On the spin L-function of Ikeda’s lifts

The recent construction of Siegel modular forms of degree 2n from elliptic cusp forms by Ikeda is interpreted as a special case of Langlands functoriality. This is then used to express the (full) spin L–function of an Ikeda lift as a product of symmetric power L–functions for GL(2). As a corollary one obtains the desired analytic properties of these spin L–functions provided the analogous properties of the symmetric power L–functions are known. Introduction Let F be a classical Siegel modular form of degree n, assumed to be a cuspidal eigenform. There are (at least) two different L–functions attached to F , the standard L–function, which is an Euler product of degree 2n+ 1, and the spin L–function, an Euler product of degree 2. It is explained in [AS] how these two L–functions are related to the two “smallest” irreducible finite-dimensional representations of the group Spin(2n+ 1,C). While the expected analytic properties of the standard L–function, in particular the analytic continuation and functional equation, were shown to hold by Böcherer [Bö], very little seems to be known about the spin L–function beyond degree 2, where Andrianov has complete results (see [An]). One goal of this note is to get insight into the analytic properties of the spin L–function at least for a small class of Siegel modular forms, namely the lifts constructed by Ikeda in [Ik]. Let f ∈ S2k(SL(2,Z)) be a cuspidal elliptic eigenform. Given an integer n ≡ k mod 2, Ikeda constructs a cuspidal eigenform F ∈ Sk+n(Sp(4n,Z)) of degree 2n such that the (finite part of the) standard L–function of F is given by