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Spectral Reciprocity and Hybrid Subconvexity Bound for triple product $L$-functions

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Pith's one-line read Triple product L-functions admit an explicit hybrid level-aspect subconvexity bound over any number field.

desk verdict Solid extension of the HMN23 reciprocity program with explicit hybrid exponents; the advertised saving rests on an imported local lower bound that needs verification before the exponent is trusted. read the letter →

arxiv 2501.04022 v1 pith:MXNYRXAR submitted 2024-12-27 math.NT

Xinchen Miao This is my paper · ORCID
classification math.NT MSC 11F7011M4111F72
keywords subconvexitytripleproductL-functionsspectralreciprocitylevelaspectautomorphicformsanalyticconductoramplificationmethod
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

Over a fixed number field, the paper proves a subconvexity bound at the central point for triple product L-functions of three automorphic representations of PGL$_2$. The bound takes the form $Q_f^{1/4+\varepsilon}P_f^{-\delta}$ with an explicit positive saving exponent $\delta$; unconditionally one may take $\delta=1/60$, and under the Ramanujan conjecture $\delta=1/28$. The saving is measured against a finite conductor parameter $P_f$ that also tracks the self-twists, so the result remains meaningful when the three conductors are jointly ramified and the analytic conductor drops. This makes the hybrid level-aspect saving explicit and covers the conductor-dropping range, which is the situation previous methods left open.

What carries the argument

The machinery is a spectral reciprocity formula for the twisted first moment $$\mathcal M(\pi_1,\pi_2,a,m,n,\mathfrak l)=\sum_{\pi} \lambda_\pi(\mathfrak l)\,\frac{L(\tfrac12,\pi\otimes\pi_1\otimes\pi_2)}{\Lambda^*(1,\pi,\mathrm{Ad})} f(\pi_\infty)H(\pi,a,m,n),$$ together with the analogous Eisenstein contribution. The formula is derived by studying a symmetric period $P_{\mathfrak q}(\mathfrak l,\Phi,\Phi)=\langle T_{\mathfrak l}\Phi,\Phi\rangle$ with $\Phi=\varphi_1\varphi_2^{\mathfrak q}$; expanding it in the level aspect via the Plancherel formula and regrouping the same inner product as $\langle\varphi_1^{\mathfrak p}\varphi_1,\varphi_2^{\mathfrak p}\varphi_2\rangle$ moves the Hecke action to the dual side and produces a moment of self triple products. Bounding that dual moment with local period-integral estimates and then applying the amplifier yields the subconvex bound.

What would settle it

Evaluate the local triple product integral $I_v^T$ for the paper's test vectors at a non-archimedean place where the residue field has small characteristic, for instance characteristic 2, and compare it with $Q_v^{-1/4}$; a decay faster than $Q_v^{-1/4}$ would invalidate the diagonal lower bound and collapse the subconvex saving.

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Extended reading notes

Core claim

Let $F$ be a fixed number field, $\pi_1,\pi_2$ unitary cuspidal representations of $\mathrm{PGL}_2(\mathbb{A}_F)$, and $\pi_3$ a unitary automorphic representation, with finite conductors $m,n,a$. Theorem 1.3 asserts that $$L\big(\tfrac12,\pi_1\otimes\pi_2\otimes\pi_3\big) \ll_{\varepsilon,F,\pi_{i,\infty}} $Q_f^{{1/4+\varepsilon}}$ $P_f^{{-(1/4-\theta/2)(1-2\theta_1-2\theta_2)/(7-2\theta_1-2\theta_2)}}$,$$ where $Q_f$ is the finite part of the analytic conductor of the triple product, $P_f$ is the finite part of the parameter $\prod_v Q_v^{1/2}\max_{i=1,2}C_v(\pi_i\otimes\pi_i)$, and $\theta,\theta_i$ are the best exponents toward the Ramanujan conjecture available for $\mathrm{GL}(2)$ over $F$. With $\theta=\theta_1=\theta_2=7/64$ the bound becomes $Q_f^{1/4+\varepsilon}P_f^{-1/60}$ unconditionally; under Ramanujan it becomes $Q_f^{1/4+\varepsilon}P_f^{-1/28}$. The paper also gives explicit corollaries for pairwise-coprime squarefull levels and for the case where $\pi_3$ is an Eisenstein series.

