{"id":"40413acb-9cbc-4be0-a98c-3e984444f13b","arxiv_id":"2505.18967","paper_version":3,"verdict":"ACCEPT","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"A new identity isolates all one-dimensional and Eisenstein terms in the elliptic part of the GL2 trace formula over Q when ramification at any finite set containing 2 is allowed.","lead":"This paper proves a ramified version of the beyond endoscopy isolation step for GL2 over the rational numbers. It writes the elliptic side of the trace formula as a sum of traces of one-dimensional representations and Eisenstein twists plus explicit correction terms, extending Altuğ's unramified result.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Proposition B.6's reduction to the model function Omega omits the singular prefactor multiplying V in the second summand of Psi; if that prefactored term fails the Fourier-decay estimate, Theorem 4.1's Poisson step and hence Theorem 8.10 are unsupported.","rationale":"The reader correctly identified the non-Schwartz nature of Psi and the reliance on Proposition B.6/B.8 as the weakest assumption. My stress-test agrees with that identification and sharpens it: even granting the decay estimate for the Omega model, the actual Psi contains a singular prefactor in the V-term that is not visibly covered by the reduction in Proposition B.6. The proof's own text acknowledges Psi is not Schwartz and defers the issue to Appendix B, so this is an in-manuscript admitted limitation rather than a manufactured objection. The rest of the paper is careful and internally consistent: the elementary computations in Section 2 are detailed, the approximate functional equation in Section 3 is explicit, the Kloosterman-series computation in Section 5 is straightforward, and the residue analysis in Section 6 is plausible. Those parts do not raise a comparable concern. The lack of formal verification and the importation of several standard facts from Altuğ and germ-expansion literature are consistent with the reader's moderate confidence. However, since Theorem 8.10's central identity is obtained by applying Poisson summation to the specific Psi in (B.6), the omitted verification of the V-term is load-bearing. I therefore recommend a conditional acceptance pending an explicit proof that the prefactored V-term satisfies the hypotheses of Corollary B.4, or a counterexample to that verification, which would require revisiting Theorem 4.1.","tokens_in":55842,"tokens_out":10842,"duration_ms":103158,"concrete_test":"Re-derive Proposition B.6 keeping the V-term intact: set Phi_2(t) = t^{1/2} V(C t^{1-theta}) and check whether Phi_2 is C^2 at t=0 using the contour representation V(x) = (sqrt(pi)/2 pi i) int_{(sigma)} eF(s) prod_i (...) Gamma((iota+s)/2)/Gamma((iota+1-s)/2) (pi x)^{-s} ds. Then, for a model case r=1, N=1, theta^1_q(y,1) as in (B.9), compute or bound the semilocal Fourier transform of the V-summand and verify that sum_{xi in Z_S} |widehat{Psi}(xi/k, xi/k)| converges and satisfies Lemma B.8's estimate. If Phi_2 is not C^2 or the Fourier decay is only O(|xi|^{-1}), Theorem 4.1 requires a genuinely new argument.","verdict_should_be":"CONDITIONAL","load_bearing_attack":"The proof of Theorem 4.1 applies the semilocal Poisson summation formula (Corollary B.4) to the function Psi displayed in (B.6). The second summand of Psi is not of the form handled by Proposition B.6: it contains the explicit prefactor k f^2 / sqrt(|x^2 - 4N|_infty |y^2 - 4N|'_q) multiplying V(k f^2 / (|x^2 - 4N|^{1-theta}_infty |y^2 - 4N|'^{1-theta}_q)). Proposition B.6 only proves the required decay for Omega(x,y) = theta_infty prod theta_q Phi(1/(|x^2 - 4N|^theta_infty |y^2 - 4N|'^theta_q)) with Phi rapidly decreasing. The sentence 'Since F and V have rapid decay ... we only need to consider a more general case Omega' implicitly absorbs the singular prefactor into Phi. To do this one would need Phi(t) = t^{1/2} V(C t^{1-theta}) to be smooth on [0,infty) with all derivatives rapidly decreasing. This is not shown. If Phi has fractional-power behavior near t = 0, the second derivative used in the integration-by-parts step of Lemma B.8 may be unbounded, so the claimed bound |widehat{Omega_v}(xi,eta)| << (1+|xi|_infty)^{-2} prod(1+|eta_i|)^{-3} can fail. Since Theorem 4.1 is derived solely from this Poisson summation step, and Theorem 8.10 combines Theorem 4.1 with the later contour computations, the central identity rests on this unverified reduction.