Uniform subconvexity bounds for $GL(2)\times GL(2)$ $L$-functions in the spectral aspect

Zhao Xu

REVIEW 4 major objections 5 minor 53 references

Uniform subconvexity bounds for $GL(2)\times GL(2)$ $L$-functions in the spectral aspect

T0 review · 4 major / 5 minor · reviewed 2026-08-05 · deepseek-v4-flash

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Pith's one-line read The paper proves a uniform spectral-aspect subconvexity bound of Burgess type, exponent 3/4, for GL(2)×GL(2) Rankin–Selberg L-functions when one form is dihedral or has level 1, and derives a quantum-ergodicity shrinking result for dihedral

desk verdict Genuinely new uniform second-moment theorem with a long real proof, but the advertised Burgess bounds rest on an unproved level extension and an announced L4 bound; send it to a serious referee rather than accepting it as is. read the letter →

arxiv 2509.05968 v1 pith:DCHKTWYR submitted 2025-09-07 math.NT

Zhao Xu This is my paper · ORCID

classification math.NT MSC 11F1211F6611F72

keywords Rankin–SelbergL-functionsspectralaspectsubconvexityMaassnewformsdihedralformssecondmomentquantumuniqueergodicityL4-normestimates

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

The paper aims to prove that the central value of a Rankin–Selberg L-function attached to two Maass forms can be bounded by a power that is uniform in both spectral parameters, not just one. Its main theorem is a second-moment estimate over a spectral window: the average size of |L(1/2,f×g)|^2 is controlled by the L4-norm of g plus an error in the window width. Combining that estimate with known L4-norm results yields a Burgess-type exponent 3/4 when one form is dihedral or has level 1, where prior bounds were either non-uniform in the two parameters or weaker. An application shrinks the known equidistribution domain for dihedral Maass forms, proving mass equidistribution on hyperbolic balls whose radius decays like a small power of the spectral parameter.

What carries the argument

The workhorse is a spectral second-moment identity. After the Kuznetsov trace formula and Voronoi summation, the problem reduces to a shifted convolution sum D_g(s,1,1;r)—a Dirichlet series over pairs (m,n) with n−m=r weighted by Fourier coefficients of g—whose spectral decomposition expresses it as sums over Maass forms φ of triple products ⟨|g|^2,φ⟩. The Watson–Ichino formula converts those triple products into central values of triple-product L-functions, and the L4-norm of g enters through the diagonal. A hypergeometric function F arising from the Bessel integral is handled by a Pochhammer-series expansion, and the uniform asymptotic expansion of J_{2it} controls the oscillatory phase. T

What would settle it

Run the claimed second-moment estimate in the regime t_g ≪ T^{1+ε}/H for a level-q family with q>1 and compare with a direct computation of the mixed moment Σ L(1/2,φ)L(1/2,φ×ad g); if that mixed moment exceeds the stated bound at some squarefree q, the proof of Theorem 1.1 fails. Separately, compute the L4-norm of a level-one Maass form at large t_g; if ||g||_4 is not t_g^ε, the Burgess bound for level-one inputs lacks its required input.

Watch

Extended reading notes

Core claim

The central claim is Theorem 1.1: for squarefree levels and a real primitive nebentypus, the second moment of L(1/2,f×g) over f in a spectral window T−H≤t_f≤T+H is bounded by O(T^{1+ε}(T+t_g)|T−t_g|^{1/2}||g||_4^2 + T^{1+ε}H), under conditions that keep H smaller than |T−t_g| and T+t_g ≪ (TH)^{3/4−ε}. In the special case t_g ≪ T^{1+ε}/H, the bound improves to O(T^{3/2+ε}+T^{1+ε}H). Feeding in known L4-norm bounds for dihedral forms, and an announced L4 bound for level-one forms, the paper derives uniform Burgess-type subconvexity bounds |L(1/2,f×g)| ≪ |t_f+t_g|^{3/4+ε}. It also proves that for dihedral Maass newforms, quantum unique ergodicity holds on hyperbolic balls of radius t_g^{−δ} for

Load-bearing premise

The load-bearing premise is that a level-one mixed moment estimate extends to every squarefree level with no loss, together with an announced L4-norm bound for level-one Maass forms; the paper states the former is 'clear' without proof and cites the latter as announced.

Editorial extensions

If this is right

Uniform Burgess-type subconvexity bounds for dihedral Maass forms: |L(1/2,f×g)| ≪ |t_f+t_g|^{3/4+ε}.
The same 3/4 exponent holds when one of f,g has level 1, conditional on the announced L4 bound.
The second-moment theorem yields a shrinking QUE result for dihedral Maass forms on hyperbolic balls of radius t_g^{−δ} for any δ<1/12.
The method converts any future improvement of ||g||_4 bounds directly into improved spectral-aspect subconvexity exponents.

