REVIEW 3 major objections 4 minor 5 references
Relative Trace Formula And Simultaneous Nonvanishing for GL_3 x GL_2 and GL_3 x GL_1 L-functions
T0 review · 3 major / 4 minor · reviewed 2026-08-12 · deepseek-v4-flash
Pith's one-line read For any fixed Hecke character and GL(2) cusp form with coprime conductors, infinitely many GL(3) cusp forms have both central L-values nonzero.
desk verdict A genuinely new nonvanishing theorem for GL(3), with a real but fixable gap in the spectral-side expansion that a referee should flag. read the letter →
The pith
A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.
The reading
What carries the argument
The central object is the relative trace formula identity specialized to the two period integrals attached to the $\mathrm{GL}(1)$ and $\mathrm{GL}(2)$ subgroups, which by Rankin-Selberg theory represent the two central L-values. The test function is assembled place by place: a supercuspidal matrix coefficient at a fixed split place kills the continuous spectrum, Iwahori-level components at a large prime $N$ set the level, and the ramified components match the conductors of $\chi$ and $\sigma$. The Bruhat decomposition of $\mathrm{GL}(3)$ splits the geometric side into a small-cell term and two regular orbital-integral terms; the proofs of Sections 5 and 6 show the regular terms vanish for large level, while the small-cell term factors into local integrals that grow like $N_F(N)^3$. This factorization is what converts a lower bound on a single geometric integral into a lower bound on the spectral average of L-values.
What would settle it
A concrete check would be to fix a coprime pair $(\chi,\sigma)$, implement the paper's test function, and compute the level-$N$ spectral average in (7.2); the proof predicts growth of order $N_F(N)^3$, so observing growth like $N_F(N)^{3-\varepsilon}$ rather than $N_F(N)^3$ would break the main-term estimate.
Extended reading notes
Core claim
The paper establishes Theorem A: given a number field $F$, a unitary Hecke character $\chi$ of $F^\times\backslash\mathbb{A}_F^\times$, and a unitary cuspidal automorphic representation $\sigma$ of $\mathrm{GL}(2)/F$ with relatively prime conductors, infinitely many unitary cuspidal automorphic representations $\pi$ of $\mathrm{PGL}(3)/F$ satisfy $L(1/2,\pi\times\chi)L(1/2,\pi\times\sigma)\neq 0$. The proof constructs a global test function $f$ with a fixed supercuspidal component at a split place $v_0$, an Iwahori-level component at a large prime ideal $N$, and components at the ramified places of $\chi$ and $\sigma$. In the relative trace formula, the geometric side decomposes by Bruhat cells into the small-cell term $J_{\mathrm{Small}}$ and two regular orbital integrals $J_{\mathrm{Reg}}^{\mathrm{I}}$ and $J_{\mathrm{Reg}}^{\mathrm{II}}$; the latter two vanish for large $N_F(N)$, while $J_{\mathrm{Small}}(f,\varphi,(0,0))$ is bounded below by a positive constant times $N_F(N)^3$. The spectral side therefore has an average of products of the two period integrals, hence of the two central L-values, of size at least $N_F(N)^3$, which forces infinitely many individual products to be nonzero.
Load-bearing premise
The argument assumes that the kernel's spectral expansion can be restricted to an orthonormal basis of forms fixed by the chosen level subgroup, with all other forms contributing nothing; if that restricted expansion is not exact, the geometric main term no longer equals the L-value average and the conclusion would not follow.
Editorial extensions
If this is right
- For any fixed coprime pair $(\chi,\sigma)$, the average of $L(1/2,\pi\times\chi)L(1/2,\pi\times\sigma)$ over the level-$N$ $\mathrm{GL}(3)$ cuspidal spectrum is at least proportional to $N_F(N)^3$, so the product cannot vanish for all but finitely many $\pi$.
- The type I and type II regular orbital integrals vanish identically for large level, so the small-cell term alone carries the arithmetic information in the relative trace formula.
- Every nonvanishing $\pi$ produced has a fixed supercuspidal component at $v_0$, an Iwahori-invariant vector at the large prime $N$, and is unramified at all other finite places, so the result holds within a sparse, explicitly described family.
- Replacing $\sigma$ by its contragredient and invoking the functional equation converts the spectral identity into the exact form of Theorem A, with $L(1/2,\pi\times\sigma)$ rather than $L(1/2,\tilde{\pi}\times\sigma)$.
Reading between the lines
- Beyond the paper: the same Bruhat-cell split suggests an analogous simultaneous-nonvanishing statement for $\mathrm{GL}(n+1)\times\mathrm{GL}(1)$ and $\mathrm{GL}(n+1)\times\mathrm{GL}(n)$, with the geometric side expected to be harder as the rank grows.
- Beyond the paper: because the argument does not use positivity of the individual periods, a quantitative strengthening would require extra bounds on individual period integrals rather than following from the averaged identity alone.
