Fourier-Jacobi expansion and the Ikeda lift

Abstract

In this article, we consider a Fourier-Jacobi expansion of Siegel modular forms generated by the Ikeda lift. There are two purposes of this article: first, to give an expression of L-function of certain Siegel modular forms of half-integral weight of odd degree; and secondly, to give a relation among Fourier-Jacobi coefficients of Siegel modular forms generated by the Ikeda lift.

Miyawaki’s F12 spinor L-function conjecture

“…Hence it does not cover the eigenvalues λ F 12 (p) necessary to study the spinor L-function of F 12 . To avoid this gap we choose a different approach to study the spinor and standard L-functions of a Hecke eigenform, namely, we use Hayashida's description of Ikeda lifts [7] in the frame work of Jacobi forms and Fourier-Jacobi expansions of Siegel modular forms. It is in the spirit of Yamazaki's proof [15] of the Maass relations of Siegel Eisenstein series and Eichler and Zagier's description of the Maass Spezialschar [5].…”

“…g) be an Ikeda lift of the primitive newform g ∈ S 2k with κ = m + k. Then Hayashida [7] proved that there exists an operator (2.7)…”

“…In this paper we can manifest his belief in a theorem. Our proof builts on results of Andrianov [1], Yamazaki [15], Ikeda [8][9], Hayashida [7] and others.…”

Degenerate principal series and Langlands classification

“…If ε ( 1 2 , π ) = (−1) d(n−1)/2 , then there should be a Hilbert-Siegel cusp form of weight κ + n 2 whose p-component is the small nontempered representation A + n,ψp (π p ). When F = Q and π is everywhere unramified, such a Siegel cusp form is constructed in [8,7].…”

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

“…Let E (m+1) k be the Siegel Eisenstein series of weight k and of degree m + 1. (For the definition of the Siegel Eisenstein series, see, for example,[6].) Suppose that m > 0 and let e , Hayashida defined the generalized Cohen Eisenstein series E(m) k−1/2 as E (m) k−1/2 = σ m (e(m+1) k,, where σ m is the Ibukiyama isomorphism.…”