Reading between the lines

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

If the asserted level extension of the mixed-moment bound is proved carefully, the main theorem becomes unconditional except for the L4 input; this is the first spot a referee would send the author back to.
The same second-moment framework may apply to holomorphic forms or to higher-rank Rankin–Selberg convolutions, replacing the L4-norm of g by an appropriate L^p norm.
A proof of the announced level-1 L4 bound would make the level-1 corollary unconditional and would likely improve the QUE shrinking exponent beyond 1/12.
The balance between H and L in Theorem 1.1 suggests a wider hybrid regime where the subconvexity exponent should interpolate between 3/4 and the Weyl exponent 2/3; the paper does not explore this.

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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 headline take: this is a real paper with a genuinely new uniform second-moment theorem, but the advertised Burgess subconvexity bounds rest on an unproved level extension and an announced L4 bound. Send it to a good referee; do not accept as is.

What is actually new: Theorem 1.1 is not a repackaging of Lau-Liu-Ye or Jutila-Motohashi. It handles both spectral parameters uniformly and puts an L4 norm of g on the right-hand side. The proof is a long, serious calculation. The treatment of the J-Bessel and hypergeometric terms in Sections 5.1-5.3 contains real work, and the author is transparent about what is borrowed and what is new. If the corollaries hold, the Burgess-type bound uniform in both spectral parameters and the shrinking QUE result with delta < 1/12 are substantial.

The soft spots are exactly where the reader and the stress-test point. Lemma 5.1 is [18, Prop 6.1] for q=1, and the footnote says it is clear the tools work for general squarefree level. That is not a proof. Lemma 5.1 is not decorative: it produces the C2 bound in the t_g << T^{1+eps}/H regime, and that bound feeds into the second assertion of Theorem 1.1 and, through (1.1), into the corollaries. If the level extension fails, the advertised results lose their support. This is a load-bearing gap, not an aesthetic one. The stress-test gets this right.

Two more gaps, in decreasing order of severity. Corollary 1.2 is promised but the proof section only sketches Corollary 1.3, and that sketch uses Ki's announced L4-norm bound [28] as if it were available. The extension of Watson-Ichino to non-primitive characters is also asserted. None of these are internal contradictions, and each may be routine, but together they make the paper conditional.

I believe the main moment calculation is not circular: the target subconvexity bound never appears as an input, and the L4 bounds and large sieve estimates are external. The paper is honest about its dependencies.

Who this is for: analytic number theorists working on subconvexity and moments. It deserves a serious referee, and I would want to see it in a refereed venue after revision. My recommendation: invite a revision with the level extension of Lemma 5.1 proved, a real proof of Corollary 1.2, and a status check on Ki's bound or a replacement.

Referee Report

4 major / 5 minor

Summary. The paper studies a second moment of GL(2)xGL(2) Rankin--Selberg L-functions L(1/2, f x g) in the spectral aspect, with a uniform treatment of both spectral parameters t_f and t_g. The main technical result, Theorem 1.1, bounds a spectral average of |L(1/2, f x g)|^2 over f in a short interval around T, in terms of the L4-norm of g and two main terms. The proof follows the Sarnak--Lau--Liu--Ye approach: Kuznetsov trace formula, Voronoi summation, uniform asymptotics of J-Bessel functions, a shifted convolution sum, spectral decomposition, and large sieve estimates. The advertised consequences are Burgess-type subconvexity bounds L(1/2, f x g) << |t_f+t_g|^{3/4+epsilon} when one of the forms is dihedral (Corollary 1.2) or when one form has level 1 (Corollary 1.3), together with a shrinking-set QUE result for dihedral Maass forms (Theorem 1.5).

Significance. If correct, the paper would give a genuinely uniform spectral-aspect subconvexity bound of Burgess strength for GL(2)xGL(2), a major open-type problem that has seen substantial recent progress (Nelson, Blomer--Jana--Nelson, etc.). The method is a natural and ambitious extension of Sarnak's and Lau--Liu--Ye's work, and the use of L4-norm bounds to control triple products is a plausible and interesting idea. The paper also gives a detailed treatment of the Bessel and hypergeometric phases, which is a substantial technical contribution. The promised shrinking-set QUE result for dihedral forms would also be new. However, the announced results rely on several external or only footnoted statements: the level-q extension of [18, Prop. 6.1], Ki's announced L4 bound, and an extension of the Watson--Ichino formula to non-primitive nebentypus. The proof of Corollary 1.2 is not supplied at all. These are load-bearing gaps rather than cosmetic issues.

