Pith. sign in

REVIEW 3 major objections 3 minor 29 references

Level aspect subconvexity for $\textrm{GL(2)}\times \textrm{GL(2)}$ $\textrm{L}$-functions

T0 review · 3 major / 3 minor · reviewed 2026-08-11 · deepseek-v4-flash

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗ Review JSON

Pith's one-line read For each newform $f$ of prime level $p$, the paper establishes $L(1/2, f\otimes g)\ll p^{1/2-1/524+\varepsilon}$ for every fixed level-one eigenform $g$.

desk verdict Sharp new method and exponent for GL2×GL2 level-aspect subconvexity, but the amplifier lower bound (4.3) is algebraically wrong for non-quadratic characters, so the stated uniformity is not yet proven. read the letter →

arxiv 2412.12410 v1 pith:ZKR3KWFF submitted 2024-12-16 math.NT

Keshav Aggarwal , Sumit Kumar , Chung-Hang Kwan , Wing Hong Leung , Junxian Li , Matthew P. Young This is my paper · ORCID
classification math.NT MSC 11F6611F6711L0511N37
keywords levelaspectsubconvexityRankin–SelbergL-functionsdeltamethodKloostermanfractionsDirichletcharactersautomorphic
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 proves a subconvexity bound for the central value of the Rankin–Selberg $L$-function $L(1/2, f\otimes g)$ as the level of $f$ runs over primes $p$: for every $\varepsilon>0$, $L(1/2, f\otimes g)\ll p^{1/2-1/524+\varepsilon}$, uniformly in the nebentypus of $f$ and with $g$ any fixed level-one cusp form or Eisenstein series, with $f$ holomorphic or Maass. This improves the previously best exponent $1/1413$ and, because the proof uses only $\mathrm{GL}(1)$ harmonic analysis, it covers weight-one forms whose $L$-functions include class-group and Artin $L$-functions. The argument obtains the saving by applying an amplifier, Voronoi summation, Poisson summation, and a trilinear bound on Kloosterman fractions; it does not solve shifted convolution problems and is independent of any approximation to the Ramanujan conjecture.

What carries the argument

The engine is a delta-symbol method, a smooth-weight version of the circle method, applied to an amplified short sum of Hecke eigenvalues $S(N)=\sum_{n\asymp N}\lambda_f(n)\lambda_g(n)$ with amplifier length $L=p^{1/151}$. An amplifier $A(L)=\sum_{\nu\le L}\gamma_\nu(|\lambda_f(\nu)|^2-\chi(\nu)\lambda_f(\nu^2))$ is meant to recover the size of the moments; then Voronoi summation converts the opening exponential sums into sums of Ramanujan sums, and a Poisson summation step evaluates the correlation of two Ramanujan sums. That correlation produces a trilinear exponential sum whose kernel is a Kloosterman fraction $e(am/n)$; the final cancellation comes from a known bilinear bound on such fractions, refined by an extra savings when a third variable is averaged. This trilinear refinement is what supplies the specific exponent $1/524$.

What would settle it

Compute, for a primitive non-quadratic character $\chi\pmod p$ and $L=p^{1/151}$, the prime character sum $\sum_{L/2<\nu\le L}\chi(\nu)^2$; a single prime $p$ for which this sum is $o(L^{1-\varepsilon})$ would invalidate equation (4.3) and with it the amplifier lemma (Lemma 4.1) on which the proof rests.

Watch

Extended reading notes

Core claim

The main theorem states that for prime $p$, $f$ a Hecke newform of level $p$ with any central character $\chi\pmod p$, and $g$ a fixed Hecke eigenform of level $1$, one has $L(1/2, f\otimes g)\ll p^{1/2-1/524+\varepsilon}$, with the implied constant depending on $g$, $\varepsilon$, and the archimedean parameter of $f$. The proof is uniform: $g$ may be cuspidal or Eisenstein, $f$ holomorphic or Maass, and $\chi$ primitive or trivial, and the same estimate extends to $L(1/2+it, f\otimes g)$ with polynomial dependence on $t$. Unlike earlier moment-method proofs, the argument never solves a shifted convolution problem and is free of any dependence on the Ramanujan-conjecture parameter $\theta$, because all cancellation is extracted from arithmetic sums—Ramanujan sums and Kloosterman fractions—after Voronoi and Poisson summation.

