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Double square moments and subconvexity bounds for Rankin-Selberg L-functions of holomorphic cusp forms

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Abstract

Let f and g be holomorphic cusp forms of weights k1 and k2 for the congruence subgroups Γ0(N1) and Γ0 (N2), respectively. In this paper the square moment of the Rankin-Selberg L-function for f and g in the aspect of both weights in short intervals is bounded, when kε1 ≪ k2 ≪ k11-ε. These bounds are the mean Lindelöf hypothesis in one case and subconvexity bounds on average in other cases. These square moment estimates also imply subconvexity bounds for individual L(12 + it, f × g) for all g when f is chosen outside a small exceptional set. In the best case scenario the subconvexity bound obtained reaches the Weyl-type bound proved by Lau et al. (2006) in both the k1 and k2 aspects.

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Acknowledgements

The first author was supported by National Natural Science Foundation of China (Grant No. 11531008), Ministry of Education of China (Grant No. IRT16R43) and Taishan Scholar Project of Shandong Province. The second author was supported by National Natural Science Foundation of China (Grant No. 11601271), China Postdoctoral Science Foundation (Grant No. 2016M602125) and China Scholarship Council (Grant No. 201706225004). The second author is grateful to the Department of Mathematics, The University of Iowa, for hospitality in his visit during which the present work was done. The authors thank anonymous referees for numerous helpful suggestions.

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  1. These authors contributed equally to this work.

Authors and Affiliations

  1. School of Mathematics, Shandong University, Jinan, 250100, China

    Jianya Liu

  2. School of Mathematics and Statistics, Shandong University, Weihai, 264209, China

    Haiwei Sun

  3. Department of Mathematics, The University of Iowa, Iowa City, IA, 52242-1419, USA

    Haiwei Sun & Yangbo Ye

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  1. Jianya Liu
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  2. Haiwei Sun
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  3. Yangbo Ye
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Correspondence to Yangbo Ye.

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Liu, J., Sun, H. & Ye, Y. Double square moments and subconvexity bounds for Rankin-Selberg L-functions of holomorphic cusp forms. Sci. China Math. 63, 823–844 (2020). https://doi.org/10.1007/s11425-018-9380-6

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  • DOI: https://doi.org/10.1007/s11425-018-9380-6

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