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First moments of ${\rm{GL}} (3) \times {\rm{GL}} (2)$ and ${\rm{GL}} (2)$ $L$-functions and their applications

T0 review · 3 major / 4 minor · reviewed 2026-08-10 · deepseek-v4-flash

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Pith's one-line read A first-moment method yields simultaneous level-aspect subconvexity for self-dual GL(3)×GL(2) L-functions, plus weight-aspect Lindelöf averages.

desk verdict This paper has a serious and genuinely interesting level-aspect subconvexity package, but Theorem 1.4 as written rests on an unproved approximate-functional-equation step; it deserves refereeing, not desk rejection. read the letter →

arxiv 2501.15886 v4 pith:JV6M7JUS submitted 2025-01-27 math.NT

Fei Hou This is my paper · ORCID
classification math.NT MSC 11F6711F6611L05
keywords automorphicformsfirstmomentRankin-SelbergL-functionssubconvexityGL(3)×GL(2)levelaspectweightVoronoisummation
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 new estimates for central values of automorphic L-functions attached to a self-dual GL(3) form (a symmetric-square lift) and a GL(2) newform. The main level-aspect result is a subconvex bound for L(1/2,F⊗f) that saves a power of the conductor simultaneously in both level parameters, in the range $M^{{13/64+ε}}$ ≤ q ≤ $M^{{11/40−ε}}$, which the paper states is the first such instance for self-dual GL(3)×GL(2) L-functions. In the weight aspect, it establishes a Lindelöf-on-average bound for the first moment of the degree-8 product L(1/2,f)L(1/2,F⊗f) of size $K^{{1+ε}}$, and an asymptotic formula for the first moment of L(1/2,F⊗f) with power-saving error O($K^{{−1/4+ε}}$). If correct, these results give the expected non-vanishing of these central values and improve on what the spectral large sieve alone provides.

What carries the argument

The engine is a harmonic first-moment average: a Petersson trace formula with weights $ω_f^{{-1}}$ turns sums over f into a diagonal term plus averages of Kloosterman sums. On the off-diagonal, a GL(3) Voronoĭ summation formula in the level aspect (Lemma 2.1) is applied; the proof switches between its unramified case (c,M)=1 and its ramified case N|c by a reciprocity law taken from reference [6], which lets the argument use Voronoĭ twice even when the additive twist shares a factor with the level $q^{2}$. The resulting exponential sums are bounded by bilinear-form estimates for Kloosterman sums (Lemmas 2.6–2.7) and by a square-root cancellation estimate for sums of GL(3) Fourier coefficients against √n-phases (Lemma 2.3). In the weight aspect, the first-moment average is fused with an average over even weights k ~ K; the Bessel function J_{k−1} is summed over k using Lemmas 2.8–2.9, and the exact 2π in the exponential phase (Remark 2.10) makes the phase α = 2√ℓ/c rational, which is what allows Lemma 2.3 to apply.

What would settle it

Compute, for a fixed newform f of weight k ~ K, the difference between ∑_{$K^{{1−δ}}$ ≤ ℓ ≤ $K^{{1+ε}}$} λ_f(ℓ)/√ℓ and the main term of the standard approximate functional equation for L(1/2,f); if the ratio is not uniformly bounded below by a positive constant, the ℓ-summation step in the proof of Theorem 1.4 is invalid.

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Extended reading notes

Core claim

The paper's central claim is that the first moment of the central value L(1/2,F⊗f), where F is the symmetric-square lift of a GL(2) newform of square-free level q and f runs over holomorphic newforms of prime level M, can be estimated with enough uniformity to yield a simultaneous level-aspect subconvex bound: for $M^{{13/64+ε}}$ ≤ q ≤ $M^{{11/40−ε}}$ and M > q^δ, one has L(1/2,F⊗f) ≪ $X^{{1/2−η}}$ for an explicit power saving η, with X ≍ $M^{{3/2}}$$q^{2}$ the conductor size; the paper states this is the first such simultaneous level-aspect subconvexity for self-dual GL(3)×GL(2) L-functions. In the weight aspect, it proves a Lindelöf-on-average bound of size $K^{{1+ε}}$ for the first moment of the degree-8 product L(1/2,f)L(1/2,F⊗f) over even weights K ≤ k ≤ 2K, and an asymptotic formula L(1,F)K/4 \hat{W}(0) + O($K^{{−1/4+ε}}$) for the first moment of L(1/2,F⊗f). A corollary is that L(1/2,F⊗f) ≠ 0 for some newform f of sufficiently large weight.