Load-bearing premise

The proof needs the amplifier sum $A(L)=\sum_{\nu\le L}\gamma_\nu(|\lambda_f(\nu)|^2-\chi(\nu)\lambda_f(\nu^2))$ to be $\gg L^{1-\varepsilon}$, and the paper's derivation of that lower bound reduces to a prime sum weighted by $\chi(\nu)^2$; for non-quadratic characters no unconditional lower bound of this strength is known, and without it the amplification lemma fails.

Editorial extensions

If this is right

  • Weight-one holomorphic cusp forms are covered, so the result applies to $L$-functions attached to odd two-dimensional Artin representations and to class-group $L$-functions of imaginary quadratic fields.
  • If the conjectural mean-square lower bound $\sum_{\nu\le L}\gamma_\nu|\lambda_f(\nu)|^2\gg L^{1-\varepsilon}$ is assumed, the same argument sharpens the exponent from $1/524$ to $1/302$.
  • The estimate extends to $L(1/2+it, f\otimes g)$ with polynomial dependence on $t$ and on the archimedean parameter of $f$.
  • The method is expected to work for general levels and arbitrary central characters, not only prime level.
  • The bound does not depend on $\theta$, so it is unaffected by the quality of the available approximation to the Ramanujan conjecture.

Reading between the lines

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

  • Inference: if the obstructing character-sum lower bound becomes available, the same proof immediately upgrades the exponent to $1/302$.
  • Inference: the Ramanujan-sum correlation step is independent of the particular forms $f$ and $g$, so this Poisson-summation device should transfer to other delta-method level-aspect subconvexity problems.
  • Inference: because the route avoids $\mathrm{GL}(2)$ spectral theory, it suggests that weight-one subconvexity can be pursued without developing the full spectral machinery; a testable next step is a hybrid aspect bound along these lines.
Share X Bluesky LinkedIn Reddit HN

Signed reviews

No signed human review yet.

Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

3 major / 3 minor

Summary. The paper claims a level-aspect subconvexity bound L(1/2, f \otimes g) \ll p^{1/2 - 1/524 + \varepsilon} for a Hecke newform f of prime level p with arbitrary central character \chi, and a fixed level-one eigenform g that may be holomorphic, Maass, or Eisenstein. The method uses the DFI delta symbol, Voronoi summation, Cauchy-Schwarz amplification, Poisson summation, and the Bettin-Chandee trilinear Kloosterman-fraction bound. The claimed exponent improves Harcos's \delta = 1/1413 and is stated uniformly in the nebentypus, in the type of g, and without recourse to the Ramanujan conjecture.

Significance. If correct, the result is a significant advance: it is the first delta-method treatment of the level-aspect GL(2) x GL(2) subconvexity problem that is uniform over all nebentypi and over cuspidal/Eisenstein choices of g, and it avoids the spectral machinery used in earlier moment-method proofs. The paper is careful and detailed: the exponent arithmetic is checkable, the use of external results (DFI delta lemma, Bettin-Chandee, Voronoi, stationary phase) is explicit, and the argument does not circularly assume the target bound. The main technical innovation, the exploitation of cancellation in correlations of Ramanujan sums, is clearly explained and is independent of the automorphic forms involved. However, the unconditional amplifier lower bound in Section 4.1 is load-bearing and is not justified for non-quadratic nebentypi; this must be repaired before the claimed uniformity over all central characters is established.