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper generalizes Altuğ's \"Beyond Endoscopy\" isolation theorem for GL2 over Q to the case with ramification at a finite set S={∞,q_1,...,q_r} containing 2. The main result, Theorem 8.10, asserts an identity expressing the elliptic part I_ell(f^n) of the trace formula as the sum of traces of all one-dimensional representations, minus one half of the traces of their Eisenstein twists, plus explicit correction terms Σ(□), Σ(0), and Σ(ξ≠0). The proof proceeds by computing the local orbital integrals and Shalika germs, deriving an approximate functional equation for the relevant partial Zagier L-function, applying a semilocal Poisson summation formula on R×∏Q_{q_i}, analyzing the resulting Kloosterman-type Dirichlet series, and then isolating the contributions of the residues at s=0, s=-1/2, and s=1/2. The paper is largely self-contained and includes detailed appendices for the local orbital integral computations and the semilocal analysis.","tokens_in":56189,"tokens_out":38222,"duration_ms":286580,"significance":"If the main identity is correct, the paper resolves a problem that has been open since Altuğ's unramified work, giving a fully ramified isolation formula over Q with arbitrary ramification at a finite set containing 2. The result is of clear interest to the Beyond Endoscopy program. The paper's strengths include the explicit, parameter-free local computations, the use of an approximate functional equation to justify Poisson summation, the treatment of the nonsmooth integrand in Appendix B, and the explicit residue calculations that isolate the one-dimensional and Eisenstein contributions. These are substantial technical achievements and the presentation is generally careful. However, several load-bearing algebraic and analytic steps in Sections 4 and 6 and in Appendix B need to be made fully rigorous or corrected; these are detailed in the major comments.","major_comments":[{"comment":"This cancellation does not appear to hold for the displayed formulas. Writing A=|x^2∓1|_∞ and B=|y^2∓4nq^ν|'_q, the residue at s=-1/2 of the first contour integral contributes, after the prefactor 4√n q^{ν/2}, a term proportional to eF(-1/2)(4nq^ν)^{-ϑ/2}(AB)^{-ϑ/2}, while the residue at s=1/2 of the second contour integral contributes a term proportional to eF(1/2)(4nq^ν)^{(1-ϑ)/2}(AB)^{-(1+ϑ)/2}. Using eF(-1/2)=-eF(1/2), the sum is a nonzero multiple of (AB)^{-(1+ϑ)/2}-(AB)^{-ϑ/2} (up to powers of nq^ν), which vanishes only for special values of AB. The proof should be corrected or the formulas in Theorem 6.1 and Theorem 8.10 should include the additional residual terms. This is load-bearing because the isolation of the one-dimensional and Eisenstein terms relies on this cancellation.","section":"Section 6, proof of Theorem 6.1, around the claim \"Since eF is odd, the third and the last term in the bracket cancel.