Load-bearing premise

The weight-aspect proof relies on summing over the short window $K^{{1−δ}}$ ≤ ℓ ≤ $K^{{1+ε}}$ to recover the L(1/2,f) factor for each newform, but it supplies no explicit partition of unity or approximate functional equation showing this short window reproduces every coefficient; if that recovery fails at the level of individual weights, the Lindelöf bound in Theorem 1.4 does not follow.

Editorial extensions

If this is right

  • Corollary 1.3 gives a power saving over the convexity bound for L(1/2,F⊗f) simultaneously in both level aspects in the range M^{13/64+ε} ≤ q ≤ M^{11/40−ε}, the first such result for self-dual GL(3)×GL(2) L-functions.
  • Theorem 1.4 provides a Lindelöf-on-average bound for the first moment of the degree-8 L-function L(1/2,f)L(1/2,F⊗f), beating the O(K^{2+ε}) bound that the spectral large sieve alone supplies.
  • Theorem 1.5 gives an asymptotic formula with power-saving error O(K^{−1/4+ε}) for the first moment of L(1/2,F⊗f).
  • Corollary 1.6 states that for any self-dual GL(3) Maaß form F there exists a newform f of sufficiently large weight with L(1/2,F⊗f) ≠ 0.
  • The paper indicates the method extends in principle to square-free level M with additional bookkeeping, and identifies the reciprocity–Voronoĭ switch as the way the level-aspect collusion deadlock is overcome.

Reading between the lines

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

  • If the short-ℓ recovery step in §4.1 is made explicit with a partition of unity, the same framework should yield a subconvex bound in the weight aspect for L(1/2,f)L(1/2,F⊗f), since the Lindelöf average already saves a full power of K relative to the large sieve.
  • The reciprocity–Voronoĭ switching mechanism is a general recipe for level-aspect subconvexity: whenever the additive twist shares a prime with the level, a one-dimensional reciprocity law can restore the unramified case, so similar hybrid bounds may hold for other GL(3)×GL(2) families, including non-self-dual F.
  • The sensitivity to the 2π phase in the Bessel average suggests the weight-aspect result is arithmetic rather than analytic: only this normalization makes the exponential phase rational, so the same proof would not work for a differently scaled Bessel average.
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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 studies first moments of GL(3)×GL(2) Rankin–Selberg L-functions, with the GL(3) form taken to be a self-dual symmetric square lift. The main results are: (i) a twisted first-moment estimate (Theorem 1.2) which, via the amplifier method and Lapid's nonnegativity, yields a hybrid level-aspect subconvexity bound for L(1/2, F⊗f) in a simultaneous range M^{13/64+ε} ≤ q ≤ M^{11/40−ε} (Corollary 1.3); (ii) a Lindelöf-on-average bound for the first moment of L(1/2,f)L(1/2,F⊗f) in the weight aspect (Theorem 1.4); and (iii) an asymptotic formula for the first moment of L(1/2,F⊗f) with error O(K^{-1/4+ε}) (Theorem 1.5). The paper also includes a detailed root-number computation and several auxiliary lemmas on Voronoĭ summation, bilinear forms, and Bessel functions.

Significance. If correct, the subconvexity result in the simultaneous level aspects would be a significant breakthrough: it appears to be the first such result for self-dual GL(3)×GL(2) L-functions, and the method overcomes the ramified-Voronoĭ obstruction by a reciprocity step. The weight-aspect results are also of interest, since a Lindelöf average bound of the type (1.6) is stronger than what the classical large sieve and Cauchy–Schwarz give. The paper ships no code, but the proof is built on standard tools (Petersson formula, Voronoĭ summation, exponential-sum bounds) and cites prior art appropriately. The claimed results are falsifiable and, if established, would be a substantial contribution.

