On the Triple Product Formula: Real Local Calculations

Woodbury, Michael

doi:10.1007/978-3-319-69712-3_16

Cited by 3 publications

(2 citation statements)

References 16 publications

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“…We should point out that in [Hsi17], he used Proposition 4.5 to compute the local period integrals for the nonarchimedean places under quite general settings. The computation for the real place was also treated in [Woo16].…”

Section: Introduction and Main Resultsmentioning

confidence: 99%

Special value formula for the twisted triple product $L$-function and an application to the restricted $L^2$-norm problem

Yang¹

2018

Preprint

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We establish explicit Ichino's formulae for the central values of the triple product L-functions with emphasis on the calculations for the real place. The key ingredient for our computations is Proposition 4.5 for the real place which is the main novelty of this article. As an application we prove the optimal upper bound of a sum of restricted 2-norms of the L 2 -normalized newforms on certain quadratic extensions with prime level and bounded spectral parameter following the methods in [Blo13]. Contents 1. Introduction and main results 1 2. Proof of Theorem 1.2 3 3. Special value formulae 12 4. Local calculations 16 5. Proof of Proposition 4.5 28 References 40Here the L-functions are complete L-functions.

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Section: Introduction and Main Resultsmentioning

confidence: 99%

Special value formula for the twisted triple product $L$-function and an application to the restricted $L^2$-norm problem

Yang¹

2018

Preprint

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“…When k 1 = k 2 , Watson [28] shows that I * ∞ = 1. For the general case, Woodbury [29] and Cheng [5] calculated for the real local place and show that I * ∞ = 2 −k2+k1 . We have that…”

Section: Integral Triple Product Identitiesmentioning

confidence: 99%

Dissipation of correlations of holomorphic cusp forms

Constantinescu¹

2021

Preprint

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We obtain a generalisation of the Quantum Unique Ergodicity for holomorphic cusp forms on SL 2 (Z)\H in the weight aspect. We show that correlations of masses coming from off-diagonal terms dissipate as the weight tends to infinity. This corresponds to classifying the possible quantum limits along any sequence of Hecke eigenforms of increasing weight.Our new ingredient is to incorporate the spectral theory of weight k automorphic functions to the method of Holowinsky-Soundararajan. For Holowinsky's shifted convolution sums approach, we need to develop new bounds for the Fourier coefficients of weight k cusp forms. For Soundararajan's subconvexity approach, we use Ichino's formula for evaluating triple product integrals.

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Special value formula for the twisted triple product L-function and an application to the restricted L ²-norm problem

Cheng

2020

Forum Mathematicum

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We establish explicit Ichino’s formulae for the central values of the triple product L-functions with emphasis on the calculations for the real place. The key ingredient for our computations is Proposition 8 which generalizes a result in [P. Michel and A. Venkatesh, The subconvexity problem for {\rm GL}_{2}, Publ. Math. Inst. Hautes Études Sci. 111 2010, 171–271]. As an application we prove the optimal upper bound of a sum of restricted {L^{2}}-norms of the {L^{2}}-normalized newforms on certain quadratic extensions with prime level and bounded spectral parameter following the methods in [V. Blomer, On the 4-norm of an automorphic form, J. Eur. Math. Soc. (JEMS) 15 2013, 5, 1825–1852].

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On the Triple Product Formula: Real Local Calculations

Cited by 3 publications

References 16 publications

Special value formula for the twisted triple product $L$-function and an application to the restricted $L^2$-norm problem

Special value formula for the twisted triple product $L$-function and an application to the restricted $L^2$-norm problem

Dissipation of correlations of holomorphic cusp forms

Special value formula for the twisted triple product L-function and an application to the restricted L ²-norm problem

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On the Triple Product Formula: Real Local Calculations

Cited by 3 publications

References 16 publications

Special value formula for the twisted triple product $L$-function and an application to the restricted $L^2$-norm problem

Special value formula for the twisted triple product $L$-function and an application to the restricted $L^2$-norm problem

Dissipation of correlations of holomorphic cusp forms

Special value formula for the twisted triple product L-function and an application to the restricted L 2-norm problem

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Product

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Special value formula for the twisted triple product L-function and an application to the restricted L ²-norm problem