major comments (3)
  1. [§4.1, Eq. (4.3)] The lower bound A(L) \gg L^{1-\varepsilon} is not proved for non-quadratic \chi and is in fact algebraically incorrect as written. From (3.6), \lambda_f(\nu) = \chi(\nu)\overline{\lambda_f(\nu)}, so with \alpha = \lambda_f(\nu) we have |\alpha|^2 = \overline{\chi(\nu)} \alpha^2 = \chi(\nu)^{-1}\alpha^2, not \chi(\nu)\alpha^2. Substituting the Hecke relation \lambda_f(\nu^2)=\alpha^2-\chi(\nu) into A(L)=\sum_\nu \gamma_\nu(|\lambda_f(\nu)|^2-\chi(\nu)\lambda_f(\nu^2)) gives \sum_\nu \gamma_\nu[(\chi(\nu)^{-1}-\chi(\nu))\alpha^2 + \chi(\nu)^2], not \sum_\nu\gamma_\nu. Even the intermediate line \sum_\nu\gamma_\nu\chi(\nu)(\lambda_f(\nu)^2-\lambda_f(\nu^2)) equals \sum_\nu\gamma_\nu\chi(\nu)^2, which has no prime-number-theorem lower bound for a character of order greater than 2. Consequently Lemma 4.1's division by A(L) is unsupported for arbitrary \chi, and Proposition 4.2 and Theorem 1.1 are not established in the stated generality. A likely repair is to replace a_2(\nu) = -\chi(\nu) by -\chi(\nu)^{-1} (equivalently -\overline{\chi}(\nu)), which makes the identity exact, but this change must be traced through the b_j weights used in Lemmas 4.7, 4.9, and 4.19.
  2. [§4.1, definition of b_2] There is an internal inconsistency in the amplifier definition: a_2(\nu) is defined as -\chi(\nu), but the alternative expression A_j(L)=\sum_{\ell\in L_j} b_j(\ell)\lambda_f(\ell) sets b_2(\ell)=\chi(\sqrt{\ell}), which has the opposite sign. The later estimates mostly use |b_j(\ell)|, so the sign may be harmless once absolute values are taken, but the two displayed definitions of A_2(L) are not equal as written and should be reconciled.
  3. [Lemma 4.7] The asserted bound S_0(N,A_1) \ll p^{-2024} is not a defined order of magnitude; the paper's convention of 'very small' means \ll_M p^{-M} for every M, and the constant 2024 appears to be a placeholder. The proof also asserts that the n-sum 'is essentially empty' for j=1 or c/c'=\nu^2 without displaying the truncation argument; this should be stated as a precise 'very small' estimate with the relevant ranges from (4.13).
minor comments (3)
  1. [§4.6, after Eq. (4.19)] In the decomposition of T'(N,A_j), the displayed condition 'n_1\ell_2 \neq n_1\ell_1' should read 'n_1\ell_2 \neq n_2\ell_1'.
  2. [§4.10, Lemma 4.17] In the prime-power analysis, the notation m_0' = m_0/q_{10} is not defined globally; it should be m_0' = m_0/(c_{10},c_{20}) or the local prime-power version should be introduced explicitly.
  3. [§1.1, Remark 1.4] The improved exponent under assumption (1.3) is helpful for orientation, but the reader would benefit from a one-sentence reminder that (1.3) is not used in the unconditional proof and that the unconditional amplifier is designed precisely to avoid it.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: the proof reduces the target bound to external analytic-number-theory estimates, not to its own conclusion.

full rationale

Walk of the derivation: Theorem 1.1 is reduced via the approximate functional equation (3.3) to sums S(N) (3.4). The proof then applies the DFI delta-method identity (Lemma 3.1), Voronoi summation (Lemma 3.2), Cauchy–Schwarz, Poisson summation, and the external trilinear Kloosterman-fraction bound of Bettin–Chandee (Theorem 1.8). Each of these is a stated, parameter-free input with assumptions that do not include the subconvexity bound being proved. The one self-citation, [Leu24, Lemma 3.1], supplies a delta-symbol identity; it is a standard tool, not a hidden form of the Rankin–Selberg L-function bound, and it is not used to forbid alternatives or to carry the main subconvexity content. No fitted values, data, or 'prediction' of a fitted quantity occur. The contested amplifier lower bound in (4.3) is an algebraic and number-theoretic assertion inside the proof; even if a reviewer were to conclude it is unjustified for non-quadratic characters, that would be a correctness defect, not a circular reduction, because the bound is not obtained by assuming the theorem being proved. Accordingly no circular step is present and the score is 0.

Assumptions & free parameters 1 free parameters · 6 assumptions · 0 invented entities

No new objects are postulated; the paper relies on standard analytic number theory input. The main ledger entry is the amplifier length L, chosen to optimize the bound.

free parameters (1)
  • Amplifier length L = L = p^{1/151} (chosen to optimize the error terms in Proposition 4.2)
    L is chosen by hand to balance the error terms; the final exponent δ=1/524 depends on this choice.
assumptions (6)
  • domain assumption Prime number theorem for primes in short intervals over characters (used in equation (4.3))
    The lower bound A(L) ≫ L^{1-ε} relies on summing χ^2(ν) over primes ν ≤ L. For nontrivial χ^2 and L = p^{1/151}, this is not a standard PNT consequence.
  • domain assumption Bettin-Chandee trilinear Kloosterman fraction bound (Theorem 1.8)
    External theorem used as a black box in Lemma 4.19 to obtain cancellation in the trilinear form.
  • domain assumption Voronoi summation for GL2 (Lemma 3.2, [KMV02])
    Standard tool used to dualize the sums over the coefficients of f and g.
  • domain assumption DFI delta method with smooth weight (Lemma 3.1, [Leu24])
    External result used to detect the equality m = nℓ.
  • domain assumption Approximate functional equation for L(s, f⊗g) (equation 3.3)
    Reduces the L-value to a finite sum of S(N).
  • domain assumption Iwaniec's spectral second moment bound (3.8)
    Used to control sums of |λ_f(n)|^2.