major comments (3)
  1. [§4.1, Eqs. (4.13) and (1.6)] The transition from (4.13) to the Lindelöf bound (1.6) is not justified. Equation (4.13) is an upper bound for the fixed-ℓ twisted sum Σ_f ω_f^{-1} λ_f(ℓ) L(1/2,F⊗f). After dividing by √ℓ and summing over ℓ in [K^{1−δ}, K^{1+ε}], the proof asserts that one obtains (1.6), i.e. Σ_f ω_f^{-1} L(1/2,f) L(1/2,F⊗f). However, this would require the identity Σ_{ℓ} λ_f(ℓ)/√ℓ ≈ L(1/2,f) for each f (or on average). No approximate functional equation, smooth partition of unity, or dual-side argument is supplied. The short interval [K^{1−δ}, K^{1+ε}] omits the range n ≪ K^{1−δ} and completely ignores the dual sum from the functional equation of L(s,f). Moreover, summing an upper bound for the twisted moment does not produce an upper bound for the untwisted first moment; the order of the ℓ-sum and the f-sum cannot be interchanged in the asserted way. As written, Theorem 1.4 does not follow from the given proof.
  2. [§4.1, Eq. (4.6)] The diagonal contribution from the Petersson trace formula appears to be miscomputed. In (4.4)–(4.5), the sum over f is Σ_f ω_f^{-1} λ_f(ℓ) λ_f(n). After applying the Petersson formula, the diagonal in (n,ℓ) should give δ(n,ℓ) (or an appropriate normalization of it), so that the n-sum collapses to a single term A_F(ℓ,1)/√ℓ V*(ℓ/Y) multiplied by the weight sum over k—with no additional factor λ_f(ℓ). Yet (4.6) retains λ_f(ℓ). This suggests either a different normalization of the Hecke operators (which is not stated) or an error in the handling of the trace formula. If the factor is spurious, the claimed asymptotic (1.6) is not established even if the ℓ-sum step were valid.
  3. [§4.2, Theorem 1.5] The proof of the asymptotic formula (1.7) relies on the claim that the contribution (4.12) is O(Y^{-100}) when ℓ=1. Earlier in §4.1 the same quantity is only estimated by O(Y^{1/6+ε}) (with no indication of a saving in the ℓ=1 special case). The transition from a generic O(Y^{1/6+ε}) bound to a negligible bound for ℓ=1 is not justified. A separate analysis of the phase and the range of C for ℓ=1 is needed before Theorem 1.5 can be accepted.
minor comments (4)
  1. [§2.4, Eq. (2.4)] The Langlands parameters are written with three occurrences of the same symbol α1; the second and third should be α2 and α3 (the displayed formula reads α1 = −ν1−2ν2+1, α1 = −ν1+ν2, α1 = 2ν1+ν1+1, which is inconsistent). The third expression also appears to contain a typo (ν1 instead of ν2).
  2. [General presentation] There are numerous typos and infelicities: 'completly', 'quannity', 'theses', 'bar ely', 'Hek e', 'Maaß' vs 'Maass', and inconsistent spelling of 'Voronoĭ'. The notation {1,ℓ1ℓ2,ℓ1^2ℓ2^2} is not defined in a single place; the meaning of ℓ as a set element should be clarified early.
  3. [§3.1, around Eq. (3.7)] The expansion of the amplifier A_f in (3.7) is not fully explained; in particular, the term x(ℓ^2)^2 1_{ℓ1=ℓ2} appears without a defining comment. A reader needs to infer the combinatorics of the square of the sum over primes in P_L.
  4. [§2.3, Eq. (2.6)] The root-number formula ε(F⊗f) = −i^k λ_f(M)/√M ∏_{p|N} ε(Π_{F,p})^2 is stated after a local computation that gives ε(Π_{F,p}⊗π_{f,p}) = -λ_f(p)√p for p|M, but the passage from the local factors to the global product is not spelled out. It is presumably standard, but a brief explanation would help.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: no output is defined from or fitted to the claimed inputs; the sole self-citation is non-load-bearing, and the (4.13) to (1.6) step is a correctness gap rather than a circular reduction.