major comments (4)

[§5.4.1, Lemma 5.1 (footnote on p. 29-30)] Lemma 5.1 states a second-moment bound for L(1/2, phi x ad g) and L(1/2, phi) at level q_1, quoting [18, Proposition 6.1]. The original proposition is stated for q=1, and the paper only says, in a footnote, that 'the tools there ... work well in our case'. This extension is not proved. The bound is used in (5.42)-(5.43) to estimate C_2, leading to the new term N T^{3/2} in (1.2), and Theorem 1.1-(1.2) is then used in Corollaries 1.2 and 1.3. Since the level-q spectral large sieve and Li's result [34, Theorem 2] may require additional arguments at squarefree level with nebentypus, this is a nontrivial gap in a load-bearing step. A complete proof or a precise reduction to [18] with all level dependencies tracked is needed.
[§7, after Corollary 1.2] Corollary 1.2 is stated as the advertised dihedral Burgess-type bound, but no proof is given. Section 7 says 'We only state a sketch proof of Corollary 1.3' and then proves (1.7) and (1.8); Corollary 1.2 is not mentioned again. The paper needs a real proof of Corollary 1.2, or an explicit statement that it follows from the proof of Corollary 1.3 by symmetry and the dihedral L4-norm results of Luo and Humphries--Khan, with the necessary conditions on D, q, chi, and psi checked. As written, one of the two central claims in the abstract is unproved.
[§7 and §1 (Corollary 1.3, Ki's bound (1.5), Watson--Ichino extension)] The level-1 Burgess bound (1.8) in Corollary 1.3 uses Ki's announced L4 bound ||g||_4 << t_g^epsilon, which is not proved in this manuscript. Additionally, the paragraph before Corollary 1.3 states that the assumption that psi is primitive can be dropped and that the Watson--Ichino formula holds in a general setting with level-dependence left implicit, but no precise theorem or proof is given. This matters because Theorem 1.1 is stated for primitive real psi, while Corollary 1.3 concerns trivial nebentypus of arbitrary squarefree level. The application of (1.1) to that setting requires the non-primitive version of the Watson--Ichino formula, and this extension is load-bearing for the corollary. Please supply the missing statement and proof, or state the corollary conditionally on these external results.
[§5.4.1, equations (5.42)-(5.45)] In the case t_g << T^{1+epsilon}/H, the estimate (5.42)-(5.43) for C_2 is the key point that gives the unconditional T^{3/2} bound. The passage from (5.42) to (5.43) uses Lemma 5.1 in the form 'V_0^3(t_g^2+V_0^2)'. Since V_0 can be as large as T/H, the final N T^{3/2} is obtained only for t_g << T/H. This is consistent with the statement, but the proof of (5.42) also uses the upper bound (5.33) for the triple product, whose uniformity in q_1 and t_phi is only sketched. Please verify that the implied constants are uniform in all parameters when Lemma 5.1 is applied at level q_1.

minor comments (5)

[§1, p. 2] Typo: 'spectral parameterst f' needs a space. Also 'thd functional equation' in §2.1 and 'a a' in §2.3 should be corrected.
[§2.3, Lemma 2.5] The notation L_{q_2}(1, sym^2 phi) and L_{q_2}(1/2, phi) is introduced only in (2.5) and is used without being explicitly recalled in Lemma 2.5. A short reminder would improve readability.
[§4, p. 15] The sentence 'we may enlarge B*(q, chi) to B([q,D], chi)' should use the spectral basis notation consistently; as written it is confusing because B*(D, chi) was defined for newforms.
[§7, proof of Corollary 1.3] In the proof of (1.8), the choices H = T_0^{4/3+epsilon} T^{-1} and H = T^{1/3+epsilon} should be checked explicitly against the hypotheses of Theorem 1.1, in particular H <= T^{1-epsilon} and T_0 << (T H)^{3/4-epsilon}. The argument is plausible but the verification is omitted.
[§8, Theorem 1.5] The proof is only a sketch, and the role of the dihedral assumption in the equality L(1/2, phi x ad g_xi) = L(1/2, phi x psi) L(1/2, phi x g_xi^2) should be stated more explicitly. Also, the reference to Jutila--Motohashi for a subconvexity bound is marked with a footnote that itself asserts a level extension; this should be either proved or made conditional.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: the moment proof does not use the target subconvexity bound as an input; unproved level extensions are rigor gaps, not self-reference.