- Beyond the paper: the fixed supercuspidal component at $v_0$ is a simplifying device, so a natural test is whether the result persists when the level condition is imposed without that component, at the cost of handling the continuous spectrum.
Signed reviews
Editorial analysis
A structured set of objections, weighed in public.
Referee Report
Summary. The paper proves Theorem A: for a unitary Hecke character χ of GL(1) and a unitary cuspidal automorphic representation σ of GL(2)/F, there are infinitely many unitary cuspidal automorphic representations π of PGL(3)/F such that the central values L(1/2, π×χ) and L(1/2, π×σ) are simultaneously nonzero. The proof uses Jacquet's relative trace formula with a test function that is a supercuspidal matrix coefficient at a fixed finite split place v0, an Iwahori-level factor at a large prime ideal N, and local factors matching the ramification of χ and σ. The geometric side is split into a small-cell main term, which is shown to grow like N_F(N)^3, and two regular orbital integrals, which are shown to vanish for sufficiently large N_F(N). Comparing with the spectral side yields a lower bound for a sum of products of period integrals, from which the existence of infinitely many π is inferred.
Significance. If the proof is correct, the result is a valuable and relatively clean application of the relative trace formula to simultaneous nonvanishing of central L-values in two families attached to the same GL(3) representation. The test function is explicit and has no free parameters, and the main term is obtained as a product of local factors rather than by fitting constants. The paper also states its scope carefully, noting where positivity is lacking and where the method would become harder. However, the current draft contains a load-bearing gap in the spectral expansion and one essential estimate is imported from the authors' earlier work rather than proved here; these issues need to be fixed before the argument is complete.
major comments (3)
- [§3, Eqs. (3.1), (3.9), (3.10)] The spectral expansion (3.9) is restricted to an orthonormal basis B^{I(N)}_π of I(N)-invariant vectors, where I(N) is defined in (3.1) to contain GL3(O_{v0}) at the supercuspidal place v0. Since π_{v0} is supercuspidal, every cuspidal π occurring in the spectral side has V^{GL3(O_{v0})}=0, so B^{I(N)}_π is empty. The displayed identity (3.9) would then make the spectral side (3.10) identically zero, which contradicts the lower bound JSmall(f,φ,0) ≫ N_F(N)^3 from Proposition 4.1 via (3.12). The sentence preceding (3.9) explains why only the cuspidal spectrum appears, but it does not justify restricting to I(N)-invariant vectors: such a restriction requires the test function to be bi-I(N)-invariant, whereas the local factor f_{v0} is a matrix coefficient that is only bi-invariant under the smaller group K'^◦_{v0}[1] appearing in Lemma 3.1. Please replace I(N) by the correct compact open subgroup at v0 and prove that the non-invariant vectors do not contribute to the periods, or otherwise justify (3.9).
- [§7, Eq. (7.3)] The bound (7.3), asserting that the sum over the K_fin-invariant basis is O(1), is stated with the justification 'similarly to the calculation in [MRY23, §13]' and is not proved in this paper. This estimate is load-bearing: it is exactly what forces the I(N)-level contribution to be the only part that can grow like N_F(N)^3. Since [MRY23] treats the different group U(2,1)×U(1,1), the transfer of that calculation to the present GL(3)×GL(2) setting is not automatic. The authors should either include the necessary calculation or state the precise external theorem and verify its hypotheses, especially after the subgroup in (3.9) is corrected.
- [§7, final deduction] The step from the lower bound (7.2) to the existence of a single π with L(1/2,π×χ)L(1/2,π×σ)≠0 is too terse. One needs to justify that the number of terms in the sum is O(N_F(N)^3) (or otherwise dominated by the growth rate) so that a lower bound on the absolute value of the sum implies a lower bound on some individual term, and one needs to spell out that a nonzero value of the projected period P1(π(f)φ,χ) implies the corresponding Rankin-Selberg L-value is nonzero. These points are standard in the RTF literature, but they are part of the logical chain and should be stated explicitly.
minor comments (4)
- [§4, end of proof of Proposition 4.1] The sentence 'Therefore, (4.2) follows from (4.2), (4.3), (4.4) and (4.6)' should refer to (4.1) as the conclusion, not (4.2).
- [§3, notation] The compact open subgroup I(N) is defined twice: once in (3.1) as I_N[1]∏_{v<∞, v∤N} K_v and once in (3.9) as ∏_{v<∞} K_v[e_v(N)]. These agree only if e_v(N) is interpreted as the valuation of N, but the notation should be unified and the equality made explicit.
- [§2, Remark 2.2] There is a wording error: 'One may then asked for how such π satisfy (2.1)' should be 'One may then ask how many such π satisfy (2.1)' or similar.
- [§3, Lemma 3.1 and (3.2)] The embedding of K'^◦_{v0}[1] into GL3(F_{v0}) via g ↦ diag(g,1) should be stated explicitly, since the invariance properties of W_{v0} and f_{v0} with respect to this embedded subgroup are used implicitly in the proof of (3.3).
