Special value formula for the twisted triple product L-function and an application to the restricted L2-norm problem

Yao Cheng

doi:10.1515/forum-2018-0292

Article

Special value formula for the twisted triple product L-function and an application to the restricted L²-norm problem

Yao Cheng

Published/Copyright: October 10, 2020

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From the journal Forum Mathematicum Volume 33 Issue 1

Abstract

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 GL2, Publ. Math. Inst. Hautes Études Sci. 111 2010, 171–271]. As an application we prove the optimal upper bound of a sum of restricted L2-norms of the L2-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].

Keywords: Special value formula

MSC 2010: 11F67; 11F41

Communicated by Jan Bruinier

A Whittaker functions of GL2 over archimedean local fields

The purpose of this appendix is to prove Proposition 1 and equation (2.4) for the archimedean case. So we let F=ℝ or ℂ in this appendix. Let ψ be a non-trivial additive character of F and let π be an irreducible admissible generic representation of GL2⁢(F) with the minimal weight k≥0. We may assume π to be a constituent of ρ⁢(μ,ν) for some characters μ,ν.

A.1 Whittaker functional

Let (rψ,𝒮⁢(F2)) be the Weil representation of GL2⁢(F) defined in [37, Section 1] and let 𝒮⁢(F2,ψ) be the subspace defined by (6.12). For Φ∈𝒮⁢(F2), we define its partial Fourier transform by

(A.1)Φ∼⁢(x,y)=∫FΦ⁢(x,u)⁢ψ⁢(y⁢u)⁢𝑑u.

Here du is self-dual with respect to ψ. Note that 𝒮⁢(F2,ψ) is invariant under the partial Fourier transform and we have (rψ⁢(g)⁢Φ)∼=ρ⁢(g)⁢Φ∼ ([37, Proposition 1.6]). For Φ∈𝒮⁢(F2) and g∈GL2⁢(F), we define

(A.2)WΦ⁢(g)=μ⁢(det⁢(g))⁢|det⁢(g)|F12⁢∫F×rψ⁢(g)⁢Φ⁢(t,t-1)⁢μ⁢ν-1⁢(t)⁢d×⁢t.

Then WΦ∈𝒲⁢(ψ) in the notation of Section 2.4. Following [37], we set 𝒲⁢(μ,ν;ψ)={WΦ∣Φ∈𝒮⁢(F2,ψ)}. Then we have 𝒲⁢(μ,ν;ψ)=𝒲⁢(ν,μ;ψ) by [37, Proposition 3.4]. Moreover, by [37, Lemmas 5.13.1, 6.3.1], if |μ⁢ν-1⁢(y)|=|y|Fr for some r>-1, then L⁢(fΦ∼):=WΦ defines an GL2⁢(F)-isomorphism from ρ⁢(μ,ν) onto 𝒲⁢(μ,ν;ψ), where fΦ=fΦ(s)|s=12 is the Godement section defined in Section 6.3.2.^[10] When r≤-1, the map L still gives rise to an GL2⁢(F)-isomorphism when ρ⁢(μ,ν) is irreducible, but now one needs to use the analytic continuation of the Godement sections. In particular, 𝒲⁢(μ,ν;ψ)=𝒲⁢(π,ψ) if π≅π⁢(μ,ν) is a principal series representation. On the other hand, if π is a discrete series representation of GL2⁢(ℝ) (so that k≥2), then 𝒲⁢(π,ψ)⊂𝒲⁢(μ,ν;ψ) and WΦ∈𝒲⁢(π,ψ) if and only if

∫ℝxn⁢∂m∂⁡y⁢Φ⁢(x,0)⁢𝑑x=0

for every non-negative integers n,m with m+n=k-2 ([37, Corollary 5.14]).

In the following, we assume that ψ is given by (6.1). We also fix the choices of Haar measures dt and d×⁢t as follows. Let dt be the usual Lebesgue measure on ℝ and be twice of the usual Lebesgue measure on ℂ. On F×, we take d×⁢t=ζF⁢(1)⁢|t|F-1⁢d⁢t. Recall that

𝒮⁢(ℝ2,ψ)=⊕a,b≥0ℂ⁢x1a⁢x2b⁢e-π⁢(x12+x22) and 𝒮⁢(ℂ2,ψ)=⊕a,b,c,d≥0ℂ⁢x1a⁢x¯1b⁢x2c⁢x¯2d⁢e-2⁢π⁢(x1⁢x¯1+x2⁢x¯2).