Load-bearing premise

The load-bearing premise is the imported lower bound on the local triple product integral at ramified places, namely $I_v^T(\varphi_{1,v},\varphi_{2,v},\varphi_{3,v})\gg Q_v^{-1/4}$; if the true decay is worse for some ramified or small-residue-characteristic place, the diagonal term would shrink and the final saving exponent would degrade or disappear.

Editorial extensions

If this is right

  • For any fixed number field, the central value of a triple product $L$-function has an explicit power saving in the level-aspect parameter $P_f$, not just a saving in the full conductor $Q_f$.
  • The saving remains available when the three finite conductors are jointly ramified, i.e. in the conductor-dropping range where $Q_f$ can be much smaller than the product of the individual conductor powers.
  • Unconditionally the exponent is $\delta=1/60$; assuming the Ramanujan conjecture for $\mathrm{GL}(2)$ over $F$ it improves to $\delta=1/28$.
  • With pairwise coprime conductors the bound becomes an explicit expression in the norms of the three level ideals, and a stronger form holds when the two cuspidal levels are squarefull.
  • The same method covers $\pi_3$ equal to an Eisenstein series, giving subconvexity for $L(\tfrac12,\pi_1\otimes\pi_2\otimes\chi)$.

Reading between the lines

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

  • A testable extension is to let the archimedean parameters grow together with the levels; the archimedean weight in the reciprocity formula suggests the saving should persist, but that range is not carried by the present statement.
  • The reciprocity identity should also yield lower bounds or non-vanishing statements for the first moment by evaluating the dual side asymptotically, a direction the paper does not pursue.
  • Because the bound is uniform in the fixed number field, it may feed into equidistribution statements for Hecke eigenvalues against triple product periods, where a subconvex exponent in the level aspect is the standard input.
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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

5 major / 5 minor

Summary. The paper develops a spectral reciprocity formula for twisted first moments of triple product L-functions over a fixed number field and uses it, together with an amplifier, to prove a hybrid level-aspect subconvexity bound for L(1/2, π1⊗π2⊗π3). The main theorem gives an explicit saving P_f^{-δ} with δ > 1/60 unconditionally when θ=θ1=θ2=7/64, and allows joint ramification and conductor dropping. The argument builds on the period integral approach of Ichino-Watson, the GL2 subconvexity machinery of Michel–Venkatesh, and the local estimates of Hu–Michel–Nelson and Blomer–Brumley–Khayutin, with the latter imported from preprints.

Significance. If the proof is correct, the paper provides a genuinely explicit hybrid subconvexity bound for triple product L-functions over a general number field, improving on the qualitative results in [HMN23] and being new even over Q. The allowance for joint ramification and conductor dropping is a valuable technical feature. The paper is not fully self-contained, but it identifies the key local and global inputs and connects them in a plausible way, which is a useful contribution to the literature.