Circularity Check
No significant circularity: the relative trace formula derivation is self-contained; the only self-citation is a technical bound (7.3) that is not load-bearing, and the spectral-expansion gap at (3.9) is a correctness issue, not a circular reduction.
full rationale
The derivation chain is not circular. The spectral identity (3.10) and the geometric decomposition (3.12) are a genuine relative trace formula: the spectral side is expressed through Rankin-Selberg period integrals, whose relation to L(1/2, pi x chi) and L(1/2, pi x sigma) is supplied by the independent integral representations of Jacquet-Piatetski-Shapiro-Shalika, while the geometric side is analyzed directly in Proposition 4.1 and Corollaries 5.3 and 6.3. No parameter is fitted to the target L-values, and the nonzero lower bound JSmall(f,phi,0) >> N_F(N)^3 is computed from local orbital integrals rather than assumed. The only citation to the authors' own prior work is the estimate (7.3), attributed to [MRY23, Section 13]; this is a technical bound used to separate the I(N)-level contribution from the K_fin-invariant contribution, and the theorem's conclusion is not defined in terms of it. In fact, because the test function at v0 is a matrix coefficient of a fixed supercuspidal representation, the K_fin-invariant sum in (7.3) is empty for the spectrum selected by f_v0, so the cited bound is not load-bearing for the existence claim. Separately, there is a serious unproved step at (3.9): the spectral expansion is written over an orthonormal basis of I(N)-invariant vectors even though I(N) contains GL3(O_v0) and the supercuspidal component pi_v0 has no such fixed vectors; if this is not repaired, the equality (3.10) is broken. That is a technical gap in the written proof, but it is not a circularity, because the problematic restriction is not derived from the conclusion and does not make the claimed L-value nonvanishing a restatement of an input.
Assumptions & free parameters
assumptions (3)
- domain assumption The spectral expansion of the RTF kernel can be written as a sum over an orthonormal basis of I(N)-invariant vectors only, with non-invariant vectors contributing zero to the period integrals.
- domain assumption The matrix coefficient of a supercuspidal representation at v0 kills the non-cuspidal spectrum, so only cuspidal representations appear.
- standard math Rankin-Selberg integrals of JPSS represent central L-values with the chosen test vectors, including local factors at ramified places.
Cite this review
Pith. "Pith review of Relative Trace Formula And Simultaneous Nonvanishing for GL_3 x GL_2 and GL_3 x GL_1 L-functions." pith.science (2026). https://pith.science/paper/RCTDGZQN
@misc{pith2026241112609,
author = {Pith},
title = {Pith review of: Relative Trace Formula And Simultaneous Nonvanishing for GL_3 x GL_2 and GL_3 x GL_1 L-functions},
year = {2026},
howpublished = {\url{https://pith.science/paper/RCTDGZQN}},
note = {Machine review of arXiv:2411.12609}
}
abstract
Fix a Dirichlet character $\chi$ and a cuspidal GL$(2)$ eigenform $\phi$ with relatively prime conductors. Then we show that there are infinitely many cusp forms $\pi$ on GL$(3)$ such that $L(1/2, \pi \times \chi)$ and $L(1/2, \pi \times \phi)$ are simultaneously non-zero. We achieve this by use of Jacquet's Relative Trace Formula. We derive an expression of the average over the GL$(3)$ cuspidal spectrum as a sum of a non-zero main term and two subsidiary terms which are forced to be zero for large enough level by use of a suitable test function. This modest article is dedicated to the memory of Harish Chandra, on the occasion of his hundredth birthday.
Reference graph
Works this paper leans on
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Herv\'e Jacquet, Ilja Iosifovitch Piatetski-Shapiro, and Joseph Shalika. Automorphic forms on GL (3) . I . Ann. of Math. (2) , 109(1):169--212, 1979
work page 1979
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[2]
Jacquet, I
H. Jacquet, I. I. Piatetskii-Shapiro, and J. A. Shalika. Rankin- S elberg convolutions. Amer. J. Math. , 105(2):367--464, 1983
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[3]
Philippe Michel, Dinakar Ramakrishnan, and Liyang Yang. Bessel periods on U (2, 1) U(1,1) , relative trace formula and non-vanishing of central L -values. arXiv preprint arXiv:2309.08490 , 2023
work page Pith review arXiv 2023
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[4]
Average values of modular L -series via the relative trace formula
Dinakar Ramakrishnan and Jonathan Rogawski. Average values of modular L -series via the relative trace formula. Pure Appl. Math. Q. , 1(4, Special Issue: In memory of Armand Borel. Part 3):701--735, 2005
work page 2005
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[5]
Liyang Yang. Relative trace formula, subconvexity and quantitative nonvanishing of R ankin- S elberg L -functions for GL(n+1) GL(n) . arXiv preprint arXiv:2309.07534 , 2023
work page Pith review arXiv 2023
Reviewed August 12, 2026 · model on record in the stance chip above.
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