The following lemma is the key of this appendix.

Lemma 1.

Let notations be as above.

Suppose that F=ℝ and μν-1=|⋅|2⁢ssgnk for some s∈ℂ and k∈ℤ. Let
Φ⁢(x1,x2)=x1a⁢x2b⁢e-π⁢(x12+x22)
for some integers a,b≥0. Then
WΦ⁢((y001))={2⁢μ⁢(y)⁢|y|12-s-a-b2⁢ya⁢Ks+a-b2⁢(2⁢π⁢|y|)if a-b≡k⁢(mod⁢2),0𝑜𝑡ℎ𝑒𝑟𝑤𝑖𝑠𝑒.
Suppose that F=ℂ and μ⁢ν-1⁢(y)=|y|ℂ2⁢s⁢(y/|y|ℂ12)k for some s∈ℂ and k∈ℤ. Let
Φ⁢(x1,x2)=x1a⁢x¯1b⁢x2c⁢x¯2d⁢e-2⁢π⁢(x1⁢x¯1+x2⁢x¯2)
for some integers a,b,c,d≥0. Then
WΦ⁢((y001))={4⁢μ⁢(y)⁢|y|ℂ12-s-a+b-c-d4⁢ya⁢y¯b⁢K2⁢s+a+b-c-d2⁢(4⁢π⁢|y|ℂ12)if a-b-c+d+k=0,0𝑜𝑡ℎ𝑒𝑟𝑤𝑖𝑠𝑒.

Proof.

Let α,β be positive real numbers and s∈ℂ. We first show that

(A.3)∫0∞e-α⁢(β⁢r2+r-2)⁢rs-1⁢𝑑r=β-s4⁢Ks2⁢(2⁢α⁢β12).

Here dr is the usual Lebesgue measure on ℝ>0 and β12 and β1/4 are positive real numbers. By changing the variable r2↦r, we find that

∫0∞e-α⁢(β⁢r2+r-2)⁢rs-1⁢𝑑r=12⁢∫0∞e-α⁢(β⁢r+r-1)⁢rs2-1⁢𝑑r=12⁢∫0∞e-α⁢β12⁢(β12⁢r+β-12⁢r-1)⁢rs2-1⁢𝑑r.

Then (A.3) now follows from changing the variable β12⁢r↦r again and (2.5).

Recall that for y∈F×, we have

rψ⁢((y001))⁢Φ⁢(x1,x2)=Φ⁢(y⁢x1,x2).

It follows that Φ⁢(y⁢t,t-1) is equal to

ya⁢ta-b⁢e-π⁢(y2⁢t2+t-2) or ya⁢y¯b⁢ta-c⁢t¯b-d⁢e-2⁢π⁢(|y⁢t|ℂ+|t|ℂ-1)

according to F=ℝ or ℂ. We only compute the case F=ℂ as the other case is similar (and easier). Note that d×⁢t=2⁢π-1⁢d×⁢r⁢d⁢θ if we use the polar coordinate t=r⁢ei⁢θ. Here d×⁢r=d⁢rr and dr is the usual Lebesgue measure on ℝ>0. Let β=|y|ℂ. By definition, we have

WΦ⁢((y001))=μ⁢(y)⁢|y|12⁢∫ℂ×Φ⁢(y⁢t,t-1)⁢μ⁢ν-1⁢(t)⁢d×⁢t

=μ⁢(y)⁢|y|ℂ12⁢ya⁢y¯b⁢∫02⁢πei⁢(a-b-c+d+k)⁢θ⁢𝑑θ⁢∫0∞e-2⁢π⁢(β⁢r2+r-2)⁢r4⁢s+a+b-c-d-1⁢2⁢π-1⁢𝑑r.