\""},{"comment":"The definition of Σ(□) is not meaningful for terms with T^2∓4nq^ν=0. In that case f^2 divides 0 for every f∈Z^{(S)}, so the sum over f is infinite, and the denominators |T^2∓4nq^ν|'_{∞,q} vanish. Such T occur whenever n is a square and the ν_i are even, so this is not a vacuous issue. The paper should either prove that these terms vanish by a limiting argument (each summand tends to 0 because F and V have rapid decay) or explicitly exclude δ=0 from Σ(□) and explain why the Poisson summation step in the proof of Theorem 4.1 is unaffected. As written, the main formula has undefined entries.","section":"Theorem 4.1 and Theorem 8.10, definition of Σ(□)"},{"comment":"The reduction to the model function Ω omits a justification for the second summand in the displayed function Ψ in (B.6). That summand contains the singular prefactor k f^2/(|x^2-4N|_∞|y^2-4N|'_q) multiplying V(k f^2/(|x^2-4N|^{1-ϑ}_∞|y^2-4N|'^{1-ϑ}_q)). The reduction can be made to work by writing the second summand as Φ(1/(|x^2-4N|^ϑ_∞|y^2-4N|'^ϑ_q)) with Φ(t)=C t^{1/ϑ} V(C t^{(1-ϑ)/ϑ}) and noting that on the support of θ the variable t stays bounded away from 0, so the rapid-decay hypothesis of Proposition B.6 is satisfied. This argument is not given in the manuscript, and without it the application of Corollary B.4 to Ψ is not justified. The proof should spell out this absorption and verify that the resulting Φ has the required smoothness and decay on the relevant range.","section":"Appendix B, Proposition B.6 and its proof"}],"minor_comments":[{"comment":"In the first row of the table, \"Theorem 1.1 (Theorem 1.1)\" should read \"Theorem 1.1 ([Alt15, Theorem 1.1])\" to avoid confusion.","section":"Section 1.2, comparison table"},{"comment":"The contour C_v is defined using a zero-free region of ζ(s+1) near s=0; this is fine, but the choice of v should be made quantitative or at least its existence justified by the fact that ζ(1)≠0.","section":"Section 6, Theorem 6.1 statement"},{"comment":"The notation T^2∓4nq^ν≠□ is used to exclude squares; since 0 is a square, the subsequent insertion of Σ(□) makes the zero-discriminant case ambiguous, as noted in the major comments.","section":"Section 4, equation (4.1)"},{"comment":"The convergence estimate after the display \"≪1+∑_{I⊆{1,...,r}}...\" is terse; a few more details on how the archimedean and nonarchimedean valuations of elements of Z_S are simultaneously controlled would improve readability.","section":"Appendix B, proof of Proposition B.6"}],"recommendation":"major_revision","confidential_remarks":"The stress-test concern about the omitted prefactor in Proposition B.6 is addressable in principle, but the more serious issue is the residue cancellation in the proof of Theorem 6.1, which appears incorrect from the displayed formulas. I would ask the author to provide a fully detailed residue bookkeeping for the two contour shifts, possibly checking the identity on a simple test case with explicit functions, before the paper can be accepted. The paper is technically ambitious and likely correct in conception, but the central isolation step needs to be verified."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Colleague,\n\nThis is a serious paper: it generalizes Altuğ's unramified isolation formula for GL2/Q to the ramified case, with arbitrary local components at a finite set S containing 2. The main identity, Theorem 8.10, isolates the one-dimensional and Eisenstein contributions inside the elliptic part of the trace formula. If correct, it completes the isolation step of the beyond-endoscopy program for GL2/Q, which had been stuck in the unramified setting for a decade. That is a real contribution, not a repackaging.\n\nThe paper is technically detailed and honest. The local orbital integral computations in Appendix A are careful, the approximate functional equation for the partial Zagier L-function is fully worked out, and the contour-shifting in Section 6 is explicit. The correction of Altuğ's Eisenstein coefficient in Remark 8.9 is useful. The citation pattern is proper: the debt to Altuğ and Langlands is clearly stated, and the newly proven identities are genuinely new.