major comments (5)
  1. [§4.1, Prop. 4.2; §5.3; §6.1] The lower bound I_T^v(φ1,v, φ2,v, φ3,v) ≫ Q_v^{-1/4} is imported verbatim from [HMN23, Theorem 3.22] and is used in §5.3 to conclude ℓ(π3,m,n,a) ≫ Q_f^{-1/4}, which is the only mechanism that makes the diagonal term in the amplifier argument of §6.1 non-negligible. However, the test vectors fixed in §4 are not the same as those in the quoted theorem: φ1,v is replaced by π1,v(diag(1, ϖ^s))φ0_{1,v} with s = c(π2⊗π3)/2 (plus a bounded shift in small residue cardinality), and the conductor-dropping cases c(π2⊗π3) < c(π2)+c(π3) are exactly where local lower bounds are delicate. The paper does not verify that the hypotheses of [HMN23, Thm 3.22] hold for these choices. Since the final saving exponent in Theorem 1.3 is proportional to the exponent -1/4 in this lower bound, this is a load-bearing gap.
  2. [§5.2, Eq. (5.6)] The symmetric relation (5.6) is asserted with the single sentence 'By the Hecke relation (3.5), we have the following symmetric relation' and no derivation. The relation involves a specific constant q^{1/2}ζ_q(1)/ζ_q(2) and a weighted sum over k of periods with Hecke operators, and it is the foundation of the reciprocity formula used throughout the paper. Please provide a complete derivation or a precise reference to a source that proves this exact identity.
  3. [§5.2, between (5.8) and (5.17)] The bound for the generic term G_{p^{v-2k}}(q, Ψ1, Ψ2) is compressed into a single sentence: the paper states that 'From above discussion, especially Proposition 4.2, Proposition 5.1, Remark 5.2, convexity bound..., we obtain an upper bound' and then gives the final estimate. The combination of the local triple product bounds, the spectral expansion, the Cauchy-Schwarz step, and the Weyl law is not shown. This is the core analytic estimate of the paper and should be written out in detail.
  4. [§6.1 and Remark 5.2] In the amplification step, the paper uses a 'slightly stronger version of Theorem 1.1' in which the exponent on ℓ is 3/2 rather than 3/2 + 2θ1 + 2θ2. The justification in Remark 5.2 is a sketch that covers k=1,2,4, but the cases with θ_i appearing in the exponent are not fully handled, and the reduction to k=1,2,4 is not justified rigorously. Since the final saving exponent depends on this strengthened bound, this step needs a complete proof.
  5. [§1, paragraph defining θ] The paper states that the best exponent toward Ramanujan-Petersson for GL(2) over a fixed number field F satisfies 0 ≤ θ ≤ 7/64. The bound 7/64 is the Kim-Sarnak bound for F=Q; to my knowledge it is not established for a general number field (the standard uniform bound is 1/4 − 1/9, due to Blomer–Brumley). If 7/64 is not available for the number fields considered, the 'unconditional' claim in Theorem 1.3 with θ=7/64 is not justified. Please provide a reference for the 7/64 bound over arbitrary number fields, or restrict the statement accordingly.
minor comments (5)
  1. [Abstract] In the abstract, 'coprimes' should be 'coprime', and the sentence 'The estimation becomes a reciprocity formula between different moments of L-functions' is unclear; the intended meaning is that the estimation is based on a reciprocity formula.
  2. [§4] The notation 'a' is overloaded: on page 4 it denotes the ideal a, while in Section 4 the phrase 'we may have a ≤ 1' seems to refer to a different quantity. Please disambiguate.
  3. [§5.2] The sentences 'This is the phenomenon of the spectral reciprocity formula' and 'We get a close and interesting relation between different type of L-functions with different spectral length' are informal and should be replaced by precise mathematical statements.
  4. [§1 and §5.3] The paper refers to 'Section 6.3 Choice of test vectors and Proposition 6.5 in [HMN23]' in the introduction and to 'Section 6.4 and Assumption 5.3 in [HMN23]' in Section 6.1, but the present paper's Section 6 contains only Section 6.1. Please update the cross-references to avoid confusion.
  5. [§6.1] The reduction assuming (mna)^4 ≥ Qf ≥ (mna)^{1/2} is stated as following from 'Section 6.4 and Assumption 5.3 in [HMN23]' without explanation. Please spell out this reduction or provide a precise reference.

Circularity Check

0 steps flagged · score 2.0 of 10

No significant circularity: the reciprocity and amplification argument is self-contained, and the only self-citation is a non-load-bearing reduction note.

full rationale

The derivation chain is not circular. The twisted first moment M in (1.3) is bounded from above in Theorem 1.1 using the spectral reciprocity identity and local upper bounds; the diagonal contribution is separately lower bounded in Section 6.1 using the imported local lower bound I_T^v(φ1,v,φ2,v,φ3,v) >> Q_v^{-1/4} (Proposition 4.2). These are independent ingredients: the lower bound is a local statement about test vectors, not the global subconvexity bound being proved, and it is imported from HMN23, a set of authors with no overlap with the present author. The amplification step then combines the upper bound on the amplified moment with the lower bound on the π3 diagonal term; this is the standard, non-circular mechanism for extracting an individual bound from a moment estimate. The paper does not fit parameters and then rename them as predictions; all exponents are explicit functions of the Ramanujan-bound parameters θ, θ1, θ2. The only self-citation is [Miao24], which is mentioned in Section 5.1 when both π1 and π2 are unramified at all finite places; this is followed immediately by an independent argument showing the constant term C1 vanishes, so the self-citation is not load-bearing. The real risk flagged by the reader is correctness, not circularity: Proposition 4.2 is quoted, not proved, and if the local lower bound degrades at jointly ramified places, the exponent in Theorem 1.3 would shrink. That is a verification gap relative to an external result, not a reduction of the paper's conclusion to its own assumptions.