Now the assertion follows from (A.3). This completes the proof. ∎

A.2 Explicit formulae

Now we prove Proposition 1. We have two cases depending on F=ℝ or ℂ.

A.2.1 F=ℝ

Suppose that F=ℝ. As we mentioned, the proof of the first case is given by [12, Lemma 3.3] so that we assume in the following that π≅π⁢(μ,ν) is a principal series representation. Let n be a non-negative integer and let Φn∈𝒮⁢(ℝ2,ψ) be the element such that

Φn⁢(x1,x2)∼=(x1+i⁢x2)n⁢e-π⁢(x12+x22).

It follows from the relation (rψ⁢(g)⁢Φ)∼=ρ⁢(g)⁢Φ∼ that rψ⁢(k⁢(θ))⁢Φn⁢(x1,x2)=ei⁢n⁢θ⁢Φn⁢(x1,x2). Suppose now that n≡k⁢(mod⁢2). We first show that WΦn defines a non-zero weight n element in 𝒲⁢(π,ψ), i.e. WΦn is a non-zero generator of 𝒱ψ⁢(π,n). Indeed, we have

ρ⁢(k⁢(θ))⁢WΦn=Wrψ⁢(k⁢(θ))⁢Φn=ei⁢k⁢θ⁢WΦn

by the above observation. It remains to show that WΦn is non-zero. Let μν-1=|⋅|2⁢ssgnk for some s∈ℂ. Since π is a principal series representation, the map L is an GL2⁢(F)-isomorphism and we have L⁢(fΦn∼)=WΦn. To show that WΦn≠0, it suffices to show that fΦn∼≠0. However, one checks easily that

fΦn∼⁢(I2)=(-1)n⁢ζℝ⁢(2⁢s+n+1)≠0.

We now compute Φk and Φ2. By easy computations, we find that

Φk⁢(x1,x2)=(x1+x2)k⁢e-π⁢(x12+x22) and Φ2⁢(x1,x2)=(x12+2⁢x1⁢x2+x22-12⁢π)⁢e-π⁢(x12+x22).

It then follows from Lemma 1 (1) that Wπ in Proposition 1 (2) is equal to 2-1⁢WΦk. On the other hand, if k=0, then we have

ρ⁢(V+)⁢Wπ=2-1⁢ρ⁢(V+)⁢WΦ0=2-1⁢Wrψ⁢(V+)⁢Φ0

and by a direct calculation that rψ⁢(V+)⁢Φ0=-2⁢π⁢Φ2. This shows Proposition 1 (2).

A.2.2 F=ℂ

Suppose that F=ℂ and π≅π⁢(μ,ν) is a principal series representation. Since π⁢(μ,ν)≅π⁢(ν,μ), we may assume that μ⁢ν-1⁢(ei⁢θ)=ei⁢k⁢θ by the uniqueness of the Whittaker model. Let n≥k be an integer with the same parity. We describe how to construct a non-zero element in 𝒱ψ⁢(π,n) following [36, Section 18]. The construction is similar to the case when F=ℝ. Write n=k+2⁢m for some non-negative integer m and put

Ψ→k,n⁢(x1,x2)=(x2⁢X-x1⁢Y)m⁢(x¯1⁢X+x¯2⁢Y)k+m⁢e-2⁢π⁢(x1⁢x¯1+x2⁢x¯2)∈𝒮⁢(ℂ2,ψ)⊗ℒn⁢(ℂ).

One checks easily that

(A.4)Ψ→k,n⁢((x1,x2)⁢u)=ρn⁢(u-1)⁢Ψ→k,n⁢((x1,x2))

for every u∈SU⁢(2). The partial Fourier transform (A.1) can be extended to 𝒮⁢(ℂ2,ψ)⊗ℒn⁢(ℂ) by performing the transform on the coefficients. Let Φ→k,n∈𝒮⁢(ℂ2,ψ)⊗ℒn⁢(ℂ) so that Φ→k,n∼=Ψ→k,n. The Weil representation (rψ,𝒮⁢(ℂ2)) of GL2⁢(ℂ) can be extended to the space 𝒮⁢(ℂ2)⊗ℒn⁢(ℂ) in a similar way as well as the integration (A.2) and the Godement section defined in Section 6.3.2. By (A.4), we see that

fΨ→k,n:=fΨ→k,n(s)|s=12∈(ℬ⁢(μ,ν)⊗ℒn⁢(ℂ))SU⁢(2).