how reviews work

0 comments
Cite this review

Pith. "Pith review of Level aspect subconvexity for $\textrm{GL(2)}\times \textrm{GL(2)}$ $\textrm{L}$-functions." pith.science (2026). https://pith.science/paper/ZKR3KWFF

@misc{pith2026241212410,
  author       = {Pith},
  title        = {Pith review of: Level aspect subconvexity for $\textrmGL(2)\times \textrmGL(2)$ $\textrmL$-functions},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/ZKR3KWFF}},
  note         = {Machine review of arXiv:2412.12410}
}
abstract

Let $f$ be a newform of prime level $p$ with any central character $\chi\, (\bmod\, p)$, and let $g$ be a fixed cusp form or Eisenstein series for $\hbox{SL}_{2}(\mathbb{Z})$. We prove the subconvexity bound: for any $\varepsilon>0$, \begin{align*} L(1/2, \, f \otimes g) \ll p^{1/2-1/524+\varepsilon}, \end{align*} where the implied constant depends on $g$, $\varepsilon$, and the archimedean parameter of $f$. This improves upon the previously best-known result by Harcos and Michel. Our method ultimately relies on non-trivial bounds for bilinear forms in Kloosterman fractions pioneered by Duke, Friedlander, and Iwaniec, with later innovations by Bettin and Chandee.

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

29 extracted references · 20 canonical work pages

  1. [1]

    Weyl bound for GL(2) in t -aspect via a simple delta method

    Keshav Aggarwal. Weyl bound for GL(2) in t -aspect via a simple delta method. J. Number Theory , 208:72--100, 2020

    work page 2020

  2. [2]

    On the subconvexity problem for L -functions on GL(3)

    Valentin Blomer and Jack Buttcane. On the subconvexity problem for L -functions on GL(3) . Ann. Sci. \' E c. Norm. Sup\' e r. (4) , 53(6):1441--1500, 2020

    work page 2020

  3. [3]

    Trilinear forms with K loosterman fractions

    Sandro Bettin and Vorrapan Chandee. Trilinear forms with K loosterman fractions. Adv. Math. , 328:1234--1262, 2018

    work page 2018

  4. [4]

    Uniform subconvexity and symmetry breaking reciprocity

    Valentin Blomer and Rizwanur Khan. Uniform subconvexity and symmetry breaking reciprocity. J. Funct. Anal. , 276(7):2315--2358, 2019

    work page 2019

  5. [5]

    Distribution of mass of holomorphic cusp forms

    Valentin Blomer, Rizwanur Khan, and Matthew Young. Distribution of mass of holomorphic cusp forms. Duke Math. J. , 162(14):2609--2644, 2013

    2013

  6. [6]

    W. Duke, J. B. Friedlander, and H. Iwaniec. Bounds for automorphic L -functions. II . Invent. Math. , 115(2):219--239, 1994

    work page 1994

  7. [7]

    W. Duke, J. B. Friedlander, and H. Iwaniec. A quadratic divisor problem. Invent. Math. , 115(2):209--217, 1994

    1994

  8. [8]

    W. Duke, J. Friedlander, and H. Iwaniec. Bilinear forms with K loosterman fractions. Invent. Math. , 128(1):23--43, 1997

    work page 1997

Show all 29 references
  1. [9]

    W. Duke, J. B. Friedlander, and H. Iwaniec. Bounds for automorphic L -functions. III . Invent. Math. , 143(2):221--248, 2001

    2001

  2. [10]

    W. Duke, J. B. Friedlander, and H. Iwaniec. The subconvexity problem for A rtin L -functions. Invent. Math. , 149(3):489--577, 2002

    2002

  3. [11]

    Subconvex bounds for automorphic L -functions and applications

    Gergely Harcos. Subconvex bounds for automorphic L -functions and applications . Doctoral dissertation, Hungarian Academy of Sciences, 2011

    2011

  4. [12]

    The subconvexity problem for R ankin- S elberg L -functions and equidistribution of H eegner points