\n\nNow the soft spot. The proof of Theorem 4.1 applies the semilocal Poisson summation formula to the function Ψ in (B.6). The second summand of Ψ contains an explicit prefactor 1/sqrt(|x^2-4N| |y^2-4N|') multiplying V(...). Appendix B reduces to a model function Ω that has no such prefactor, and the reduction is not justified. To absorb the prefactor one would need a lemma showing that t^{1/2θ} V(C t^{1-1/θ}) and its derivatives have sufficient decay near t=0; that lemma is absent. The prefactor is likely harmless because V decays super-polynomially in its argument, and the argument blows up exactly where the prefactor does. But as written, Proposition B.6 does not cover the actual Ψ, and Theorem 8.10 rests on this step. This is a genuine gap, not cosmetic, though I suspect it is repairable.\n\nMinor: the abstract's claim of \"fully resolving the problem\" overstates the scope; this is the isolation step, not the full asymptotic for cuspidal contributions.\n\nThe paper should go to a serious referee with expertise in Poisson summation on adelic semilocal spaces. If the prefactor issue is resolved, I would accept. Worth serious engagement.","headline":"A serious, largely convincing generalization of Altuğ's isolation formula to the ramified case, with one load-bearing technical gap in the semilocal Poisson summation that needs a missing lemma.","tokens_in":56723,"tokens_out":8089,"would_cite":true,"duration_ms":70078,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["11F72","11F70"],"pacs":[],"model":"deepseek-v4-flash","headline":"For GL_2 over Q with ramification at any finite set containing 2, the elliptic part of the trace formula equals the one-dimensional traces minus half the Eisenstein traces plus explicit remainder terms.","keywords":["beyond endoscopy","trace formula","GL_2","Poisson summation","orbital integrals","Zagier L-function","Kloosterman sums","Eisenstein series"],"falsifier":"Evaluate both sides of equation (4.1) for the smallest ramified case, $S=\\{\\infty,2\\}$ with $n=1$ and $f_2$ the characteristic function of the Iwahori subgroup: if the two sides differ, the semilocal Poisson summation step (Proposition B.6 / Lemma B.8) fails and Theorem 8.10 collapses. More narrowly, checking Lemma B.8's claimed decay $(1+|\\eta_i|_{q_i})^{-3}$ in the nonarchimedean Fourier variable for the germ expansion (2.6) near $y_i=2\\sqrt{N}$ would settle the load-bearing estimate directly.","tokens_in":55631,"feed_emoji":"🧮","tokens_out":17040,"duration_ms":120113,"temperature":0.7,"pith_summary":"This paper proves an exact identity for the elliptic part of the trace formula for $\\mathrm{GL}_2$ over $\\mathbb{Q}$ when the test function is ramified at an arbitrary finite set of places $S$ containing $2$. The identity splits the elliptic part into the trace contribution of all one-dimensional automorphic representations, minus half the contribution of their twists of the Borel-induced representation $\\xi_0$, plus three explicit remainder terms. This generalizes the previously solved unramified case and answers a question about the ramified setting that had remained open for over a decade. If the identity is correct, it completes the first step of the beyond-endoscopy strategy over $\\mathbb{Q}$: isolating the residual and Eisenstein parts of the spectral side inside the geometric side of the trace formula, for arbitrary ramification.","feed_headline":"Trace formula identity splits one-dimensional and Eisenstein parts","feed_subtitle":"For GL_2 over Q with any ramification containing 2, the elliptic trace formula splits into computable parts.","key_machinery":"The load-bearing mechanism is a semilocal Poisson summation formula on $Q_S=\\mathbb{R}\\times\\mathbb{Q}_{q_1}\\times\\cdots\\times\\mathbb{Q}_{q_r}$ (Corollary B.4), applied to a function $\\Psi$ that is explicitly not Schwartz class because of singularities at $y_i=\\pm2\\sqrt{N}$ in the nonarchimedean coordinates. To make the summation legitimate, the paper first rewrites the partial Zagier $L$-function $L^S(1,\\delta)$ using an approximate functional equation built from an auxiliary function $F$ (defined via the