ) on g//G. The manuscript must supply explicit bounds showing that the Jacobian of this coordinate change, together with the support and decay of the pulled-back test functions, does not produce singularities or slower decay that would render the error terms non-uniform when the level or ramification at 3 grows. Without such estimates the main-term extraction cannot be guaranteed to proceed with the claimed precision.

Authors: We agree that the uniformity of the error terms with respect to growing ramification at 3 requires explicit verification of the Jacobian and the behavior of the test functions. In the revised manuscript we will add a new subsection (following the coordinate change in Section 4) that computes the Jacobian determinant of the map (T, N) → (T, Δ) explicitly; it is a polynomial expression in the coordinates that is bounded away from zero and infinity on the compact support of the relevant test functions. We will further derive uniform decay estimates showing that the pulled-back functions retain rapid decay without introducing singularities or conductor-dependent factors, using the algebraic nature of the change and the Schwartz-class properties at places in S. These additions will confirm that the main-term extraction proceeds with the stated precision uniformly in the ramification parameter. revision: yes
Referee: The error-term analysis after the variable change must be checked for dependence on the ramification at 3. If the new coordinates introduce factors that grow with the conductor or with the weight, the claimed uniformity of the asymptotic (and therefore the limit of the simple trace formula) would require additional justification.

Authors: The error terms following the change of variables are controlled by the volume forms on the Hitchin-Steinberg base and the rapid decay of the test functions, both of which remain independent of the ramification at 3 once the test functions are fixed. We will revise the error-term analysis in Section 5 to include explicit tracking of all factors arising from the Jacobian and the measure transformation, demonstrating that no growth with conductor or weight occurs. The uniformity follows from the separation of the ramification into the smooth test functions at places in S and the standard bounds from the trace formula. We will make these dependencies fully explicit in the revision to address the concern directly. revision: partial

Circularity Check

0 steps flagged

No circularity: derivation applies standard Poisson summation and coordinate change on Hitchin-Steinberg base without reducing to fitted inputs or self-citation chains

full rationale

The paper explicitly states its approach differs entirely from Altuǧ's prior work and relies on applying a second Poisson summation with respect to the determinant to reach the Hitchin-Steinberg base g//G, followed by the change of variables from (T, N) to (T, Δ) to enable asymptotic analysis. These steps are presented as direct applications of standard trace formula machinery rather than any redefinition of the target elliptic asymptotic in terms of itself or a fitted parameter. No load-bearing self-citation to the author's own prior results is invoked to justify uniqueness or to force the main term; the derivation is self-contained against external benchmarks of Poisson summation and analytic continuation on the base. The weakest assumption (validity of the summation and preservation of decay under the coordinate change) is an analytic hypothesis, not a definitional equivalence that collapses the claimed asymptotic to the input data by construction. Therefore the central result—an explicit asymptotic for the elliptic part and the resulting limit for the simple trace formula—does not reduce to its inputs.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The derivation rests on standard analytic number theory tools rather than new fitted constants or postulated objects.

axioms (2)

standard math The Poisson summation formula applies to the relevant adelic integrals over GL2.
Invoked explicitly for the second summation with respect to the determinant.
domain assumption The change of variables (T, N) to (T, Delta) on the Hitchin-Steinberg base is bijective and preserves the measure and analytic continuation properties needed for the asymptotics.
Used to perform the analysis in the new coordinates after the Poisson step.

pith-pipeline@v0.9.0 · 5756 in / 1249 out tokens · 56159 ms · 2026-05-19T00:04:03.852612+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Foundation/RealityFromDistinction.lean reality_from_one_distinction unclear

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unclear
Relation between the paper passage and the cited Recognition theorem.

We apply a second Poisson summation with respect to the determinant, obtaining a formula on the Hitchin-Steinberg base g//G. By changing variables from (T,N) to (T,Δ) on g//G, we perform analysis in the new coordinates.
IndisputableMonolith/Foundation/ArithmeticFromLogic.lean LogicNat ≃ Nat recovery unclear

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unclear
Relation between the paper passage and the cited Recognition theorem.

Theorem 1.1 (Contribution of the elliptic part) ... = A X^{3/2} + B(X log X - X) + ... + O(X^{7/8 + 3ϱ/4 + ε})

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