Beyond endoscopy for the symmetric square representation: The simple trace formula case

Yuhao Cheng

REVIEW 2 major objections 4 minor 16 references

Beyond endoscopy for the symmetric square representation: The simple trace formula case

T0 review · 2 major / 4 minor · reviewed 2026-08-01 · deepseek-v4-flash

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗ Review JSON

Pith's one-line read This paper proves that the averaged, ramified trace formula for the symmetric square of GL(2) over Q equals three explicit constants plus an error term controlled by the Ramanujan bound, and that this limit detects dihedral automorphic form

desk verdict A genuinely new geometric asymptotic for the symmetric square, but the spectral/dihedral corollary is not proved as written. read the letter →

arxiv 2607.25383 v1 pith:2JR6YSCV submitted 2026-07-28 math.NT math.RT

Yuhao Cheng This is my paper · ORCID

classification math.NTmath.RT MSC 11F7011F72

keywords beyondendoscopysymmetricsquaresimpletraceformulatransformedKloostermansumsRamanujanconjecturedihedralformsL-functionsautomorphic

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

This paper establishes an asymptotic formula for the averaged, ramified trace formula attached to the symmetric square representation of GL(2) over Q. Under a technical condition on the local test functions, the average of the cuspidal trace equals three explicit constants times X, with an error term governed by the best known bound toward the Ramanujan conjecture. The formula is then converted into an identity relating those constants to an average of residues of L-functions of the form L(s,Sym^2)/L(2s,chi^2), evaluated at s=1. Because this residue is nonzero precisely for dihedral automorphic forms, the asymptotic yields a quantitative way to detect dihedral forms in the automorphic spectrum. The proof goes through a second Poisson summation, a full computation of transformed Kloosterman sums, and a residue analysis of the associated Dirichlet series.

What carries the argument

The load-bearing object is the decomposition of the elliptic part of the simple trace formula into a xi=0 contribution and a xi≠0 contribution after a second Poisson summation in the determinant variable. The xi=0 term is handled via analytic continuation of a Kloosterman-type Dirichlet series built from the partial generalized Kloosterman sums, while the xi≠0 term is evaluated through explicit formulas for the transformed Kloosterman sum (a two-variable exponential sum) and its Euler product, which is recognized as L^S(s,chi(delta/·))/L^S(2s,chi^2). Contour shifting to the line Res=1/2 yields the main terms as residues and the error term as a vertical integral, controlled by the Ramanujan b

What would settle it

Compute the xi=0 contribution S_{xi=0}(X) for a test function that violates Assumption 1.1, e.g. with no supercuspidal component at infinity, and check whether the square term Sigma_n(square) vanishes. If Sigma_n(square) is nonzero, the identity I_cusp = I_ell fails and the asymptotic formula of Theorem 1.2 cannot hold without extra terms. Concretely, the triple-pole residue at u=1 in Proposition 4.7 is only shown to cancel under the supercuspidal condition; evaluating that residue for a non-supercuspidal function would give a direct counterterm.

Watch

Extended reading notes

Core claim

The central claim is Theorem 1.2: for any epsilon>0, the sum over n<X coprime to S of I_cusp(f^{n^2}) equals AX+BX+CX+O(X^{4/3 rho + 5/6 + epsilon}), where A, B, C are explicit constants built from zeta values, local integrals of orbital integrals, and transformed Kloosterman data, and rho<1/8 is a bound toward the Ramanujan conjecture. As a corollary, the average over automorphic representations of res_{s=1} L^S(s,pi,Sym^2)/L^S(2s,chi_pi^2) weighted by local traces equals A+B+C; this quantity is nonzero in general and therefore detects dihedral forms, which are exactly the representations for which the symmetric-square L-function has a pole at s=1.

Load-bearing premise

The entire proof rests on Assumption 1.1: one local component of the test function is supercuspidal at the archimedean place and another is supercuspidal or elliptic-supported at a chosen finite prime, and only the case where these are infinity and q_1 is fully worked out, with all other configurations left for the reader.