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

The central claim rests on imported local estimates from HMN23, BBK22, Hu17 and on standard trace/formula results. No numerical parameters are fitted to data: the theta constants are fixed bounds from the literature, and the ideal c is an absolutely bounded test-vector choice. No new entities are postulated.

assumptions (7)
  • domain assumption Ramanujan-Petersson bounds: 0 <= theta, theta1, theta2 <= 7/64 for GL(2) over F.
    Used throughout (Theorem 1.1, amplification) to control Hecke eigenvalues |lambda_pi(q)| <= tau(q) q^theta and local exponents; unconditional bound 7/64 is imported from [BB11].
  • domain assumption Local integral estimates: lower bound I_T^v >> Q_v^{-1/4} and upper bounds (4.2), (4.3).
    Quoted from [HMN23, Theorem 3.22]; controls the diagonal and off-diagonal terms in the moment and reciprocity.
  • domain assumption Local bounds for translated newvectors: Proposition 5.1 (I_T << k^4 p^{-k(1-2theta2)+...}).
    Quoted from [BBK22, Proposition 11.4] and sketched for a special case in Remark 5.2; used to bound the amplified generic sum.
  • standard math Ichino-Watson formula (Proposition 3.1) connecting the triple product period to the central value.
    Established in [Ich08] and [MV10]; foundational for all period computations in the paper.
  • standard math Spectral decomposition of L^2(X) and the Plancherel formula at level K0(c[m,n,a]).
    Used to expand the symmetric period; from [GJ79] and [Zac20, Theorem 2.8].
  • domain assumption The archimedean spectral parameters of pi can be treated as absolutely bounded.
    Argued via rapid decay of f(pi_infty) and a weak Weyl law; if false, the spectral sums are not finite and the explicit constants fail.
  • domain assumption All prime ideals under consideration do not divide the discriminant Delta_F.
    Assumed at the start of Section 2; avoids complications from ramification in the base field.

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Pith. "Pith review of Spectral Reciprocity and Hybrid Subconvexity Bound for triple product $L$-functions." pith.science (2026). https://pith.science/paper/MXNYRXAR

@misc{pith2026250104022,
  author       = {Pith},
  title        = {Pith review of: Spectral Reciprocity and Hybrid Subconvexity Bound for triple product $L$-functions},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/MXNYRXAR}},
  note         = {Machine review of arXiv:2501.04022}
}
abstract

Let $F$ be a number field with adele ring $\mathbb{A}_F$, $\pi_1, \pi_2$ be two unitary cuspidal automorphic representations of $\mathrm{PGL}_2(\mathbb{A}_F)$ with finite analytic conductor. We study the twisted first moment of the triple product $L$-function $L(\frac{1}{2}, \pi \otimes \pi_1 \otimes \pi_2)$ and the Hecke eigenvalues $\lambda_\pi (\mathfrak{l})$, where $\pi$ is a unitary automorphic representation of $\mathrm{PGL}_2(\mathbb{A}_F)$ and $\mathfrak{l}$ is an integral ideal coprimes with the finite analytic conductor $C(\pi \otimes \pi_1 \otimes \pi_2)$. The estimation becomes a reciprocity formula between different moments of $L$-functions. Combining with the ideas and estimations established in [HMN23] and [MV10], we study the subconvexity problem for the triple product $L$-function in the level aspect and give a new explicit hybrid subconvexity bound for $L(\frac{1}{2}, \pi \otimes \pi_1 \otimes \pi_2)$, allowing joint ramifications and conductor dropping range.