modified Bessel function $K_0$) and a variance function $V$, producing an absolutely convergent series. A modified $p$-adic norm $|y|'_p$ (with $|y|'_p\\asymp|y|_p$ and $p^{k_\\gamma}=|T^2-4N|'^{-1/2}_p$) controls the germ expansions at the center, and the generalized Kloosterman sums $\\mathrm{Kl}^S_{k,f}(\\xi,m)$ encode the arithmetic of the summation. Contour shifts in the Mellin representation of $F$ and $V$, with residues at the poles of $\\tilde{F}(s)$ and of ratios of Riemann zeta functions, then isolate the $\\xi=0$ terms, which the Weyl integration formula and Fourier inversion on $\\widehat{\\mathbb{Z}^\\times_{S,\\mathrm{fin}}}$ identify with the traces of the one-dimensional representations and their Eisenstein twists.","core_discovery":"The paper's central claim is Theorem 8.10: for every finite set $S=\\{\\infty,q_1,\\ldots,q_r\\}$ with $2\\in S$, and every suitable test function $f^n$ whose local components outside $S$ are Hecke operators and whose ramified components are arbitrary smooth compactly supported functions, the elliptic part of the trace formula satisfies $$I_{\\mathrm{ell}}(f^n)=\\sum_\\mu \\operatorname{Tr}(\\mu(f^n))-\\frac{1}{2}\\sum_\\mu \\operatorname{Tr}((\\xi_0\\otimes\\mu)(f^n))-\\Sigma(\\square)+\\Sigma(0)+\\Sigma(\\xi\\neq0),$$ where $\\mu$ runs over all one-dimensional (Grossencharacter-type) representations of $G(\\mathbb{Q})\\backslash G(\\mathbb{A})^1$ and $\\xi_0$ is the representation induced from the trivial character of the Borel subgroup. The remainder terms are explicit: $\\Sigma(\\square)$ collects the square-discriminant contributions, $\\Sigma(0)$ is the zero-frequency term produced by the contour shift, and $\\Sigma(\\xi\\neq0)$ contains the nonzero-frequency terms of the semilocal Poisson summation. The paper further shows that the first term equals the sum of traces of all one-dimensional representations (Theorem 7.7) and that the second equals half the sum of traces of their Eisenstein twists (Theorem 8.8), so the identity genuinely isolates these two families inside the elliptic part.","pith_inferences":["The semilocal Poisson summation combined with an approximate functional equation is a transferable template: the same combination should apply to higher-rank groups or to central-value problems for twisted partial $L$-functions, where the singularities of the test function near elliptic tori take the same schematic form.","The computation of the one-dimensional term via Fourier inversion on $\\widehat{\\mathbb{Z}^\\times_{S,\\mathrm{fin}}}$ suggests that with ramification the isolated spectrum is naturally parameterized by characters of an $S$-unit class group; characterizing that group explicitly for each $S$ is a direct extension.","Because the identity is exact and explicit, it could be tested numerically for $S=\\{\\infty,2\\}$ with small $n$, using the closed-form local orbital integrals of Section 2 and Appendix A; a successful numerical check would independently confirm the Poisson summation step, while a failure would pinpoint the load-bearing estimate."],"forward_implications":["For every ramification set $S\\ni 2$, the residual (one-dimensional) part of the spectral side is expressible as an explicit finite sum of traces of one-dimensional representations, each computable from local data.","The same formula gives the Eisenstein contribution as an explicit sum over the same one-dimensional characters twisted by the Borel-induced representation $\\xi_0$, with the coefficient $-1/2$ fixed.","Since $a_{\\pi,\\mathrm{Sym}^j}(p)=a_\\pi(p^j)$ for primes $p$, the theorem also isolates the $p$-th Fourier coefficients of higher symmetric powers from the elliptic part.","The remainder terms $\\Sigma(\\square)$, $\\Sigma(0)$, and $\\Sigma(\\xi\\neq0)$ are written as explicit convergent series of generalized Kloosterman sums and contour integrals of zeta ratios, so the identity is fully explicit and