A Mean Value Result for a Product of Gl(2) and Gl(3)  ‐functions

Balkanova, Olga; Bhowmik, Gautami; Frolenkov, Dmitry; Raulf, Nicole

doi:10.1112/s002557931900010x

Cited by 3 publications

(9 citation statements)

References 23 publications

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“…This possibility does not seem to be within the scope of Michel-Venkatesh's period formalism. Such moment was recently studied in [1,2]. It should be possible to generalize these results over general number fields in the light of this possible application of Theorem 2.10.…”

mentioning

confidence: 87%

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On Motohashi’s formula

2022

Trans. Amer. Math. Soc.

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We complement and offer a new perspective of the proof of a Motohashi-type formula relating the fourth moment of L L -functions for G L 1 GL_1 with the third moment of L L -functions for G L 2 GL_2 over number fields, studied earlier by Michel-Venkatesh and Nelson. Our main tool is a new type of pre-trace formula with test functions on M 2 ( A ) M_2(\mathbb {A}) instead of G L 2 ( A ) GL_2(\mathbb {A}) , on whose spectral side the matrix coefficients are replaced by the standard Godement-Jacquet zeta integrals. This is also a generalization of Bruggeman-Motohashi’s other proof of Motohashi’s formula. We give a variation of our method in the case of division quaternion algebras instead of M 2 M_2 , yielding a new spectral reciprocity, for which we are not sure if it is within the period formalism given by Michel-Venkatesh. We also indicate a further possible generalization, which seems to be beyond what the period method can offer.

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mentioning

confidence: 87%

“…(1) In the case Ω | A × = 1, Ω can be viewed as an automorphic representation of F × \E × E 1 , where E 1 is the F-subgroup of elements in E × with norm 1. Then we have the theta lift Θ 1 (Ω) to the metaplectic group Mp via the Weil representation r 1 of E 1 × Mp.…”

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confidence: 99%

On Motohashi’s formula

2022

Trans. Amer. Math. Soc.

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“…It follows from and the Phragmen–Lindelöf principle that for any

ε > 0

and

ℜ u > 0

the following upper bound holds

{scriptL}_{n} false(1 / 2 + u false) ≪ {| n |}^{\max (θ (1 - 2 ℜ u), 0) + ε} .

The generalised Dirichlet

L

‐function

{scriptL}_{n} false(s false)

shows up in the Fourier expansion of a linear combination of half‐integral weight Eisenstein series. To be more precise, we briefly summarise the principal arguments of [, Section 2]: let

{normalΓ}_{0} false(4 false)

be the Hecke congruence subgroup of level

4

and

ν

be the weight

1 / 2

multiplier system related to the theta series

θ false(z false) : = y^{1 / 4} \sum_{m \in Z} e^{2 π i m^{2} z} .

It is well know that

{normalΓ}_{0} false(4 false)

has three cusps which we denote by

{fraktura}_{1} = \infty

{fraktura}_{2} = 0

and

{fraktura}_{3} = 1 / 2

. Then for a cusp

fraktura

{normalΓ}_{0} false(4 false)

, we define

E_{fraktura} false(z; s; 1 / 2 false)

to be the Eisenstein series of weight

1 / 2 ...

…”

Section: Generalised Dirichlet L‐functionsmentioning

confidence: 99%

“…The mixed moment with an extra smooth average over weight was studied in by combining an explicit formula for the first moment of symmetric square

L

‐functions and an approximate functional equation for the Hecke

L

‐function. This approach along with the Liouville–Green method appeared to be quite effective, producing an asymptotic formula with an arbitrary power saving error term.…”

Section: Introductionmentioning

confidence: 99%

“…However, the same problem without the extra smooth averaging is much more difficult since in this case

L

‐functions are averaged over the family of size

k

instead of

k^{2}

which means that the non‐diagonal terms require more involved analysis. For this reason, we modify the methods of , relying now entirely on analytic continuation. More precisely, we prove an explicit formula for the mixed moment , which contains the diagonal main term of size

\log k

, the non‐diagonal main term of size

k^{- 1 / 2}

and dual mixed moments weighted by

_{3} F_{2}

hypergeometric functions.…”

Section: Introductionmentioning

confidence: 99%

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Mixed moment of GL(2) and GL(3)L‐functions

Balkanova

Bhowmik

Frolenkov

et al. 2019

Proc. London Math. Soc.

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Let frakturf run over the set H4k of primitive cusp forms of level one and weight 4k, k∈N. We prove an explicit formula for the mixed moment of the Hecke L‐function L(f,1/2) and the symmetric square L‐function L(prefixsym2f,1/2), relating it to the dual mixed moment of a double Dirichlet series and the Riemann zeta function weighted by the 3F2 hypergeometric function. Analysing the corresponding special functions by means of the Liouville–Green approximation followed by the saddle point method, we prove that the initial mixed moment is bounded above by log3k.

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A Voronoi summation formula for non-holomorphic Maass forms of half-integral weight

Balkanova,

Frolenkov

2023

Monatsh Math

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A Mean Value Result for a Product of Gl(2) and Gl(3) ‐functions

Cited by 3 publications

References 23 publications

On Motohashi’s formula

On Motohashi’s formula

Mixed moment of GL(2) and GL(3)L‐functions

A Voronoi summation formula for non-holomorphic Maass forms of half-integral weight

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