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Reference graph

Works this paper leans on

40 extracted references · 35 canonical work pages

  1. [1]

    Level reciprocity in the twisted second moment of R ankin- S elberg L -functions

    Nickolas Andersen and Eren Mehmet Kiral. Level reciprocity in the twisted second moment of R ankin- S elberg L -functions. Mathematika , 64(3):770--784, 2018

    work page 2018

  2. [2]

    Subconvexity for twisted L -functions on (3)

    Valentin Blomer. Subconvexity for twisted L -functions on (3) . American Journal of Math. , 134(5):1385-1421, 2012

    work page 2012

  3. [3]

    On the R amanujan conjecture over number fields

    Valentin Blomer and Farrell Brumley. On the R amanujan conjecture over number fields. Ann. of Math. (2) , 174(1):581--605, 2011

    work page 2011

  4. [4]

    The mixing conjecture under GRH

    Valentin Blomer, Farrell Brumley and Ilya Khayutin. The mixing conjecture under GRH. arXiv preprint arXiv: 2212.06280 , 2022

    arXiv 2022

  5. [5]

    Motohashi's fourth moment identity for non-archimedean test functions and applications

    Valentin Blomer, Peter Humphries, Rizwanur Khan, and Micah Milinovich. Motohashi's fourth moment identity for non-archimedean test functions and applications. arXiv preprint arXiv: 1902.07042 , 2019

    arXiv 1902

  6. [6]

    Valentin Blomer, Subhajit Jana and Paul D. Nelson. Local integral transforms and global spectral decomposition. arXiv preprint arXiv: 2404.10692 , 2024

    work page arXiv 2024

  7. [7]

    Twisted moments of L -functions and spectral reciprocity

    Valentin Blomer and Rizwanur Khan. Twisted moments of L -functions and spectral reciprocity. Duke Math. J. , 168(6):1109--1177, 2019

    work page 2019

  8. [8]

    Valentin Blomer, Xiaoqing Li, and Stephen D. Miller. A spectral reciprocity formula and non-vanishing for L -functions on GL (4) GL (2) . J. Number Theory , 205:1--43, 2019

    work page 2019

Show all 40 references
  1. [9]

    The Second Moment of Twisted Modular L -Functions

    Valentin Blomer and Djordje Milićević. The Second Moment of Twisted Modular L -Functions. Geom. Funct. Anal. , 25(2) 453-516, 2015

    2015

  2. [10]

    Automorphic forms and representations , volume 55 of Cambridge Studies in Advanced Mathematics

    Daniel Bump. Automorphic forms and representations , volume 55 of Cambridge Studies in Advanced Mathematics . Cambridge University Press, Cambridge, 1997

    1997

  3. [11]

    Stephen S. Gelbart. Automorphic forms on ad\`ele groups . Princeton University Press, Princeton, N.J.; University of Tokyo Press, Tokyo, 1975. Annals of Mathematics Studies, No. 83

    1975

  4. [12]

    A relation between automorphic representations of (2) and (3)

    Stephen Gelbart and Herv\'e Jacquet. A relation between automorphic representations of (2) and (3) . Annales Scientifiques de I'ENS Volume 11 Issue 4, page 471-542

  5. [13]

    Forms of GL (2) \ from the analytic point of view

    Stephen Gelbart and Herv\'e Jacquet. Forms of GL (2) \ from the analytic point of view. In Automorphic forms, representations and L -functions ( P roc. S ympos. P ure M ath., O regon S tate U niv., C orvallis, O re., 1977), P art 1 , Proc. Sympos. Pure Math., XXXIII, pages 213...

    1977

  6. [14]

    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

  7. [15]

    Coefficients of M aass forms and the S iegel zero

    Jeffrey Hoffstein and Paul Lockhart. Coefficients of M aass forms and the S iegel zero. Ann. of Math. (2) , 140(1):161--181, 1994. With an appendix by Dorian Goldfeld, Hoffstein and Daniel Lieman

    1994

  8. [16]

    Level aspect subconvexity for R ankin- S elberg L -functions

    Roman Holowinsky and Ritabrata Munshi. Level aspect subconvexity for R ankin- S elberg L -functions. In Automorphic representations and L -functions , volume 22 of Tata Inst. Fundam. Res. Stud. Math. , pages 311--334. Tata Inst. Fund. Res., Mumbai, 2013

    2013

  9. [17]

    Triple product formula and the subconvexity bound of triple product L -function in level aspect

    Yueke Hu. Triple product formula and the subconvexity bound of triple product L -function in level aspect. Amer. J. Math. , 139(1):215--259, 2017

    2017

  10. [18]

    Triple product formula and mass equidistribution on modular curves of level N

    Yueke Hu. Triple product formula and mass equidistribution on modular curves of level N . Int. Math. Res. Not. , 9, 2899--2943, 2018