Moreover, fΨ→k,n⁢(I2)≠0 by [36, Lemma 18.4] and the assumption that π≅π⁢(μ,ν) is a principal series representation. It follows that WΦ→k,n:=L⁢(fΨ→k,n) defines a non-zero element in 𝒱ψ⁢(π,n). More precisely, WΦ→k,n is given by the integral (A.2) with Φ replaced by Φ→k,n. To prove Proposition 1 (3), it remains to compute Φ→k,k and Φ→0,2. By routine calculations, we find that

Φ→k,k⁢(x1,x2)=∑j=0k(kj)⁢(--1)k-j⁢x¯1j⁢x2k-j⁢e-2⁢π⁢(x1⁢x¯1+x2⁢x¯2)⁢Xj⁢Yk-j

and

Φ→0,2⁢(x1,x2)=(--1)⁢x¯1⁢x¯2⁢e-2⁢π⁢(x1⁢x¯1+x2⁢x¯2)⁢X2+(12⁢π-x1⁢x¯1-x2⁢x¯2)⁢e-2⁢π⁢(x1⁢x¯1+x2⁢x¯2)⁢X⁢Y+(-1)⁢x1⁢x2⁢e-2⁢π⁢(x1⁢x¯1+x2⁢x¯2)⁢Y2.

It then follows from Lemma 1 (2) that W→π=2-2⁢μ⁢ν⁢(-1)⁢(-1)k⁢WΦ→k,k. This shows Proposition 1 (3). Similarly when k=0, we have W→π(2)=2-2⁢WΦ→0,2. This completes the proof of Proposition 1.

A.3 Majorization of Whittaker functions

We prove (2.4) when F=ℝ or ℂ. We begin with a lemma.

Lemma 2.

Let s∈C with |Re(s)|=r and let r1>0 be such that r1≥r. Let φ be a function on R× defined by φ⁢(y)=yr1⁢Ks⁢(|y|). Then we have

φ⁢(y)≪r,r1exp⁢(-12⁢|y|).

Proof.

Since Ks⁢(z)=K-s⁢(z) and Ks⁢(|y|)≪KRe⁡(s)⁢(|y|), we may assume that s=r. Then the lemma follows immediately from the asymptotic form of Kr⁢(|y|). Indeed, when |y|→∞, we have

Kr⁢(|y|)∼π2⁢|y|⁢e-|y|,

while when 0<|y|≪1, we have

Kr⁢(|y|)∼{-ln⁢(|y|2)-γif r=0,Γ⁢(r)2⁢(2|y|)rif r>0.

Here γ stands for the Euler’s constant. This finishes the proof. ∎

Proposition 3.

Suppose that π is unitary. Let ϵ>0 and W∈W⁢(π,ψ). Then

W⁢((y001)⁢h)≪π,W,ϵ|y|F12-λ⁢(π)-ϵ⁢exp⁢(-d⁢|y|F1d)

for every h∈K. Here d=[F:R].

Proof.

Suppose that F=ℂ and π≅π⁢(μ,ν) with μ⁢ν-1⁢(y)=|y|ℂ2⁢s⁢(y/|y|ℂ12)k for some s∈ℂ and k∈ℤ. By the right K-finiteness, it suffices to prove the assertion when h is the identity element. By Lemma 1, we may assume

W⁢((y001))=μ⁢(y)⁢|y|ℂ12-s-a+b-c-d4⁢ya⋅y¯b⋅K2⁢s+a+b-c-d2⁢(4⁢π⁢|y|ℂ12).

Here a,b,c,d are non-negative integers. Note that |μ⁢(y)|=|y|ℂRe⁡(s) by (6.3). Let |y|ℂ=r2 for some r>0. We have

W⁢((y001))≪r1+a+b+c+d2⁢K2⁢Re⁡(s)+a+b-c-d2⁢(4⁢π⁢r).

