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Pullback formulae for nearly holomorphic Saito–Kurokawa lifts

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Abstract

We give explicit pullback formulae for nearly holomorphic Saito–Kurokawa lifts restrict to product of upper half-plane against with product of elliptic modular forms. We generalize the formula of Ichino to modular forms of higher level and free the restriction on weights. The explicit formulae provide non-trivial examples for the refined Gan–Gross–Prasad conjecture for \((\mathrm{SO}_5,\mathrm{SO}_4)\) in the non-tempered cases. As an application, we obtain Deligne’s conjecture for critical values of certain automorphic L-functions for \(\mathrm{{GL}}_3 \times \mathrm{{GL}}_2\). We also expect to apply our pullback formulae to construct two-variables p-adic L-functions for \(\mathrm{{GL}}_3 \times \mathrm{{GL}}_2\) in the future.

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Acknowledgements

The results of this paper are part of the author’s Ph.D. thesis in National Taiwan University. The author would like to thank my advisor Ming-Lun Hsieh for the encouragement and help during the Ph.D. program. This work would been impossible without his guidance and insight on automorphic forms. The author also would like to thank Atsushi Ichino for the suggestions and sharing his program code for numerical examples. Finally the author thanks the referee for the comments on the previous version of this paper.

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  1. Institute of Mathematics, Academia Sinica, 6F, Astronomy-Mathematics Building, No. 1, Sec. 4, Roosevelt Road, Taipei, 10617, Taiwan, ROC

    Shih-Yu Chen

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Chen, SY. Pullback formulae for nearly holomorphic Saito–Kurokawa lifts. manuscripta math. 161, 501–561 (2020). https://doi.org/10.1007/s00229-019-01111-2

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