full rationale

The paper's derivation chain is not circular. The main bounds in Theorem 1.2 (Proposition 3.3) and Theorem 1.4 are obtained by applying external summation machinery (Petersson trace formula, GL(3) Voronoi summation, Ren–Ye's square-root estimate, and the Kim–Sarnak bound) to approximate functional equations; the parameters L and X are optimized to balance the resulting error terms rather than fitted to force the desired inequalities. The only self-citation, [20], appears in a methodological aside in §3.1.1 ('see, for example, [43, 20] for relevant details') and is not load-bearing: no theorem from [20] is used to establish the main results. The proof of Theorem 1.4 does contain a notable unproved transition: after (4.13), the text asserts 'after divided by a factor √ℓ on both sides of (4.13), we sum over K^{1−δ} ≪ ℓ ≪ K^{1+ε}... obtaining' (1.6); this identifies a short unweighted sum of λ_f(ℓ)/√ℓ with L(1/2,f), which would require an approximate functional equation or dual side and is not supplied. That is a correctness gap, not a circular reduction: the target quantity is neither defined as that partial sum nor obtained by inverting a fitted parameter. No equation in the paper makes the conclusion equal to its input by construction, so the circularity score is 0.

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

The central claims rest on standard theorems of analytic number theory; no constant is fitted to data and no new entity is postulated. The amplifier length L and weight W are optimization variables, not free fitted parameters. The paper's novelty is the combination of these tools, especially the reciprocity step that bypasses the level-modulus collusion in the GL(3) Voronoi formula.

assumptions (7)
  • standard math Approximate functional equation for L(1/2,F tensor f) and L(1/2,f), as in (4.2) and the discussion before (3.9).
    Used to express central values as short Dirichlet sums with smooth weights; cited to Iwaniec-Kowalski and Holowinsky-Munshi-Qi.
  • standard math Petersson trace formula with harmonic weights (Lemma 2.4).
    Expands sums over newforms into diagonal and Kloosterman terms.
  • standard math GL(3) Voronoi summation in both unramified and ramified cases (Lemma 2.1, credited to Zhou's 2018 preprint).
    The ramified case is load-bearing because the additive twist can collide with the level q^2.
  • standard math Kim-Sarnak bound |A_F(m,n)| much less than (mn)^{7/32+epsilon}.
    Used to estimate Fourier coefficients in small-range cases.
  • standard math Ren-Ye square-root cancellation estimate for GL(3) exponential sums (Lemma 2.3).
    Provides the key cancellation for the weight-aspect Lindelof average.
  • domain assumption Lapid's nonnegativity of L(1/2,F tensor f) for self-dual F.
    Needed to pass from an amplified first moment to an individual subconvex bound in Corollary 1.3.
  • domain assumption Root number factorization epsilon(F tensor f) = -i^k lambda_f(M) sqrt(M) product_{p|N} epsilon(Pi_{F,p})^2, Eq. (2.6).
    Determines the sign in the functional equation and hence the form of the approximate functional equation used in averaging.

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Pith. "Pith review of First moments of ${\rm{GL}} (3) \times {\rm{GL}} (2)$ and ${\rm{GL}} (2)$ $L$-functions and their applications." pith.science (2026). https://pith.science/paper/JV6M7JUS

@misc{pith2026250115886,
  author       = {Pith},
  title        = {Pith review of: First moments of $\rmGL (3) \times \rmGL (2)$ and $\rmGL (2)$ $L$-functions and their applications},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/JV6M7JUS}},
  note         = {Machine review of arXiv:2501.15886}
}
abstract

Let $F$ be a self-dual Hecke-Maa\ss\ form for ${\rm{GL}}(3)$ underlying the symmetric square lift of a ${\rm{GL}}(2)$-newform of square-free level and trivial nebentypus. In this paper, we are interested in the first moments of the central values of ${\rm{GL}}(3) \times {\rm{GL}}(2)$ $L$-functions and ${\rm{GL}}(2)$ $L$-functions. As a result, we obtain an estimate for the first moment for $L(1/2, F\otimes f)$ over a family, where $F$ is of the level $q^2$, and $f\in \mathcal{B}^\ast_k(M)$ for any primes $q,M\ge 2$ such that $(q,M)=1$. We prove the subconvex bound for $L(1/2, F\otimes f)$ involving the levels aspects simultaneously in the range $M^{13/64+\varepsilon }\le q \le M^{11/40-\varepsilon}$ and $M> q^\delta$ for any $\varepsilon, \delta>0$ for the first time. Moreover, we further investigate the first moments of these $L$-functions in the weight $k$ aspect over $K\le k\le 2K$, with $K$ being a large number. As the results, we obtain a Lindel\"of average bound for the first moment of $L(1/2, f)L(1/2, F\otimes f)$ of degree 8 and an asymptotic formula for the first moment of $L(1/2, F\otimes f)$ with an error term of $O(K^{-1/4+\varepsilon})$, respectively.