Since π is unitary, we have λ(π)=|Re(s)|. In particular, λ⁢(π)-Re⁡(s)≥0. Let ϵ>0. We have

r1+a+b+c+d2⁢K2⁢Re⁡(s)+a+b-c-d2⁢(4⁢π⁢r)=r1-2⁢λ⁢(π)-2⁢ϵ⋅r2⁢λ⁢(π)+a+b+c+d2+2⁢ϵ⁢K2⁢Re⁡(s)+a+b-c-d2⁢(4⁢π⁢r)

≪a,b,c,d,λ⁢(π),ϵr1-2⁢λ⁢(π)-2⁢ϵ⁢e-2⁢r

by Lemma 2. This proves the case where F=ℂ.

Suppose that F=ℝ. If π is a discrete series representation, then the assertion follows from Proposition 1 (1). If π is a principal series representation, then it can be proved in a similar way. ∎

Acknowledgements

The present article presents part of my Ph.D. thesis and I would like to thank Professor Ming-Lun Hsieh for his support and guidance. Thanks also to Professor Sheng-Chi Liu for pointing out a possible application to the restricted L2-norm problem, and to professor P. Nelson for his comments. Finally, I would like thank the referee for many helpful comments and suggestions, which greatly improve the original draft.

References

[1] U. K. Anandavardhanan, R. Kurinczuk, N. Matringe, V. Secherre and S. Stevens, Galois self-dual cuspidal type and Asai local factors, preprint (2019), https://arxiv.org/abs/1807.07755v3. 10.4171/JEMS/1080Search in Google Scholar

[2] T. Asai, On certain Dirichlet series associated with Hilbert modular forms and Rankin’s method, Math. Ann. 226 (1977), no. 1, 81–94. 10.1007/BF01391220Search in Google Scholar

[3] W. D. Banks, Twisted symmetric-square L-functions and the nonexistence of Siegel zeros on GL⁢(3), Duke Math. J. 87 (1997), no. 2, 343–353. 10.1215/S0012-7094-97-08713-5Search in Google Scholar

[4] V. Blomer, Subconvexity for twisted L-functions on GL⁢(3), Amer. J. Math. 134 (2012), no. 5, 1385–1421. 10.1353/ajm.2012.0032Search in Google Scholar

[5] V. Blomer, On the 4-norm of an automorphic form, J. Eur. Math. Soc. (JEMS) 15 (2013), no. 5, 1825–1852. 10.4171/JEMS/405Search in Google Scholar

[6] V. Blomer and P. Maga, The sup-norm problem for PGL(4), Int. Math. Res. Not. IMRN 2015 (2015), no. 14, 5311–5332. 10.1093/imrn/rnu100Search in Google Scholar

[7] N. Burq, P. Gérard and N. Tzvetkov, Restrictions of the Laplace–Beltrami eigenfunctions to submanifolds, Duke Math. J. 138 (2007), no. 3, 445–486. 10.1215/S0012-7094-07-13834-1Search in Google Scholar

[8] W. Casselman, On Some Results of Atkin and Lehner, Math. Ann. 201 (1973), 301–314. 10.1007/BF01428197Search in Google Scholar

[9] J. Chai, Archimedean Derivatives and Rankin–Selberg Integrals, ProQuest LLC, Ann Arbor, 2012; Ph.D. thesis, The Ohio State University, 2012. Search in Google Scholar

[10] S.-Y. Chen, Pullback formulae for nearly holomorphic Saito–Kurokawa lifts, Manuscripta Math. 161 (2020), no. 3–4, 501–561. 10.1007/s00229-019-01111-2Search in Google Scholar

[11] S.-Y. Chen and Y. Cheng, On Deligne’s conjecture for certain automorphic L-functions for GL⁢(3)×GL⁢(2) and GL⁢(4), Doc. Math. 24 (2019), 2241–2297. 10.4171/dm/725Search in Google Scholar

[12] S.-Y. Chen, Y. Cheng and I. Ishikawa, Gamma factors for the Asai representation of GL2, J. Number Theory 209 (2020), 83–146. 10.1016/j.jnt.2019.08.008Search in Google Scholar