Editorial extensions

If this is right

If correct, the theorem gives a direct identity: the average over automorphic representations of res_{s=1} L^S(s,pi,Sym^2)/L^S(2s,chi_pi^2) equals the explicitly computed constant A+B+C.
The nonzero limit provides a new quantitative detection test for dihedral forms: one only needs to compute the local traces and the explicit constants to decide whether a dihedral representation contributes to the spectrum.
The error term O(X^{4/3 rho + 5/6 + epsilon}) improves as the Ramanujan bound rho decreases; with the current rho=7/64 it becomes O(X^{47/48+epsilon}).
The proof exhibits a complete template for the simple trace formula case: decompose the elliptic term by a second Poisson summation, compute the transformed Kloosterman sums exactly, recognize their Dirichlet series as a ratio of L-functions, and then extract residues.
The comparison between the sharp sum and its smooth approximation shows that the sharp asymptotic can be recovered from the smooth one once a bound toward the Ramanujan conjecture is assumed.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

If the supercuspidal assumption in Assumption 1.1 were dropped, the square term Sigma_n(square) would not vanish, and the same machinery suggests an additional hyperbolic-orbital contribution; this extra term would likely cancel against a part of the spectral side, giving a conditional test of the full Beyond Endoscopy expectation.
The explicit formulas for the transformed Kloosterman sums (Propositions 5.3-5.10) are self-contained and could be reused as a small toolkit for evaluating other quadratic-twist exponential sums in higher symmetric-power or higher-rank trace formulas.
A natural testable extension is to replace the supercuspidal condition at infinity and at q_1 by elliptic-supported conditions, or to allow arbitrary ramification at all finite places in S, and check whether the same shape AX+BX+CX persists with modified constants.
The ratio L^S(s,Sym^2)/L^S(2s,chi^2) that appears in Corollary 1.5 is itself a Shintani-type factor; the methods here could be adapted to study the analogous ratio for the standard representation, where the same residue analysis would yield a different set of constants.

Share X Bluesky LinkedIn Reddit HN

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

The one thing to know: Theorem 1.2 is a real new result, and the heavy computation behind it deserves a referee. The advertised corollary, though, is not a consequence of the paper as written. The stress-test note is right.

What is genuinely new: this is the first ramified symmetric-square case of Beyond Endoscopy via the simple trace formula, with S containing 2 and a fixed supercuspidal place. The structure is coherent: second Poisson summation, explicit transformed Kloosterman sums, the Dirichlet series analysis, and the final error-term balancing at δ = 5/6 − 2/3ϱ. The cancellation of the X^{3/2} terms via oddness of the Mellin transform of F checks out. The explicit constants A, B, C are plausible and the whole thing is a serious piece of analytic number theory.

The soft spots are real but proportionate. Assumption 1.1 is only worked out for v1 = ∞, v2 = q1; the paper says the other cases are similar and, in Theorem 4.1, leaves the computation to the reader. That is a limitation, not a fatal flaw: the stated configuration is already meaningful. More importantly, Section 10 does not prove what Corollary 1.5 claims. The formal identity ∑ aπ(n²)n^{-s} = L^S(s,π,Sym²)/L^S(2s,χ²) is fine, but the jump to the average over n and then to the spectral sum requires a Tauberian step and an interchange of an infinite sum with the limit X→∞. No uniform bound for the spectral remainders is given, and the Ramanujan-type bound ϱ<1/8 controls the geometric side, not the convergence of ∑π mπ ∏Tr(π_v(f_v)) times those remainders. So the detection of dihedral forms is an expectation, not a proved consequence.

The heavy reliance on unpublished same-author preprints is a referee burden, not an error. The cited results appear to be independently developed, and the self-citation pattern is not by itself suspicious.

My bottom line: the paper should go to a serious referee, with instructions to focus on Section 10 and on the deferred cases under Assumption 1.1. It should not be accepted as is; the corollary needs either a proof or a clear conditional statement. But the main theorem is substantial enough that ignoring the paper would be a mistake.

Referee Report

2 major / 4 minor

Summary. This paper develops a beyond-endoscopy analysis for the symmetric-square L-function on GL_2/Q in the presence of ramification at S={∞,q_1,...,q_r} (with 2∈S), under a 'simple trace formula' hypothesis (Assumption 1.1). The main theorem (Theorem 1.2) asserts that for any bound ϱ<1/8 toward Ramanujan and any function satisfying the assumption, ∑_{n<X,(n,S)=1} I_cusp(f^{n^2}) = AX+BX+CX+O(X^{4ϱ/3+5/6+ε}), where A,B,C are explicit constants from Theorems 4.1 and 8.12. The proof passes through the elliptic part of the trace formula, a second Poisson summation, exact evaluation of transformed Kloosterman sums, meromorphic continuation and residue analysis of the associated Dirichlet series, and a dyadic comparison with a smooth version. Section 10 proves a formal Euler-product identity and states that the average of the geometric side equals a spectral residue average, leading to Corollary 1.5.