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Works this paper leans on

50 extracted references · 48 canonical work pages

  1. [1]

    Bettin, On the reciprocity law for the twisted second moment of Diric hlet L-functions, Trans

    S. Bettin, On the reciprocity law for the twisted second moment of Diric hlet L-functions, Trans. Amer. Math. Soc. 368 (2016), no. 10, 6887-6914

    work page 2016

  2. [2]

    Blomer, Subconvexity for twisted L-functions on GL(3), Amer

    V. Blomer, Subconvexity for twisted L-functions on GL(3), Amer. J. Math. 134 (2012), no. 5, 1385-1421

    work page 2012

  3. [3]

    Blomer, On the 4-norm of an automorphic form , J

    V. Blomer, On the 4-norm of an automorphic form , J. Eur. Math. Soc. 15 (2013), no. 5, 1825-1852

    work page 2013

  4. [4]

    Blomer, J

    V. Blomer, J. Buttcane and P. Maga, Applications of the Kuznetsov formula on GL(3) II: the level aspect, Math. Ann. 369 (2017), 723-759

    work page 2017

  5. [5]

    Blomer and R

    V. Blomer and R. Holowinsky, Bounding sup-norms of cusp forms of large level , Invent. Math. 179 (2010), no. 3, 645-681

    work page 2010

  6. [6]

    Blomer and R

    V. Blomer and R. Khan, Twisted moments of L-functions and spectral reciprocity, Duke Math. J. 168 (2019), no. 6, 1109-1177

    work page 2019

  7. [7]

    Blomer, R

    V. Blomer, R. Khan and M. P. Young, Distribution of mass of holomorphic cusp forms , Duke Math. J. 162 (2013), no. 14, 2609-2644

    work page 2013

  8. [8]

    A. R. Booker, M. B. Milinovich and N. Ng, Subconvexity for modular form L-functions in the t aspect, Adv. Math. 341 (2019), no. 7, 299-335

    work page 2019

Show all 50 references
  1. [9]

    Corbett, Vorono ˘ ı summation forGLn: collusion between level and modulus , Amer

    A. Corbett, Vorono ˘ ı summation forGLn: collusion between level and modulus , Amer. J. Math. 143 (2021), no. 5, 1361-1395

    2021

  2. [10]

    J. B. Conrey, The mean-square of Dirichlet L-functions, arXiv preprint, arXiv:0708.2699

    arXiv

  3. [11]

    J. B. Conrey and H. Iwaniec, The cubic moment of central values of automorphic L-functions, Ann. of Math. (2) 151 (2000), no. 3, 1175-1216

    2000

  4. [12]

    Deshouillers and H

    J.-M. Deshouillers and H. Iwaniec, Kloosterman sums and fourier coefficients of cusp forms , Invent. Math. 70 (1982), no. 2, 219–288

    1982

  5. [13]

    Goldfeld, Automorphic forms and L-functions for the group GL(n, R), Cambridge Studies in Advanced Mathematics, vol

    D. Goldfeld, Automorphic forms and L-functions for the group GL(n, R), Cambridge Studies in Advanced Mathematics, vol. 99 (Cambridge University Press , Cambridge, 2006, With an appendix by K. Broughan)

    2006

  6. [14]

    Goldfeld and J

    D. Goldfeld and J. Hundley, Automorphic representations and L-functions for the general linear group. Volume II , Cambridge Studies in Advanced Mathematics. vol. 130 (Camb ridge University Press, Cambridge, 2011, With exercises and a preface by Xand er Faber)

    2011

  7. [15]

    Harcos and P

    G. Harcos and P. Michel, The subconvexity problem for Rankin-Selberg L-functions and equidis- tribution of Heegner points. II , Invent. Math. 163 (2006), 581-655

    2006

  8. [16]

    Harcos and N

    G. Harcos and N. Templier, On the sup-norm of maass cusp forms of large level, III , Math. Ann. 356 (2013), no. 1, 209-216

    2013

  9. [17]

    Harun, S

    M. Harun, S. Kumar and S. K. Singh, Hybrid subconvexity b ound for GL(3) × GL(2) L-functions: t and level aspect, Mathematika, Published online, Doi: 10.1 112/mtk.12272