full rationale

The paper's derivation chain does not reduce to its own conclusion. Theorem 1.1 is a spectral second-moment estimate over f of |L(1/2, f x g)|^2, obtained through Kuznetsov/Voronoi summation, stationary phase, and spectral decomposition of shifted convolution sums. The advertised Burgess bounds (Corollaries 1.2 and 1.3) are deduced from Theorem 1.1 by combining it with external L^4-norm results: the dihedral case uses Humphries-Khan's asymptotic (1.3), and the level-1 case uses Ki's announced bound (1.5) and Humphries-Khan's (1.6). Nowhere is the bound L(1/2, f x g) << |t_f + t_g|^{3/4+epsilon} inserted back into the moment computation; no displayed equation identifies the target with an input. The Watson-Ichino formula (Lemma 2.6) and the mixed second-moment bound (Lemma 5.1, quoted from [18, Prop. 6.1]) are external results about L(1/2, phi)L(1/2, phi x ad g), not about the original f x g family. No parameter is fitted and then renamed a prediction. The paper contains no self-citations by the author: [18], [19], and [28] are by other authors, and the L^4 bounds they provide are independent of the paper's subconvexity claim. The genuinely weak points are all correctness/rigor gaps, not circularity. (i) Lemma 5.1 is stated in [18] for q=1, and the footnote 'It is clear to see that the tools there ... work well in our case' supplies no proof of the squarefree-level extension; the C2 bound in Section 5.4.1 and hence Theorem 1.1(1.2) depend on this extension. (ii) The paper drops primitivity from the Watson-Ichino formula with the note 'this formula in fact holds in general setting (with level-dependence left implicit)', again without proof. (iii) Corollary 1.3 relies on Ki's announced L^4 norm bound (1.5). These are substantial rigor concerns that should be checked, but none is a case of the conclusion being assumed or of a fitted input being relabeled as a prediction. Therefore the circularity score is 0.

Assumptions & free parameters 0 free parameters · 7 assumptions · 0 invented entities

The paper relies on a standard toolbox of trace formulas, Bessel asymptotics, the large sieve, and external L4-norm bounds for the fixed form g. No free parameters are fitted to data, and no new mathematical entities are introduced. The main non-standard input is the asserted level extension of [18, Proposition 6.1].

assumptions (7)

standard math Kuznetsov trace formula with nebentypus (Lemma 2.7)
Used to open the second moment after enlarging the spectral basis; quoted from Michel and Blomer.
standard math Voronoi summation formula for Maass newforms (Lemma 2.8)
Used to transform the shifted sum over n; quoted from Kowalski-Michel-VanderKam.
domain assumption Watson-Ichino formula in the explicit form of Humphries-Khan (Lemma 2.6)
Relates the triple product <|g|^2, phi> to central values of L(1/2, phi) and L(1/2, phi x ad g); assumed valid for squarefree q and primitive real nebentypus.
standard math Spectral large sieve of Deshouillers-Iwaniec (Lemma 2.1)
Used to bound the spectral sums C1 and J1; quoted with extensions to general nebentypus from Drappeau and Zacharias.
domain assumption External L4-norm bounds for g: (1.3), (1.5), (1.6)
These bounds convert the moment theorem into subconvexity; (1.5) is only announced by Ki and is not proved in this paper.
ad hoc to paper Extension of [18, Proposition 6.1] from q = 1 to general squarefree q
Asserted in a footnote in Section 5.4.1 with 'It is clear to see'; needed to bound C2 and hence Theorem 1.1 for general level.
domain assumption Assumptions that q and D are squarefree and chi, psi are real primitive, with t_g real and large
These are the stated scope conditions of Theorem 1.1; the primitivity and reality of psi are used at equation (5.11) and in the Watson-Ichino application.

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Pith. "Pith review of Uniform subconvexity bounds for $GL(2)\times GL(2)$ $L$-functions in the spectral aspect." pith.science (2026). https://pith.science/paper/DCHKTWYR

@misc{pith2026250905968,
  author       = {Pith},
  title        = {Pith review of: Uniform subconvexity bounds for $GL(2)\times GL(2)$ $L$-functions in the spectral aspect},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/DCHKTWYR}},
  note         = {Machine review of arXiv:2509.05968}
}

abstract

In this paper, we study the second moment for $GL(2)\times GL(2)$ $L$-functions $L(\frac{1}{2},f\times g)$, which leads to a uniform subconvexity bound in the spectral aspect. In particular, if either $f$ or $g$ is a dihedral Maass newform, or if one of them has level $1$, we obtain a Burgess-type bound that is uniform in both $t_f$ and $t_g$, where $t_f$, $t_g$ denote the spectral parameters of $f$, $g$. As an application, we also establish a shrinking result for QUE in the case of dihedral Maass newforms.

Discussion (0). Continue with ORCID to comment.

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