Significance. If the spectral passage of Section 10 can be supplied, the paper would be a substantial new case of Langlands' beyond-endoscopy strategy for Sym^2, with explicit main terms and no fitted parameters: the input ϱ<1/8 and Assumption 1.1 are external, and the trace-formal computation is not circular. The local evaluation of transformed Kloosterman sums (Sections 5–6) and the residue analysis (Sections 4, 8) are impressive and generally coherent. The main advertised spectral/dihedral conclusion, however, is not obtained in the present text, because Section 10 stops at a formal identity and does not justify the limit interchange. The geometric asymptotic Theorem 1.2 is a credible conditional contribution in its own right.

major comments (2)

[Section 10 / Corollary 1.5] Corollary 1.5 does not follow from Theorem 1.2 as written. Theorem 10.2 proves only the formal Dirichlet-series identity ∑_{n,(n,S)=1} aπ(n^2)n^{-s}=L^S(s,π,Sym^2)/L^S(2s,χ_π^2). The paragraph after the theorem says that if the meromorphic continuation and simplicity of the pole are known, then the limit formula follows, and then asserts 'Thus by using the trace formula, the average analog is' (1.2). No Tauberian theorem is stated, no uniform bound for the individual remainders R_π(X)=(1/X)∑_{n<X}aπ(n^2)−res_{s=1} L^S(s,π,Sym^2)/L^S(2s,χ_π^2) is proved, and no argument shows that ∑_π mπ(∏_{v∈S}Trπ_v(f_v))R_π(X) tends to 0. The spectral sum is infinite and the weights are not shown to be uniformly summable in X. Since Corollary 1.5 is the advertised detection of dihedral forms, this is a load-bearing gap. It should either be filled with the required analytic estimates or explicitly labele
[§1.2, Assumption 1.1; §4, Theorem 4.1] Theorem 1.2 is stated for any S satisfying Assumption 1.1, but the proof is carried out only for the configuration v_1=∞, v_2=q_1. Assumption 1.1 says 'The other cases are similar but with different results', and Section 4 says 'We leave the computation to the reader.' The constant A in Theorem 4.1 is specific to q_1 (it contains log q_1 and local integrals over Q_{q_1}), so 'different results' means the formula is not proved for the other configurations. The statement of Theorem 1.2 and Corollary 1.5 should either be restricted to the configuration actually treated, or the analogous computations need to be included.

minor comments (4)

[Lemma 3.1] The proof of Lemma 3.1 is omitted ('We leave it to the reader'). Since this lemma justifies the second Poisson summation, a sketch or a precise statement of the analogous argument in [Che25c] should be included.
[Theorem 10.2 proof, Eq. (10.4)] The displayed series is written with p^{2us}, but the subsequent computation uses p^{us}. The identity ∑_u aπ(p^{2u})/p^{us} = (1+χπ(p)p^{-s})/((1−α^2 p^{-s})(1−β^2 p^{-s})) is the correct one; the exponent in the display should be p^{us}, not p^{2us}.
[Remark 1.4] The sentence 'Maybe one can improve the bound such that ϱ=1/4 proved in [Che25b] can be used' is confusing because Theorem 1.2 requires ϱ<1/8. If an improved theorem allowing ϱ=1/4 is envisioned, this should be stated explicitly.
[Theorem 7.1] The constants A,B,C in Theorem 7.1 clash with the main-term constants A,B,C of Theorem 1.2. Renaming the growth constants (for example α,β,γ) would avoid confusion.

Circularity Check

0 steps flagged · score 1.0 of 10

No circular reduction found; Theorem 1.2 is a conditional trace-formula computation, with the unproved Tauberian step in Section 10 a correctness gap rather than circularity.

