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Beyond endoscopy for $\mathsf{GL}_2$ over $\mathbb{Q}$ with ramification 1: Poisson summation

T0 review · 3 major / 4 minor · reviewed 2026-08-07 · deepseek-v4-flash

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Pith's one-line read 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.

desk verdict 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. read the letter →

arxiv 2505.18967 v3 pith:WDPPC74S submitted 2025-05-25 math.NT math.RT

Yuhao Cheng This is my paper · ORCID
classification math.NTmath.RT MSC 11F7211F70
keywords beyondendoscopytraceformulaGL_2PoissonsummationorbitalintegralsZagierL-functionKloostermansumsEisensteinseries
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 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.

What carries the argument

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.

What would settle it

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.

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Extended reading notes

Core claim

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.

Load-bearing premise

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.

Editorial extensions

If this is right

  • 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.

Reading between the lines

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

  • 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.
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Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

3 major / 4 minor

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.

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 (3)
  1. [Section 6, proof of Theorem 6.1, around the claim "Since eF is odd, the third and the last term in the bracket cancel."] 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.
  2. [Theorem 4.1 and Theorem 8.10, definition of Σ(□)] 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.
  3. [Appendix B, Proposition B.6 and its proof] 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.
minor comments (4)
  1. [Section 1.2, comparison table] 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.
  2. [Section 6, Theorem 6.1 statement] 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.
  3. [Section 4, equation (4.1)] 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.
  4. [Appendix B, proof of Proposition B.6] 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.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: Theorem 8.10 is derived from independent trace-formula, Poisson-summation, and residue computations, with only external citations used.

full rationale

The main identity Theorem 8.10 is not an input: it is assembled from Theorem 4.1 (Poisson summation on Q_S), Theorem 6.1 (contour shift and residue evaluation of the xi = 0 term), Theorem 7.7, and Theorem 8.8. The '1-dimensional term' (6.3) is first produced by a residue at s = 0 in the Dirichlet-series/contour calculation and only later identified with the sum over mu of Tr(mu(f^n)) via Weyl integration and Fourier inversion on the finite character group; this identification is an equality proved in Section 7, not the definition of the term. Similarly, the 'Eisenstein term' (6.4) is obtained from the exceptional residue calculation in Theorem 6.1, and the equality with one half of the sum of Tr((xi0 tensor mu)(f^n)) is then proved independently in Section 8. Naming a residue contribution '1-dimensional' or 'Eisenstein' before evaluating its trace is not circular. The external results imported are from Altug, Shelstad, Tate, Clozel, and standard references; these are not the present author's own results, and the ramified theorem is not contained in them. No uniqueness theorem is imported, and no ansatz is smuggled in through self-citation. The paper explicitly flags the delicate point in Theorem 4.1: Psi is 'not a Schwartz function', and the Poisson step is deferred to Proposition B.6. A possible gap in Lemma B.8's reduction from Psi to the model function Omega would be a correctness defect in a technical lemma, not a circular one; the theorem would be unsupported rather than equivalent to its own inputs. No fitted parameter is renamed as a prediction.

Assumptions & free parameters 0 free parameters · 6 assumptions · 0 invented entities

No invariant parameter was fitted to an expected answer. The arbitrary test functions f_{q_i} and f_infinity are inputs to the trace formula, not fitted constants. The only auxiliary parameters are the real number theta in ]0,1[ and the contour parameter v, both harmless and not data-dependent. All non-derived inputs are standard external results, mostly from Altuğ, Shalika, Shelstad, Tate and Bourbaki, none authored by the present paper. Therefore the circularity burden is minimal.

assumptions (6)
  • domain assumption The elliptic part of the Arthur-Selberg trace formula for GL2 over Q equals the sum over elliptic conjugacy classes of vol(gamma) times the product of local orbital integrals.
    Used as equation (2.1) in Section 2.1; convergence and normalization for the chosen test functions are taken from Altuğ's framework and the trace formula literature rather than reproved.
  • standard math Shalika germ expansion: for any compactly supported f on GL2(Qp), the orbital integral near the identity has the form Gamma_I f(I) + Gamma_U orb(f;U).
    Theorem 2.11 and Corollary 2.12 rest on [Sha72] and [KT23]; the paper computes the germs explicitly for GL2 but relies on the existence theorem.
  • standard math Archimedean germ regularity: orb(f_infinity;gamma) = g1(gamma) + |gamma1 gamma2|^{1/2} |gamma1 - gamma2|^{-1} g2(gamma) locally on maximal tori.
    Theorem 2.13 is cited to [She79] and [Lan80, Chapter 6]; it is used to parametrize the archimedean orbital integral by trace and determinant.
  • standard math The classical Zagier L-function L(s,delta) and the partial L-functions L^S(s,delta) admit analytic continuation and satisfy the functional equations quoted in Theorems 3.2 and 3.3.
    The approximate functional equation in Theorem 3.8 assumes these continuation properties; the paper points to references used in the proof of [Alt15, Proposition 3.1].
  • standard math Pontryagin duality and Fourier inversion hold on the semilocal group Q_S/Z_S with the specific measure normalization stated in Appendix B.
    Propositions B.1, B.2 and Theorem B.3 are proved from standard harmonic analysis on locally compact abelian groups, following Bourbaki and Tate.
  • standard math Global class field theory identifies the dual of Q^times R_{>0} prod_{p notin S} Z_p^times \ A^times with the finite group (Z_{S,fin}^times)^wedge.
    Used in Proposition 7.6 to convert sums over automorphic characters into Fourier inversion over a finite character group; finiteness is established in Corollary 7.5.

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Pith. "Pith review of Beyond endoscopy for $\mathsf{GL}_2$ over $\mathbb{Q}$ with ramification 1: Poisson summation." pith.science (2026). https://pith.science/paper/WDPPC74S

@misc{pith2026250518967,
  author       = {Pith},
  title        = {Pith review of: Beyond endoscopy for $\mathsfGL_2$ over $\mathbbQ$ with ramification 1: Poisson summation},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/WDPPC74S}},
  note         = {Machine review of arXiv:2505.18967}
}
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. The first step involves isolating specific representations, especially the residual part of the spectral side, in the elliptic part of the geometric side of the trace formula. We generalize this step to the case with ramification at $S=\{\infty,q_1,\dots,q_r\}$ with $2\in S$, thereby fully resolving the problem of isolating these representations over $\mathbb{Q}$ which remained unresolved for over a decade. Such a formula that isolates the specific representations is derived by modifying Altu\u{g}'s approach. We use the approximate functional equation to ensure the validity of the Poisson summation formula. Then, we compute the residues of specific functions to isolate the desired representations.

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Forward citations

Cited by 1 Pith paper

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

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

    math.NT 2026-07 conditional novelty 7.0 of 10

    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.

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Works this paper leans on

6 extracted references · 5 canonical work pages · cited by 1 Pith paper

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