  10. [18]

    Holowinsky, R

    R. Holowinsky, R. Munshi and Z. Qi, Hybrid subconvexity bounds for L (1 2 , Sym2f ⊗ g ) , Math. Z. 283 (2016), no. 1, 555-579

    2016

  11. [19]

    Holowinsky and P

    R. Holowinsky and P. D. Nelson, Subconvex bounds on GL(3) via degeneration to frequency zero , Math. Ann. 372 (2018), 299-319

    2018

  12. [20]

    Hou and G

    F. Hou and G. Lü, Triple correlation sums of coefficients of Maass forms for SL4(Z), Q. J. Math. 75 (2024), no. 3, 1123-1148. 30 F. HOU

    2024

  13. [21]

    Huang, Uniform bounds for GL(3) × GL(2) L-functions, J

    B. Huang, Uniform bounds for GL(3) × GL(2) L-functions, J. Inst. Math. Jussieu 23 (2024), no. 4, 1607-1650

    2024

  14. [22]

    Ivić, On sums of Hecke series in short intervals , J

    A. Ivić, On sums of Hecke series in short intervals , J. Théor. Nombres Bordeaux 13 (2001), no. 2, 453-468

    2001

  15. [23]

    Iwaniec and E

    H. Iwaniec and E. Kowalski, Analytic number theory, American Mathematical Society Colloquium Publications, 53, Amer. Math. Soc., Providence, 2004

    2004

  16. [24]

    Iwaniec, W

    H. Iwaniec, W. Luo and P. Sarnak, Low lying zeros of families of L-functions, Inst. Hautes Études Sci. Publ. Math. 91 (2001), 55-131

    2001

  17. [25]

    Jana and R

    S. Jana and R. Nunes, Spectral reciprocity for GL(n) and simultaneous non-vanishing of central L-values, arXiv preprint 2111.02297 (2024)

    2024

  18. [26]

    B. Kerr, I. E. Shparlinski, X. Wu and P. Xi, Bounds on bilinear forms with Kloosterman sums , J. Lond. Math. Soc. 108 (2023), no. 2, 578-621 (available at https://arxiv.org/pdf/2204.05038)

    2023 arXiv

  19. [27]

    Khan, Non-vanishing of the symmetric square L-function at the central point , Proc

    R. Khan, Non-vanishing of the symmetric square L-function at the central point , Proc. Lond. Math. Soc. 100 (2010), no. 3, 736-762

    2010

  20. [28]

    Khan, On the subconvexity problem for GL(3) × GL(2) L-functions, Forum Math

    R. Khan, On the subconvexity problem for GL(3) × GL(2) L-functions, Forum Math. 27 (2015), no. 2, 897-913

    2015

  21. [29]

    H. H. Kim, Functoriality for the exterior square of GL(4) and the symmetric fourth of GL(2), J. Amer. Math. Soc. 16 (2003), no. 1, 139–183, with Appendix 1 by D. Ramakrishnan, A ppendix 2 by H. H. Kim and P. Sarnak

    2003

  22. [30]

    Kumar, Subconvexity bounds for GL(3) × GL(2) L-functions in GL(2) spectral aspect, J

    S. Kumar, Subconvexity bounds for GL(3) × GL(2) L-functions in GL(2) spectral aspect, J. Eur. Math. Soc. (2023), Published online, Doi: 10.4171/JEMS/13 83

    2023 doi

  23. [31]

    Kumar, K

    S. Kumar, K. Mallesham and S. K. Singh, Sub-convexity bound for GL(3) × GL(2) L-functions: the depth aspect , arXiv preprint, arXiv:2012.12674

    2012 arXiv

  24. [32]

    Kumar, R

    S. Kumar, R. Munshi and S. K. Singh, Subconvexity bound for GL(3) × GL(2) L-functions: Hybrid level aspect, Algebra & Number Theory 18 (2024), no. 3, 477-497

    2024

  25. [33]

    E. M. Lapid, On the nonnegativity of Rankin-Selberg L-functions at the center of symmetry , Int. Math. Res. Not. 2003 (2003), no. 2, 65-75

    2003

  26. [34]

    Li, The central value of the Rankin-Selberg L-functions, Geom

    X. Li, The central value of the Rankin-Selberg L-functions, Geom. Funct. Anal. 18 (2009), no. 5, 1660-1695

    2009

  27. [35]