full rationale

I walked the claimed derivation chain. Theorem 1.2 is a conditional trace-formula identity: under Assumption 1.1, the elliptic part of the simple trace formula is reduced, via the standard-representation machinery of [Che25a,b,c], to a second Poisson summation, an explicit computation of the transformed Kloosterman sum, and residue analysis. The constants A, B, C are defined in Theorems 4.1 and 8.12 as explicit local/global integrals; they are not fitted to the quantity being asymptotically expanded, and no equation in the paper reduces by construction to its own input. The Ramanujan-bound input ϱ<1/8 is an external analytic hypothesis, and the dihedral pole criterion for Sym^2 is not used as an input to the proof. The paper is heavily self-citational, but the cited results are separate preprints whose stated assumptions do not include the symmetric-square formula; they function as external lemmas rather than as renamed versions of Theorem 1.2. The substantial weakness is in Section 10: Theorem 10.2 proves only the formal Euler product identity sum a_π(n^2)n^{-s} = L^S(s,Sym^2)/L^S(2s,χ_π^2), and the passage to Corollary 1.5 is announced as a 'philosophy' ('The theorem implies the following philosophy... Thus by using the trace formula, the average analog is...') with no Tauberian theorem, no uniform bound on the individual remainders, and no justification of the interchange of the infinite spectral sum with X→∞. That is a missing proof/correctness gap, not a circular reduction. Likewise, the statements 'The other cases are similar but with different results' (Section 1.2) and 'We leave the computation to the reader' (Section 4) are scope limitations, not circularity. Accordingly, no circular step is established; the score 1 reflects the load-bearing reliance on unpublished same-author preprints without elevating that reliance to a circularity finding.

Assumptions & free parameters 0 free parameters · 4 assumptions · 0 invented entities

The paper introduces no fit parameters (all constants A, B, C are defined from local orbital integrals and test functions) and no new entities. The main borrowings are the trace formula and Poisson-summation machinery from the author's earlier preprints, plus external results (Ramanujan bound, Weyl bound, Sym² pole criterion).

assumptions (4)

domain assumption The simple trace formula identity I_cusp(f)=I_ell(f) and the Poisson-summation identity I_ell(f^{n^2}) = Σ_{n^2}(ξ) − Σ_{n^2}(□)
Invoked in §1.3 and §3, taken from [GH24, Ch.16] and [Che25a]; the square term vanishes under Assumption 1.1 (Prop 9.1).
domain assumption Ramanujan-conjecture bound ϱ<1/8 for GL(2) automorphic forms, used as I_cusp(f^n) ≪ n^{ϱ+ε}
Used in §9 to compare S(X) with S_G(X) and to set δ=5/6−2/3ϱ; currently only 7/64 is unconditional.
standard math Analytic estimates: Weyl bound for Dirichlet L-functions, bounds for 1/L, rapid decay and oddness of \tilde{F}
Used throughout §4–§7, e.g., (4.3), (7.5), (7.7), and Section 2.5.
domain assumption Known properties of Sym² L-functions: meromorphic continuation and a pole at s=1 iff π is dihedral
Invoked in §10 and Corollary 1.5 for detecting dihedral forms; not proved in this paper.

how reviews work

0 comments

Cite this review

Pith. "Pith review of Beyond endoscopy for the symmetric square representation: The simple trace formula case." pith.science (2026). https://pith.science/paper/2JR6YSCV

@misc{pith2026260725383,
  author       = {Pith},
  title        = {Pith review of: Beyond endoscopy for the symmetric square representation: The simple trace formula case},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/2JR6YSCV}},
  note         = {Machine review of arXiv:2607.25383}
}

abstract

At the beginning of this century, Langlands introduced a strategy known as \emph{Beyond Endoscopy} to attack the principle of functoriality. Altu\u{g} studied $\mathsf{GL}_2$ over $\mathbb Q$ in the unramified setting for the standard representation. We consider the case with ramification at $S=\{\infty,q_1,\dots,q_r\}$ with $2\in S$ and derive an asymptotic formula for the symmetric square representation adding some additional conditions on the test function so that the trace formula is simple. The limit is nonzero in general and we may detect the dihedral forms by using such limit form of the trace formula which is similar to Venkatesh's thesis. The proof involves a second Poisson summation, computation of the transformed Kloosterman sum and the corresponding series, and giving an asymptotic formula for the main term by residue analysis and using technical analysis to deal with the error term.

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

16 extracted references · 3 linked inside Pith

[1]

P oisson summation and isolation of special representations , Compos

Salim Ali Altu g , Beyond endoscopy via the trace formula: 1. P oisson summation and isolation of special representations , Compos. Math. 151 (2015), no. 10, 1791--1820

2015
[2]

, Beyond endoscopy via the trace formula, II : A symptotic expansions of F ourier transforms and bounds towards the R amanujan conjecture , Amer. J. Math. 139 (2017), no. 4, 863--913

2017
[3]

, Beyond endoscopy via the trace formula--- III T he standard representation , J. Inst. Math. Jussieu 19 (2020), no. 4, 1349--1387

2020
[4]

A. I. Borevich and I. R. Shafarevich, Number theory, Pure and Applied Mathematics, vol. Vol. 20, Academic Press, New York-London, 1966, Translated from the Russian by Newcomb Greenleaf