[13] 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. 10.2307/121132Search in Google Scholar

[14] Y. Z. Flicker, Twisted tensors and Euler products, Bull. Soc. Math. France 116 (1988), no. 3, 295–313. 10.24033/bsmf.2099Search in Google Scholar

[15] Y. Z. Flicker, On zeroes of the twisted tensor L-function, Math. Ann. 297 (1993), no. 2, 199–219. 10.1007/BF01459497Search in Google Scholar

[16] W. T. Gan, Trilinear forms and triple product epsilon factors, Int. Math. Res. Not. IMRN 2008 (2008), no. 15, Article ID rnn058. 10.1093/imrn/rnn058Search in Google Scholar

[17] W. T. Gan, The Shimura correspondence a la Waldspurger, Lecture Notes of Theta Festival, Pohang, 2011. Search in Google Scholar

[18] P. B. Garrett, Decomposition of Eisenstein series: Rankin triple products, Ann. of Math. (2) 125 (1987), no. 2, 209–235. 10.2307/1971310Search in Google Scholar

[19] S. Gelbart and H. Jacquet, A relation between automorphic representations of GL⁢(2) and GL⁢(3), Ann. Sci. Éc. Norm. Supér. (4) 11 (1978), no. 4, 471–542. 10.24033/asens.1355Search in Google Scholar

[20] S. Gelbart and H. Jacquet, Forms of GL⁢(2) from the analytic point of view, Automorphic Forms, Representations and L-Functions. Part 1 (Corvallis 1977), Proc. Sympos. Pure Math. 33, American Mathematical Society, Providence, (1979), 213–251. 10.1090/pspum/033.1/546600Search in Google Scholar

[21] B. H. Gross and S. S. Kudla, Heights and the central critical values of triple product L-functions, Compos. Math. 81 (1992), no. 2, 143–209. Search in Google Scholar

[22] M. Harris and S. S. Kudla, The central critical value of a triple product L-function, Ann. of Math. (2) 133 (1991), no. 3, 605–672. 10.2307/2944321Search in Google Scholar

[23] M. Harris and S. S. Kudla, On a conjecture of Jacquet, Contributions to Automorphic Forms, Geometry, and Number Theory, Johns Hopkins University Press, Baltimore (2004), 355–371. Search in Google Scholar

[24] H. Hida, p-ordinary cohomology groups for SL⁢(2) over number fields, Duke Math. J. 69 (1993), no. 2, 259–314. 10.1215/S0012-7094-93-06914-1Search in Google Scholar

[25] J. Hoffstein and P. Lockhart, Coefficients of Maass forms and the Siegel zero. With an appendix by Dorian Goldfeld, Hoffstein and Daniel Lieman, Ann. of Math. (2) 140 (1994), no. 1, 161–181. 10.2307/2118543Search in Google Scholar

[26] M. L. Hsieh, Hida families and p-adic triple L-functions, Amer. J. Math., to appear. 10.1353/ajm.2021.0011Search in Google Scholar

[27] Y. Hu, Triple product formula and the subconvexity bound of triple product L-function in level aspect, Amer. J. Math. 139 (2017), no. 1, 215–259. 10.1353/ajm.2017.0004Search in Google Scholar

[28] A. Ichino, Trilinear forms and the central values of triple product L-functions, Duke Math. J. 145 (2008), no. 2, 281–307. 10.1215/00127094-2008-052Search in Google Scholar

[29] A. Ichino and T. Ikeda, On the periods of automorphic forms on special orthogonal groups and the Gross–Prasad conjecture, Geom. Funct. Anal. 19 (2010), no. 5, 1378–1425. 10.1007/s00039-009-0040-4Search in Google Scholar

[30] A. Ichino and K. Prasanna, Period of quaternionic Simura varieties. I, Mem. Amer. Math. Soc., to appear. Search in Google Scholar

[31] T. Ikeda, On the functional equations of the triple L-functions, J. Math. Kyoto Univ. 29 (1989), no. 2, 175–219. 10.1215/kjm/1250520261Search in Google Scholar