    2018

  11. [19]

    Mass equidistribution on the torus in the depth aspect

    Yueke Hu. Mass equidistribution on the torus in the depth aspect. Algebra and Number Theory , 14 (4), 2020

    2020

  12. [20]

    The subconvexity problem for Rankin-Selberg and triple product L -functions

    Yueke Hu, Philippe Michel, and Paul Nelson. The subconvexity problem for Rankin-Selberg and triple product L -functions. arXiv: 2207.14449v2

    arXiv

  13. [21]

    Trilinear forms and the central values of triple product L -functions

    Atsushi Ichino. Trilinear forms and the central values of triple product L -functions. Duke Math. J. , 145(2):281--307, 2008

    2008

  14. [22]

    Herve Jacquet and Joseph A. Shalika. On E uler products and the classification of automorphic representations. I . Amer. J. Math. , 103(3):499--558, 1981

    1981

  15. [23]

    Simultaneous non-vanishing of (3) (2) and (2) L -functions

    Rizwanur Khan. Simultaneous non-vanishing of (3) (2) and (2) L -functions. Math. Proc. Cambridge Philos. Soc. , 152(3):535--553, 2012

    2012

  16. [24]

    Rankin- S elberg L -functions in the level aspect

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

    2002

  17. [25]

    Subconvexity for twisted L -functions over number fields via shifted convolution sums

    Peter Maga. Subconvexity for twisted L -functions over number fields via shifted convolution sums. Acta Math. Hungar. , 151 (1) 232-257, 2017

    2017

  18. [26]

    Spectral reciprocity for the first moment of triple product L -functions and applications

    Xinchen Miao. Spectral reciprocity for the first moment of triple product L -functions and applications. preprint, 2024

    2024

  19. [27]

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

    Philippe 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

  20. [28]

    Recent progresses on the subconvexity problem

    Philippe Michel. Recent progresses on the subconvexity problem. Seminar Bourbaki , no.1190, (74), 2021-2022

    2021

  21. [29]

    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

  22. [30]

    Paul D. Nelson. Subconvex equidistribution of cusp forms: reduction to E isenstein observables. Duke Math. J. , 168(9):1665--1722, 2019

    2019

  23. [31]

    Nelson, Ameya Pitale and Abhishek Saha

    Paul D. Nelson, Ameya Pitale and Abhishek Saha. Bounds for Rankin–Selberg integrals and quantum unique ergodicity for powerful levels. J. Amer. Math. Soc. , 27 (2014), 147-191

    2014

  24. [32]

    Spectral reciprocity via integral representations

    Ramon Nunes. Spectral reciprocity via integral representations. Alg. Number Th. , 17(8):1381-1409, 2023

    2023

  25. [33]

    Archimedean L -factors on (n) (n) and Generalized Barnes Integrals

    Eric Stade. Archimedean L -factors on (n) (n) and Generalized Barnes Integrals. Israel Journal of Math. , 127(1), 201-219, 2002

    2002

  26. [34]

    Sparse equidistribution problems, period bounds and subconvexity

    Akshay Venkatesh. Sparse equidistribution problems, period bounds and subconvexity. Ann. of Math. (2) , 172(2):989--1094, 2010

    2010

  27. [35]

    Woodbury

    Michael C. Woodbury. Explicit trilinear forms and subconvexity of the triple product L -function. ProQuest LLC, Ann Arbor, MI, 2011. Thesis (Ph.D.)--The University of Wisconsin - Madison

    2011

  28. [36]

    Woodbury

    Michael C. Woodbury. Trilinear forms and subconvexity of the triple product L -functions. submitted, 2012

    2012

  29. [37]

    Woodbury

    Michael C. Woodbury. On the triple product formula: real local calculations. preprint 2017

    2017

  30. [38]

    Burgess-like subconvex bounds for GL _2 GL _1

    Han Wu. Burgess-like subconvex bounds for GL _2 GL _1 . Geom. Funct. Anal. , 24(3):968--1036, 2014

    2014

  31. [39]

    Periods and reciprocity I

    Raphaël Zacharias. Periods and reciprocity I . International Mathematics Research Notices , rnz100

  32. [40]

    Periods and reciprocity II

    Raphaël Zacharias. Periods and reciprocity II . arXiv: 1912.01512v2

    1912 arXiv

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