Moments of central $L$-values for Maass forms over imaginary quadratic fields
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- by Sheng-Chi Liu and Zhi Qi PDF
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Abstract:
In this paper, over imaginary quadratic fields, we consider the family of $L$-functions $L (s, f)$ 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 $L\big (\frac 1 2, f\big )$, which can be applied to prove that at least $33 %$ of $L\big (\frac 1 2, f\big )$ with $t_f \leqslant T$ are non-vanishing as $T \rightarrow \infty$. Our main tools are the spherical Kuznetsov trace formula and the Voronoï summation formula over imaginary quadratic fields.References
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Additional Information
- Sheng-Chi Liu
- Affiliation: Department of Mathematics and Statistics, Washington State University, Pullman, Washington 99164-3113
- MR Author ID: 825335
- Email: scliu@math.wsu.edu
- Zhi Qi
- Affiliation: School of Mathematical Sciences, Zhejiang University, Hangzhou 310027, People’s Republic of China
- MR Author ID: 963773
- ORCID: 0000-0002-2454-3291
- Email: zhi.qi@zju.edu.cn
- Received by editor(s): August 18, 2021
- Received by editor(s) in revised form: October 19, 2021
- Published electronically: January 20, 2022
- Additional Notes: Zhi Qi is the corresponding author
The first author was supported by a grant (#344139) from the Simons Foundation. The second author was supported by a grant (#12071420) from the National Natural Science Foundation of China - © Copyright 2022 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 375 (2022), 3381-3410
- MSC (2020): Primary 11F67, 11F12
- DOI: https://doi.org/10.1090/tran/8588
- MathSciNet review: 4402665