    Gergely Harcos and Philippe Michel. The subconvexity problem for R ankin- S elberg L -functions and equidistribution of H eegner points. II . Invent. Math. , 163(3):581--655, 2006

    2006

  5. [13]

    Analytic number theory , volume 53 of American Mathematical Society Colloquium Publications

    Henryk Iwaniec and Emmanuel Kowalski. Analytic number theory , volume 53 of American Mathematical Society Colloquium Publications . American Mathematical Society, Providence, RI, 2004

    2004

  6. [14]

    The spectral growth of automorphic L -functions

    Henryk Iwaniec. The spectral growth of automorphic L -functions. J. Reine Angew. Math. , 428:139--159, 1992

    1992

  7. [15]

    Subconvexity bound for GL (3) GL (2) L -functions: hybrid level aspect

    Sumit Kumar, Ritabrata Munshi, and Saurabh Kumar Singh. Subconvexity bound for GL (3) GL (2) L -functions: hybrid level aspect. Algebra Number Theory , 18(3):477--497, 2024

    2024

  8. [16]

    Kowalski, P

    E. Kowalski, P. Michel, and J. VanderKam. Rankin- S elberg L -functions in the level aspect. Duke Math. J. , 114(1):123--191, 2002

    2002

  9. [17]

    Eren Mehmet Kiral, Ian Petrow, and Matthew P. Young. Oscillatory integrals with uniformity in parameters. J. Th\' e or. Nombres Bordeaux , 31(1):145--159, 2019

    2019

  10. [18]

    Shifted convolution sums for GL(3) GL(2) averaged over weighted sets

    Wing Hong Leung. Shifted convolution sums for GL(3) GL(2) averaged over weighted sets. Mathematika , 70(2):Paper No. e12247, 27, 2024

    2024

  11. [19]

    Bounds for GL (3) GL (2) L -functions and GL (3) L -functions

    Xiaoqing Li. Bounds for GL (3) GL (2) L -functions and GL (3) L -functions. Ann. of Math. (2) , 173(1):301--336, 2011

    2011

  12. [20]

    Primes in arithmetic progressions to large moduli, and G oldbach beyond the square-root barrier

    Jared Duker Lichtman. Primes in arithmetic progressions to large moduli, and G oldbach beyond the square-root barrier. 2023. arXiv:2309.08522

    2023 arXiv

  13. [21]

    Bounds for twists of GL(3) L -functions

    Yongxiao Lin. Bounds for twists of GL(3) L -functions. J. Eur. Math. Soc. (JEMS) , 23(6):1899--1924, 2021

    1924

  14. [22]

    P. Michel. The subconvexity problem for R ankin- S elberg L -functions and equidistribution of H eegner points. Ann. of Math. (2) , 160(1):185--236, 2004

    2004

  15. [23]

    On a hybrid bound for twisted L -values

    Ritabrata Munshi. On a hybrid bound for twisted L -values. Arch. Math. (Basel) , 96(3):235--245, 2011

    2011

  16. [24]

    The circle method and bounds for L -functions--- III : t -aspect subconvexity for GL(3) L -functions

    Ritabrata Munshi. The circle method and bounds for L -functions--- III : t -aspect subconvexity for GL(3) L -functions. J. Amer. Math. Soc. , 28(4):913--938, 2015

    2015

  17. [25]

    The circle method and bounds for L -functions--- IV : S ubconvexity for twists of GL(3) L -functions

    Ritabrata Munshi. The circle method and bounds for L -functions--- IV : S ubconvexity for twists of GL(3) L -functions. Ann. of Math. (2) , 182(2):617--672, 2015

    2015

  18. [26]

    The subconvexity problem for GL _2

    Philippe Michel and Akshay Venkatesh. The subconvexity problem for GL _2 . Publ. Math. Inst. Hautes \' E tudes Sci. , (111):171--271, 2010

    2010

  19. [27]

    Paul D. Nelson. Bounds for standard L -functions, 2023. arXiv:2109.15230

    2023 arXiv

  20. [28]

    Sub-convexity problem for R ankin- S elberg L -functions, 2018

    Chandrasekhar Raju. Sub-convexity problem for R ankin- S elberg L -functions, 2018. arXiv:1807.11092

    2018 arXiv

  21. [29]

    Periods and reciprocity II

    Raphaël Zacharias. Periods and reciprocity II . 2020. arXiv:1912.01512

    2020 arXiv

Pith tools

Reviewed August 11, 2026 · model on record in the stance chip above.