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

Hidenori Katsurada ¹   ; Hisa-aki Kawamura ²

¹ Muroran Institute of Technology Mizumoto 27-1 Muroran, 050-8585, Japan
² Department 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

Acta Arithmetica, Tome 162 (2014) no. 1, pp. 1-42

Résumé

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 $f$.

Détail

DOI : 10.4064/aa162-1-1

Keywords: positive even integers cuspidal hecke eigenform kohnen plus space weight k n vargamma corresponding primitive form weight k n mathbb under shimura correspondence duke imamo ikeda lift space cusp forms weight mathbb moreover phi first fourier jacobi coefficient sigma n phi cusp form generalized kohnen plus space weight k corresponding phi under ibukiyama isomorphism explicit formula koecher maass series sigma n phi sigma n phi expressed terms usual l functions