1966
[5]

Yuhao Cheng, Beyond endoscopy for _2 over with ramification 1 : Poisson summation , preprint (2025), ArXiv:2505.18967

arXiv 2025
[6]

, Beyond endoscopy for GL _2 over with ramification 2 : Bounds towards the R amanujan conjecture , preprint (2025), ArXiv:2507.09655

arXiv 2025
[7]

, Beyond endoscopy for GL _2 over with ramification 3 : Contribution of the elliptic part , preprint (2025), ArXiv:2508.07167

arXiv 2025
[8]

, Beyond endoscopy for GL _2 over with ramification 4 : Contribution of non-elliptic parts , preprint (2026), ArXiv:2605.20719

arXiv 2026

Show all 16 references

[9]

Getz and Heekyoung Hahn, An introduction to automorphic representations: with a view toward trace formulae, Graduate Texts in Mathematics, vol

Jayce R. Getz and Heekyoung Hahn, An introduction to automorphic representations: with a view toward trace formulae, Graduate Texts in Mathematics, vol. 300, Springer, 2024

2024
[10]

Langlands, Beyond endoscopy, Contributions to automorphic forms, geometry, and number theory, Johns Hopkins Univ

Robert P. Langlands, Beyond endoscopy, Contributions to automorphic forms, geometry, and number theory, Johns Hopkins Univ. Press, Baltimore, MD, 2004, pp. 611--697

2004
[11]

10, 1879--1960

Ian Petrow and Matthew P Young, The fourth moment of Dirichlet L -functions along a coset and the Weyl bound , Duke Mathematical Journal 172 (2023), no. 10, 1879--1960

2023
[12]

E ndoscopy and B eyond

Peter Sarnak, Comments on R obert L anglands' lecture: " E ndoscopy and B eyond" , 2001, available at http://publications.ias.edu/sites/default/files/SarnakLectureNotes-1.pdf

2001
[13]

4, Citeseer, 2005, pp

, Notes on the generalized R amanujan conjectures , Harmonic analysis, the trace formula, and Shimura varieties (James Arthur, David Ellwood, and Robert Kottwitz, eds.), vol. 4, Citeseer, 2005, pp. 659--685

2005
[14]

163, American Mathematical Soc., 2015

G \'e rald Tenenbaum, Introduction to analytic and probabilistic number theory, Graduate Studies in Mathematics, vol. 163, American Mathematical Soc., 2015

2015
[15]

thesis, Princeton University, 2002

Akshay Venkatesh, Limiting forms of the trace formula, Ph.D. thesis, Princeton University, 2002

2002
[16]

577, 23--80

, Beyond endoscopy and special forms on GL(2) , Journal für die reine und angewandte Mathematik (2004), no. 577, 23--80

2004

Pith tools

Reviewed August 1, 2026 · model on record in the stance chip above.

[1] [1]

P oisson summation and isolation of special representations , Compos

Salim Ali Altu g , Beyond endoscopy via the trace formula: 1. P oisson summation and isolation of special representations , Compos. Math. 151 (2015), no. 10, 1791--1820

2015

[2] [2]

, Beyond endoscopy via the trace formula, II : A symptotic expansions of F ourier transforms and bounds towards the R amanujan conjecture , Amer. J. Math. 139 (2017), no. 4, 863--913

2017

[3] [3]

, Beyond endoscopy via the trace formula--- III T he standard representation , J. Inst. Math. Jussieu 19 (2020), no. 4, 1349--1387

2020

[4] [4]

A. I. Borevich and I. R. Shafarevich, Number theory, Pure and Applied Mathematics, vol. Vol. 20, Academic Press, New York-London, 1966, Translated from the Russian by Newcomb Greenleaf

1966

[5] [5]

Yuhao Cheng, Beyond endoscopy for _2 over with ramification 1 : Poisson summation , preprint (2025), ArXiv:2505.18967

arXiv 2025

[6] [6]

, Beyond endoscopy for GL _2 over with ramification 2 : Bounds towards the R amanujan conjecture , preprint (2025), ArXiv:2507.09655

arXiv 2025

[7] [7]

, Beyond endoscopy for GL _2 over with ramification 3 : Contribution of the elliptic part , preprint (2025), ArXiv:2508.07167

arXiv 2025

[8] [8]

, Beyond endoscopy for GL _2 over with ramification 4 : Contribution of non-elliptic parts , preprint (2026), ArXiv:2605.20719

arXiv 2026