Abstract
In this paper, we study the non-vanishing of the Miyawaki type lift in various situations. In the case of GSpin(2, 10) constructed in Kim and Yamauchi (Math Z 288(1–2):415–437, 2018), we use the fact that the Fourier coefficient at the identity is closely related to the Rankin–Selberg L-function of two elliptic cusp forms. In the case of the original Miyawaki lifts of Siegel cusp forms, we use the fact that certain Fourier coefficients are the Petersson inner product which is non-trivial. This provides infinitely many examples of non-zero Miyawaki lifts. We give explicit examples of degree 24 and weight 24. We also prove a similar result for Miyawaki lifts for unitary groups. Especially, we obtain an unconditional result on non-vanishing of Miyawaki lifts for \(U(n+1,n+1)\) for each \(n\equiv 3\) mod 4. In the last section, we prove the non-vanishing of the Miyawaki lifts for infinitely many half-integral weight Siegel cusp forms. We give explicit examples of degree 16 and weight \(\frac{29}{2}\).
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References
Arthur, J.: The endoscopic classification of representations. Orthogonal and symplectic groups. American Mathematical Society Colloquium Publications, 61. American Mathematical Society, Providence, RI, (2013)
Atobe, H.: A theory of Miyawaki liftings: The Hilbert-Siegel case, preprint, arXiv:1712.03624
Atobe, H.: Applications of Arthur’s multiplicity formula to Siegel modular forms, preprint, arXiv:1810.09089
Atobe, H., Kojima, H.: On the Miyawaki lifts of Hermitian modular forms. J. Number. Theory 185, 281–318 (2018)
Bump, D.: Automorphic Forms and Representations, Cambridge Studies in Advanced Mathematics, 55. Cambridge University Press, Cambridge (1997)
Cohen, D.M., Resnikoff, H.L.: Hermitian quadratic forms and Hermitian modular forms. Pac. J. Math. 78, 329–337 (1978)
Freitag, E.: Siegelsche Modulfunktionen. Grundlehren der Mathematischen Wissenschaften, vol. 254, Springer, Berlin, (1983)
Gan, W.T., Gross, B.H., Prasad, D.: Symplectic local root numbers, central critical L-values, and restriction problems in the representation theory of classical groups. Astérisque. No. 346, 1–109 (2012)
Garrett, P.: Pullbacks of Eisenstein series; applications, Automorphic forms of several variables (Katata, 1983), 114-137, Progr. Math., 46, Birkhauser Boston, Boston, MA, (1984)
Hayashida, S.: Lifting from two elliptic modular forms to Siegel modular forms of half-integral weight of even degree. Doc. Math. 21, 125–196 (2016)
Hentschel, M., Krieg, A., Nebe, G.: On the classification of even unimodular lattices with a complex structure. Int. J. Number Theory 8(4), 983–992 (2012)
Ibukiyama, T.: On Jacobi forms and Siegel modular forms of half integral weights. Comment. Math. Univ. St. Paul 41, 109–124 (1992)
Ikeda, T.: On the lifting of elliptic cusp forms to Siegel cusp forms of degree \(2n\). Ann. Math. 154, 641–681 (2001)
Ikeda, T.: Pullback of the lifting of elliptic cusp forms and Miyawaki’s conjecture. Duke Math. J. 131, 469–497 (2006)
Ikeda, T.: On the lifting of Hermitian modular forms. Comp. Math. 144, 1107–1154 (2008)
Ikeda, T., Yamana, S.: On the lifting of Hilbert cusp forms to Hilbert–Siegel cusp forms, to appear in Ann. Sci. Ec. Norm. Sup
Kalinin, V.L.: Analytic properties of the convolution of Siegel modular forms of genus \(n\), Mat. Sb. (N.S.) 120 (162) (1983), no. 2, 200–206, 286–287
Karel, M.: Fourier coefficients of certain Eisenstein series. Ann. Math. 99, 176–202 (1974)
Kim, H., Yamauchi, T.: Cusp forms on the exceptional group of type \(E_7\). Compositio Math. 152, 223–254 (2016)
Kim, H., Yamauchi, T.: A Miyawaki type lift for \(GSpin(2,10)\). Math. Z. 288(1–2), 415–437 (2018)
Kim, H., Yamauchi, T.: Higher level cusp forms on the exceptional group of type \(E_7\) (submitted)
Koblitz, N.: Introduction to elliptic curves and modular forms. Second edition. Graduate Texts in Mathematics, 97. Springer, New York, (1993)
Kowalski, E., Michel, P., VanderKam, J.: Rankin–Selberg L-functions in the level aspect. Duke Math. J. 114(1), 123–191 (2002)
LMFDB, the database for Siegel modular forms, available on http://www.lmfdb.org/ModularForm/GSp/Q/
Miyawaki, I.: Numerical examples of Siegel cusp forms of degree 3 and their zeta-functions. Mem. Fac. Sci. Kyushu Univ. Ser. A 46(2), 307–339 (1992)
Schmidt, R.: Some remarks on local newforms for \(GL(2)\). J. Ramanujan Math. Soc. 17, 115–147 (2002)
Tsuzuki, M.: Spectral means of central values of automorphic \(L\)-functions for \(GL(2)\), Memoirs of AMS, 235 (2015), no. 1110
Yamazaki, T.: Rankin-Selberg method for Siegel cusp forms. Nagoya Math. J. 120, 35–49 (1990)
Yamana, S.: On the lifting of Hilbert cusp forms to Hilbert-Hermitian cusp forms (submitted)
Ziegler, C.: Jacobi forms of higher degree. Abh. Math. Sem. Univ. Hamburg 59, 191–224 (1989)
Acknowledgements
We would like to thank H. Atobe, T. Ikeda, S. Hayashida, and M. Tsuzuki for helpful discussions. Special thanks are given to H. Atobe for pointing out some mistakes in an earlier version and to M. Tsuzuki for guiding the second author on the computation in Section 2.2. A part of this paper was done during our stay at MPIM in Bonn. We thank the institute for incredible hospitality. We thank the referee for helpful remarks.
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Communicated by Jens Funke.
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Henry H. Kim partially supported by NSERC #482564. Takuya Yamauchi partially supported by JSPS Grant-in-Aid for Scientific Research (C) No.15K04787.
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Kim, H.H., Yamauchi, T. Non-vanishing of Miyawaki type lifts. Abh. Math. Semin. Univ. Hambg. 89, 117–134 (2019). https://doi.org/10.1007/s12188-019-00207-6
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DOI: https://doi.org/10.1007/s12188-019-00207-6