accessible to further analysis."],"supporting_citations":[{"why":"It supplies the unramified analogue of the main identity and the proof template (Poisson summation, isolation of special representations) that this paper generalizes.","marker":"[Alt15]"},{"why":"It raises the question of isolating these representations in the ramified case, which the paper answers.","marker":"[EL22]"},{"why":"It introduces the beyond-endoscopy strategy and the framework of elliptic parts, orbital integrals, and measure normalizations used throughout.","marker":"[Lan04]"},{"why":"It provides the measure normalization for the elliptic part of the trace formula, which differs from the earlier convention and affects the constant factors.","marker":"[FL11]"},{"why":"It gives the structure of centralizers of elliptic elements and the local orbital-integral framework used in Section 2 and Appendix A.","marker":"[Kot05]"},{"why":"It provides the Shalika germ expansion used to analyze ramified orbital integrals near the center in Theorem 2.11 and Corollary 2.12.","marker":"[Sha72]"},{"why":"It supplies the Weyl integration formula used in Propositions 7.3 and 8.5 to compute local traces.","marker":"[Clo89]"},{"why":"It supplies the Pontryagin duality and Fourier inversion on the semilocal space underlying the Poisson summation formula used in Theorem 4.1.","marker":"[Bou19]"},{"why":"It provides the volume computation for elliptic contributions (residue of the Dedekind zeta function) used in Theorem 2.3 and Corollary 2.4.","marker":"[Tat67]"},{"why":"It provides the Stirling estimates used to bound the Mellin transform in Proposition 3.5 and Lemma 3.6, which underpin the approximate functional equation.","marker":"[Ten15]"}],"fun_headline_variants":["Beyond endoscopy resolves decade-old isolation problem for GL2 over Q","Poisson summation fully isolates GL2 representations with any ramification","New identity in beyond endoscopy: ramified case now fully resolved","GL2 trace formula: isolating one-dimensional and Eisenstein terms","Beyond endoscopy's first step now complete for GL2 over Q"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The argument assumes that the non-Schwartz function $\\Psi$ built from the local orbital integrals still satisfies the semilocal Poisson summation hypotheses, in particular the Fourier decay estimate of Proposition B.6 proved via Lemma B.8 involving the modified $p$-adic norm; if that estimate fails, equation (4.1) fails and the main identity collapses.","fun_headline_variants_meta":{"raw":{"variants":["Beyond endoscopy resolves decade-old isolation problem for GL2 over Q","Poisson summation fully isolates GL2 representations with any ramification","New identity in beyond endoscopy: ramified case now fully resolved","GL2 trace formula: isolating one-dimensional and Eisenstein terms","Beyond endoscopy's first step now complete for GL2 over Q"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.001336,"raw_usage":{"total_tokens":5470,"prompt_tokens":1023,"completion_tokens":4447,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":639,"completion_tokens_details":{"reasoning_tokens":4358}},"tokens_in":639,"tokens_out":4447,"duration_ms":27470,"temperature":1.0,"reasoning_tokens":4358,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-07T14:22:27.486817+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Evaluate both sides of equation (4.1) for the smallest ramified case, $S=\\{\\infty,2\\}$ with $n=1$ and $f_2$ the characteristic function of the Iwahori subgroup: if the two sides differ, the semilocal Poisson summation step (Proposition B.6 / Lemma B.8) fails and Theorem 8.10 collapses. More narrowly, checking Lemma B.8's claimed decay $(1+|\\eta_i|_{q_i})^{-3}$ in the nonarchimedean Fourier variable for the germ expansion (2.6) near $y_i=2\\sqrt{N}$ would settle the load-bearing estimate directly.","supporting_citations":[],"review_version":1}