    Li, Bounds for GL(3) × GL(2) L-functions and GL(3) L-functions, Ann

    X. Li, Bounds for GL(3) × GL(2) L-functions and GL(3) L-functions, Ann. of Math. (2) 173 (2011), no. 1, 301-336

    2011

  28. [36]

    Lin and Q

    Y. Lin and Q. Sun, Analytic twists of GL3 × GL2 Automorphic Forms, Int. Math. Res. Not. 2021 (2021), no. 19, 15143-15208

    2021

  29. [37]

    Liu, L4-norms of the holomorphic dihedral forms of large level , Abh

    S.-C. Liu, L4-norms of the holomorphic dihedral forms of large level , Abh. Math. Sem. Hamburg, 85 (2015), 53-57

    2015

  30. [38]

    S.-C. Liu, R. Masri and M. Young, Subconvexity and equidistribution of Heegner points in the level aspect, Compos. Math. 149 (2013), no. 7, 1150-1174

    2013

  31. [39]

    Luo, Central values of the symmetric square L-functions, Proc

    W. Luo, Central values of the symmetric square L-functions, Proc. Amer. Math. Soc. 140 (2012), 3313–3322

    2012

  32. [40]

    Munshi, The circle method and bounds for L-functions - II: Subconvexity for twists of GL(3) L-functions, Amer

    R. Munshi, The circle method and bounds for L-functions - II: Subconvexity for twists of GL(3) L-functions, Amer. J. Math. 137(2) (2015) 791-812

    2015

  33. [41]

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

    R. Munshi, The circle method and bounds for L-functions - III: t-aspect subconvexity for GL(3) L-functions, J. Am. Math. Soc. 28 (2015), no. 4, 913-938

    2015

  34. [42]

    Munshi, A note on Burgess bound , Geometry, Algebra, Number Theory, and Their Information Technology Applications, GANITA, Springer Proceedings in Mathematics & Statistics, 2016

    R. Munshi, A note on Burgess bound , Geometry, Algebra, Number Theory, and Their Information Technology Applications, GANITA, Springer Proceedings in Mathematics & Statistics, 2016

    2016

  35. [43]

    Munshi, Subconvexity for GL(3) × GL(2) L-functions in t-aspect, J

    R. Munshi, Subconvexity for GL(3) × GL(2) L-functions in t-aspect, J. Eur. Math. Soc. 24 (2022), no. 5, 1543-1566

    2022

  36. [44]

    P. D. Nelson, Subconvex equidistribution of cusp forms: Reduction to Eis enstein observables , Duke Math. J. 168 (2019), no. 9, 1665-1722

    2019

  37. [45]

    Ren and Y

    X. Ren and Y. Ye, Resonance of automorphic forms for GL(3), Trans. Amer. Math. Soc. 367 (2015), no. 3, 2137–2157. FIRST MOMENTS OF GL(3) × GL(2) AND GL(2) L-FUNCTIONS AND THEIR APPLICATIONS 31

    2015

  38. [46]

    Sharma and W

    P. Sharma and W. Sawin Subconvexity for GL(3)×GL(2) twists, Adv. Math. 404, Part B, 6 August 2022, 108420

    2022

  39. [47]

    I. E. Shparlinski, On sums of Kloosterman and Gauss sums , Trans. Amer. Math. Soc. 371 (2019), 8679-8697

    2019

  40. [48]

    E. T. Whittaker and G. N. Watson, A course of modern analysis , Cambridge Mathematical Li- brary. Cambridge University Press, Cambridge, 1996. An int roduction to the general theory of infinite processes and of analytic functions; with an accoun t of the principal transcendenta...

    1927

  41. [49]

    M. P. Young, The reciprocity law for the twisted second moment of Dirichl et L-functions, Forum Math. 23 (2011), no. 6, 1323-1337

    2011

  42. [50]

    Zhou, The Voronoi formula on GL(3) with ramification , arXiv preprint arXiv:1806.10786 (2018)

    F. Zhou, The Voronoi formula on GL(3) with ramification , arXiv preprint arXiv:1806.10786 (2018). School of Sciences, Xi’an University of Technology, Xi’an 71 0054, China Email address : fhou@xaut.edu.cn

    2018 arXiv

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