TY - JOUR

T1 - Non-vanishing of Miyawaki type lifts

AU - Kim, Henry H.

AU - Yamauchi, Takuya

N1 - Funding Information:
Henry H. Kim partially supported by NSERC #482564. Takuya Yamauchi partially supported by JSPS Grant-in-Aid for Scientific Research (C) No.15K04787.
Publisher Copyright:
© 2019, The Author(s), under exclusive licence to Mathematisches Seminar der Universität Hamburg.
Copyright:
Copyright 2019 Elsevier B.V., All rights reserved.

PY - 2019/10/1

Y1 - 2019/10/1

N2 - 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≡ 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 292.

AB - 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≡ 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 292.

KW - Langlands functoriality

KW - Miyawaki type lift

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U2 - 10.1007/s12188-019-00207-6

DO - 10.1007/s12188-019-00207-6

M3 - Article

AN - SCOPUS:85073984810

VL - 89

SP - 117

EP - 134

JO - Abhandlungen aus dem Mathematischen Seminar der Universitat Hamburg

JF - Abhandlungen aus dem Mathematischen Seminar der Universitat Hamburg

SN - 0025-5858

IS - 2

ER -