[32] T. Ikeda, On the location of poles of the triple L-functions, Compos. Math. 83 (1992), no. 2, 187–237. Search in Google Scholar

[33] I. Ishikawa, On the construction of twisted triple product p-adic L-functions, Ph.D. thesis, Kyoto University, 2017. Search in Google Scholar

[34] H. Iwaniec and E. Kowalski, Analytic Number Theory, Amer. Math. Soc. Colloq. Publ. 53, American Mathematical Society, Providence, 2004. 10.1090/coll/053Search in Google Scholar

[35] H. Iwaniec, W. Luo and P. Sarnak, Low lying zeros of families of L-functions, Publ. Math. Inst. Hautes Études Sci. 91 (2000), 55–131. 10.1007/BF02698741Search in Google Scholar

[36] H. Jacquet, Automorphic Forms on GL⁢(2). II, Lecture Notes in Math. 278, Springer, Berlin, 1972. 10.1007/BFb0058503Search in Google Scholar

[37] H. Jacquet and R. P. Langlands, Automorphic Forms on GL⁢(2), Lecture Notes in Math. 114, Springer, Berlin, 1970. 10.1007/BFb0058988Search in Google Scholar

[38] H. Jacquet, I. I. Piatetskii-Shapiro and J. A. Shalika, Rankin–Selberg convolutions, Amer. J. Math. 105 (1983), no. 2, 367–464. 10.2307/2374264Search in Google Scholar

[39] A. C. Kable, Asai L-functions and Jacquet’s conjecture, Amer. J. Math. 126 (2004), no. 4, 789–820. 10.1353/ajm.2004.0030Search in Google Scholar

[40] H. H. Kim, Functoriality for the exterior square of GL4 and the symmetric fourth of GL2. With appendix 1 by Dinakar Ramakrishnan and appendix 2 by Kim and Peter Sarnak, J. Amer. Math. Soc. 16 (2003), no. 1, 139–183. 10.1090/S0894-0347-02-00410-1Search in Google Scholar

[41] H. H. Kim and F. Shahidi, Functorial products for GL2×GL3 and the symmetric cube for GL2. With an appendix by Colin J. Bushnell and Guy Henniart, Ann. of Math. (2) 155 (2002), no. 3, 837–893. 10.2307/3062134Search in Google Scholar

[42] A. W. Knapp, Local Langlands correspondence: The Archimedean case, Motives (Seattle 1991), Proc. Sympos. Pure Math. 55, Amer. Math. Soc., Providence, RI, (1994), 393–410. 10.1090/pspum/055.2/1265560Search in Google Scholar

[43] M. Krishnamurthy, The Asai transfer to GL4 via the Langlands–Shahidi method, Int. Math. Res. Not. IMRN 2003 (2003), no. 41, 2221–2254. 10.1155/S1073792803130528Search in Google Scholar

[44] X. Li, S.-C. Liu and M. P. Young, The L2 restriction norm of a Maass form on SLn+1⁢(ℤ), Math. Ann. 371 (2018), no. 3–4, 1301–1335. 10.1007/s00208-017-1591-6Search in Google Scholar

[45] X. Li and M. P. Young, The L2 restriction norm of a GL3 Maass form, Compos. Math. 148 (2012), no. 3, 675–717. 10.1112/S0010437X11007366Search in Google Scholar

[46] H. Y. Loke, Trilinear forms of 𝔤⁢𝔩2, Pacific J. Math. 197 (2001), no. 1, 119–144. 10.2140/pjm.2001.197.119Search in Google Scholar

[47] S. Marshall, Lp norms of higher rank eigenfunctions and bounds for spherical functions, J. Eur. Math. Soc. (JEMS) 18 (2016), no. 7, 1437–1493. 10.4171/JEMS/619Search in Google Scholar

[48] P. Michel and A. Venkatesh, The subconvexity problem for GL2, Publ. Math. Inst. Hautes Études Sci. 111 (2010), 171–271. 10.1007/s10240-010-0025-8Search in Google Scholar

[49] G. Molteni, L-functions: Siegel type theorems and structure theorems, Ph.D. thesis, 1999. Search in Google Scholar

