LOW‐LYING ZEROS OF
            <i>L</i>
            ‐FUNCTIONS FOR MAASS FORMS OVER IMAGINARY QUADRATIC FIELDS

Liu, Sheng‐Chi; Qi, Zhi

doi:10.1112/mtk.12041

Cited by 5 publications

(5 citation statements)

References 19 publications

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“…For t real, the Bessel kernel B it pxq appears in a certain Bessel integral Hpxq in Kuznetsov which is well understood by the works of [IJ,Li1,You2,LQ1]. Moreover, the Bessel kernel B 0 pxq arises in the Hankel transform in Voronoï, and the following formulae will be crucial in our analysis:…”

Section: Preliminariesmentioning

confidence: 99%

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Proof of the Strong Ivić Conjecture for the Cubic Moment of Maass-form $L$-functions

Qi¹

2022

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In this paper, we prove the following asymptotic formula for the spectral cubic moment of central L-values:where f ranges in an orthonormal basis of (even) Hecke-Maass cusp forms, and P 3 is a certain polynomial of degree 3. It improves on the error term OpT 8{7`ε q in a paper of Ivić and hence confirms his strong conjecture for the cubic moment. This is the first time that the (strong) moment conjecture is fully proven in a cubic case. Moreover, we establish the short-interval variant of the above asymptotic formula on intervals of length as short as T ε .

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Section: Preliminariesmentioning

confidence: 99%

“…For the analysis of Hp˘x 2 q, the most important are certain integral representations. The reader is referred to [Li1, § §4, 5], [You2,§7], [LQ1,Appendix], and [Qi, §8.1] for more details, and also [LQ2,§7] for a summary (although hptq may vary in different settings).…”

Section: Refined Analysis For the Bessel Integralmentioning

confidence: 99%

Proof of the Strong Ivić Conjecture for the Cubic Moment of Maass-form $L$-functions

Qi¹

2022

Preprint

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“…Note that c 0 " γ 1 and c 2 " c 1 {2 a |d F |. By the discussions below [LQ,Lemma 2.2], it is known that the lower bound |ζ F p1 `2itq| Ï logp3 `|t|q holds, and hence ωptq Î log 2 p3 `|t|q. (3.14) 3.4.…”

Section: Automorphic Forms On Glmentioning

confidence: 99%

“…By the works in [LQ,Qi3], the Bessel integrals Hpxq and Hpzq in the Kuznetsov trace formula are well understood, and their results will be recollected in the next 2 Note that the 1 in [BHS,(3.4)] should be the characteristic function.…”

Section: Asymptotics For Bessel Kernelsmentioning

confidence: 99%

Moments of Central $L$-values for Maass Forms over Imaginary Quadratic Fields

Liu¹,

Qi²

2022

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In this paper, over imaginary quadratic fields, we consider the family of L-functions Lps, f q for an orthonormal basis of spherical Hecke-Maass forms f with Archimedean parameter t f . We establish asymptotic formulae for the twisted first and second moments of the central values L `1 2 , f ˘, which can be applied to prove that at least 33% of L `1 2 , f ˘with t f ď T are non-vanishing as T Ñ 8. Our main tools are the spherical Kuznetsov trace formula and the Voronoï summation formula over imaginary quadratic fields.

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“…Later, Iwaniec, Luo and Sarnak [19] gave densities of low-lying zeros of the standard Lfunctions and those of symmetric square L-functions associated with holomorphic elliptic cusp forms both in the weight aspect and the level aspect, assuming GRH of several Lfunctions. Inspired by their study, densities of low-lying zeros of families of automorphic L-functions have been investigated in several settings such as Hilbert modular forms ( [28]), Siegel modular forms of degree 2 ( [24], [25]), and Hecke-Maass forms ( [1], [2] [16], [29], [31], [36]). As of now, the broadest setting for low-lying zeros of automorphic L-functions was setteled by Shin and Templier [40].…”

Section: Introductionmentioning

confidence: 99%

Low-lying zeros of symmetric power $L$-functions weighted by symmetric square $L$-values

Sugiyama¹

2021

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For a totally real number field F and its adèle ring A F , let π vary in the set of irreducible cuspidal automorphic representations of PGL 2 (A F ) corresponding to primitive Hilbert modular forms of a fixed weight. Then, we determine the symmetry type of the one-level density of low-lying zeros of the symmetric power L-functions L(s, Sym r (π)) weighted by special values of symmetric square L-functions L( z+1 2 , Sym 2 (π)) at z ∈ [0, 1] in the level aspect. If 0 < z 1, our weighted density in the level aspect has the same symmetry type as Ricotta and Royer's density of low-lying zeros of symmetric power L-functions for F = Q with harmonic weight. Hence our result is regarded as a zinterpolation of Ricotta and Royer's result. If z = 0, density of low-lying zeros weighted by central values is a different type only when r = 2, and it does not appear in random matrix theory as Katz and Sarnak predicted. Moreover, we propose a conjecture on weighted density of low-lying zeros of L-functions by special L-values.In the latter part, Appendices A, B and C are dedicated to the comparison among several generalizations of Zagier's parameterized trace formula. We prove that the explicit Jacquet-Zagier type trace formula (the ST trace formula) by Tsuzuki and the author recovers all of Zagier's, Takase's and Mizumoto's formulas by specializing several data. Such comparison is not so straightforward and includes non-trivial analytic evaluations.

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LOW‐LYING ZEROS OF L ‐FUNCTIONS FOR MAASS FORMS OVER IMAGINARY QUADRATIC FIELDS

Cited by 5 publications

References 19 publications

Proof of the Strong Ivić Conjecture for the Cubic Moment of Maass-form $L$-functions

Proof of the Strong Ivić Conjecture for the Cubic Moment of Maass-form $L$-functions

Moments of Central $L$-values for Maass Forms over Imaginary Quadratic Fields

Low-lying zeros of symmetric power $L$-functions weighted by symmetric square $L$-values

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