{"id":"b0410309-0f74-4ae3-a34a-0a9e03b02785","arxiv_id":"2607.25383","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":7.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"A ramified simple-trace-formula version of Beyond Endoscopy for the symmetric square yields an asymptotic formula whose main term detects poles of the symmetric-square L-function.","lead":"This paper derives an asymptotic formula for a Beyond Endoscopy sum over n of trace-formula coefficients associated with n squared, in the symmetric-square case on GL(2) with ramification and a simple trace formula. The limit isolates poles of the symmetric-square L-function, so the authors say it can detect dihedral forms.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Section 10 proves only the formal Euler product; Corollary 1.5 requires a Tauberian step and an interchange of the spectral limit that is not supplied, so the advertised spectral/dihedral conclusion does not follow from Theorem 1.2 as written.","rationale":"The reader's weakest_assumption concerns Assumption 1.1, and that is a genuine condition: Theorem 1.2 is explicitly proved only for v1=∞, v2=q1, while the other configurations are deferred. I found no concrete flaw in the geometric trace-formula computation under that assumption: Lemma 4.2 follows from the stated Euler-product manipulation, the transformed Kloosterman sums in Sections 5–6 are computed explicitly, the estimates in Sections 7–8 are consistent, and the dyadic comparison in Section 9 balances correctly for ϱ<1/8. However, Section 10 itself labels the bridge to the spectral side as a 'philosophy', and Corollary 1.5 claims a precise averaged residue formula. The missing Tauberian/interchange step is load-bearing because the advertised purpose is detecting dihedral forms through the spectral side. This concern is distinct from, but additive to, the reader's Assumption 1.1 concern, so I only partially agree with the reader's identification. Since the geometric theorem remains conditional and the spectral corollary lacks a proof of the interchange, the conditional verdict is unchanged; the concrete test above would settle whether the concern lands.","tokens_in":61580,"tokens_out":22029,"duration_ms":213086,"concrete_test":"Work out the spectral-side identity needed by Corollary 1.5: for each X write I_cusp(f^{n^2}) = ∑_π w_π a_π(n^2) with w_π = m_π∏_{v∈S} Tr π_v(f_v), and compute the remainder R_π(X) = (1/X)∑_{n<X} a_π(n^2) − res_{s=1} L^S(s,π,Sym^2)/L^S(2s,χ_π^2). Then check whether ∑_π |w_π| |R_π(X)| → 0 as X→∞, using the best available subconvex/Weyl bounds for GL2 L-functions and the Weyl law. If the sum is not shown summable, Corollary 1.5 fails as stated; if a uniform bound can be proved, the missing lemma is supplied and the verdict can be upgraded.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The geometric asymptotic Theorem 1.2 is conditionally plausible, and under Assumption 1.1 the trace-formula computation in Sections 4–9 appears internally coherent. The unsupported step is the passage to Corollary 1.5. To obtain it, one must write I_cusp(f^{n^2}) as the spectral sum ∑_π m_π (∏_{v∈S} Tr π_v(f_v)) a_π(n^2), average over n, and take the limit X→∞ inside the infinite sum over π. Section 10 proves Theorem 10.2, which is only the formal Dirichlet-series identity ∑ a_π(n^2)n^{-s} = L^S(s,π,Sym^2)/L^S(2s,χ_π^2), and then states 'The theorem implies the following philosophy... Thus by using the trace formula, the average analog is...'. No Tauberian theorem is stated, no uniform bound for the individual remainders R_π(X) = (1/X)∑_{n<X}a_π(n^2) − res_{s=1} L^S(s,π,Sym^2)/L^S(2s,χ_π^2) is given, and no argument shows ∑_π |w_π R_π(X)| → 0. The spectral sum is infinite, and the weights m_π∏ Tr π_v(f_v) are not known to be summable uniformly in X; the bound ϱ<1/8 toward Ramanujan used in Theorem 1.2 controls the geometric side but does not by itself justify the interchange. Thus (1.2) and Corollary 1.5 are not consequences of Theorem 1.2 as written.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","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.","tokens_in":62011,"tokens_out":16144,"duration_ms":153360,"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":[{"comment":"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","section":"Section 10 / Corollary 1.5"},{"comment":"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.","section":"§1.2, Assumption 1.1; §4, Theorem 4.1"}],"minor_comments":[{"comment":"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.","section":"Lemma 3.1"},{"comment":"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}.","section":"Theorem 10.2 proof, Eq. (10.4)"},{"comment":"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.","section":"Remark 1.4"},{"comment":"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.","section":"Theorem 7.1"}],"recommendation":"major_revision","confidential_remarks":"The paper relies heavily on the author's own preprints [Che25a,b,c] (and [Che26]) for foundational identities. If those are not yet publicly available or refereed, independent verification is difficult; the editor may wish to check their status. The core missing proof is the Tauberian/interchange step in Section 10. If the author is willing to frame the spectral/dihedral conclusion as conditional and present Theorem 1.2 as the main result, the paper would be close to acceptable. I would not recommend rejection because the geometric asymptotic is a substantial, self-contained computation modulo the stated assumption."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","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.\n\nWhat 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.\n\nThe 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.\n\nThe 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.\n\nMy 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.","headline":"A genuinely new geometric asymptotic for the symmetric square, but the spectral/dihedral corollary is not proved as written.","tokens_in":62519,"tokens_out":4295,"would_cite":true,"duration_ms":51922,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["11F70","11F72"],"pacs":[],"model":"deepseek-v4-flash","headline":"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","keywords":["beyond endoscopy","symmetric square","simple trace formula","transformed Kloosterman sums","Ramanujan conjecture","dihedral forms","L-functions","automorphic forms"],"falsifier":"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.","tokens_in":61458,"feed_emoji":"🧮","tokens_out":4339,"duration_ms":43915,"temperature":0.7,"pith_summary":"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.","feed_headline":"Dihedral forms detected by a symmetric-square trace formula","feed_subtitle":"Under supercuspidal conditions, the averaged cuspidal trace equals explicit constants plus a controlled error term.","key_machinery":"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","core_discovery":"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.","pith_inferences":["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."],"forward_implications":["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."],"fun_headline_variants":["Symmetric-square trace formula spots dihedral forms","Beyond endoscopy: symmetric square detects dihedral forms","Averaged trace formula isolates dihedral representations","Symmetric-square L-function pole signals dihedral forms"],"cache_read_input_tokens":2304,"weakest_assumption_plain":"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.","fun_headline_variants_meta":{"raw":{"variants":["Symmetric-square trace formula spots dihedral forms","Beyond endoscopy: symmetric square detects dihedral forms","Averaged trace formula isolates dihedral representations","Symmetric-square L-function pole signals dihedral forms"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000229,"raw_usage":{"total_tokens":1312,"prompt_tokens":736,"completion_tokens":576,"prompt_tokens_details":{"cached_tokens":256},"prompt_cache_hit_tokens":256,"prompt_cache_miss_tokens":480,"completion_tokens_details":{"reasoning_tokens":515}},"tokens_in":480,"tokens_out":576,"duration_ms":5936,"temperature":1.0,"reasoning_tokens":515,"cache_read_input_tokens":256,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-01T02:34:47.867972+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"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.","supporting_citations":[],"review_version":1}