[50] P. D. Nelson, Equidistribution of cusp forms in the level aspect, Duke Math. J. 160 (2011), no. 3, 467–501. 10.1215/00127094-144287Search in Google Scholar

[51] P. D. Nelson, A. Pitale and A. Saha, Bounds for Rankin-Selberg integrals and quantum unique ergodicity for powerful levels, J. Amer. Math. Soc. 27 (2014), no. 1, 147–191. 10.1090/S0894-0347-2013-00779-1Search in Google Scholar

[52] T. Orloff, Special values and mixed weight triple products (with an appendix by Don Blasius), Invent. Math. 90 (1987), no. 1, 169–188. 10.1007/BF01389036Search in Google Scholar

[53] I. Piatetski-Shapiro and S. Rallis, Rankin triple L functions, Compos. Math. 64 (1987), no. 1, 31–115. 10.1007/BFb0078126Search in Google Scholar

[54] A. A. Popa, Whittaker newforms for Archimedean representations, J. Number Theory 128 (2008), no. 6, 1637–1645. 10.1016/j.jnt.2007.06.005Search in Google Scholar

[55] D. Prasad, Trilinear forms for representations of GL⁢(2) and local ϵ-factors, Compos. Math. 75 (1990), no. 1, 1–46. Search in Google Scholar

[56] D. Prasad, Invariant forms for representations of GL2 over a local field, Amer. J. Math. 114 (1992), no. 6, 1317–1363. 10.2307/2374764Search in Google Scholar

[57] D. Prasad and R. Schulze-Pillot, Generalised form of a conjecture of Jacquet and a local consequence, J. Reine Angew. Math. 616 (2008), 219–236. 10.1515/CRELLE.2008.023Search in Google Scholar

[58] D. Ramakrishnan, Modularity of the Rankin–Selberg L-series, and multiplicity one for SL⁢(2), Ann. of Math. (2) 152 (2000), no. 1, 45–111. 10.2307/2661379Search in Google Scholar

[59] A. Saha, On sup-norms of cusp forms of powerful level, J. Eur. Math. Soc. (JEMS) 19 (2017), no. 11, 3549–3573. 10.4171/JEMS/746Search in Google Scholar

[60] R. Schmidt, Some remarks on local newforms for GL⁢(2), J. Ramanujan Math. Soc. 17 (2002), no. 2, 115–147. Search in Google Scholar

[61] F. Shahidi, A proof of Langlands’ conjecture on Plancherel measures; complementary series for p-adic groups, Ann. of Math. (2) 132 (1990), no. 2, 273–330. 10.2307/1971524Search in Google Scholar

[62] J. Tate, Number theoretic background, Automorphic Forms, Representations and L-Functions. Part 2 (Corvallis 1977), Proc. Sympos. Pure Math. 33, American Mathematical Society, Providence, (1979), 3–26. 10.1090/pspum/033.2/546607Search in Google Scholar

[63] J. B. Tunnell, Local ϵ-factors and characters of GL⁢(2), Amer. J. Math. 105 (1983), no. 6, 1277–1307. 10.2307/2374441Search in Google Scholar

[64] I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series, and Products, 7th ed., Academic Press, New York, 2007. Search in Google Scholar

[65] J.-L. Waldspurger, Sur les valeurs de certaines fonctions L automorphes en leur centre de symétrie, Compos. Math. 54 (1985), no. 2, 173–242. Search in Google Scholar

[66] T. C. Watson, Rankin Triple Products and Quantum Chaos, ProQuest LLC, Ann Arbor, 2002; Ph.D. thesis, Princeton University, 2002. Search in Google Scholar

[67] M. Woodbury, On the triple product formula: Real local calculations, L-Functions and Automorphic Forms, Contrib. Math. Comput. Sci. 10, Springer, Cham (2017), 275–298. 10.1007/978-3-319-69712-3_16Search in Google Scholar

Received: 2018-11-29

Revised: 2020-09-06

Published Online: 2020-10-10

Published in Print: 2021-01-01

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https://doi.org/10.1515/forum-2018-0292

Keywords for this article

Special value formula