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Koecher–Maass series of a certain half-integral weight modular form related to the Duke–Imamoḡlu–Ikeda lift

Volume 162 / 2014

Hidenori Katsurada, Hisa-aki Kawamura Acta Arithmetica 162 (2014), 1-42 MSC: Primary 11F67; Secondary 11F46. DOI: 10.4064/aa162-1-1

Abstract

Let $k$ and $n$ be positive even integers. For a cuspidal Hecke eigenform $h$ in the Kohnen plus space of weight $k-n/2+1/2$ for $\varGamma _0(4),$ let $f$ be the corresponding primitive form of weight $2k-n$ for ${SL}_2(\mathbb {Z} )$ under the Shimura correspondence, and $I_n(h)$ the Duke–Imamoḡlu–Ikeda lift of $h$ to the space of cusp forms of weight $k$ for $Sp_n(\mathbb {Z} )$. Moreover, let $\phi _{I_n(h),1}$ be the first Fourier–Jacobi coefficient of $I_n(h)$, and $\sigma _{n-1}(\phi _{I_n(h),1})$ be the cusp form in the generalized Kohnen plus space of weight $k-1/2$ corresponding to $\phi _{I_n(h),1}$ under the Ibukiyama isomorphism. We give an explicit formula for the Koecher–Maass series $L(s,\sigma _{n-1}(\phi _{I_n(h),1}))$ of $\sigma _{n-1}(\phi _{I_n(h),1})$ expressed in terms of the usual $L$-functions of $h$ and $f$.

Authors

  • Hidenori KatsuradaMuroran Institute of Technology
    Mizumoto 27-1
    Muroran, 050-8585, Japan
    e-mail
  • Hisa-aki KawamuraDepartment of Mathematics
    Hokkaido University
    Kita 10, Nishi 8, Kita-Ku
    Sapporo, 060-0810, Japan
    and
    Department of Mathematics
    Hiroshima University
    1-7-1 Kagamiyama
    Higashi-Hiroshima, 739-8521, Japan
    e-mail

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