Transfer of Siegel cusp forms of degree 2

Details Transfer of Siegel cusp forms of degree 2 Transfer of Siegel cusp forms of degree 2

2011-06-28

arxiv.org/abs/1106.5611v2

Mathematics

Number Theory

88 pages

Scientific paper

Using Furusawa's integral representation for $\GSp_4\times\GL_2$ combined with a pullback formula involving a unitary group $\GU(3,3)$, we prove that the $L$-functions $L(s,\pi\times\tau)$ are "nice", where $\pi$ is the automorphic representation of $\GSp_4(\A)$ generated by a full level cuspidal Siegel eigenform that is not a Saito-Kurokawa lift, and $\tau$ is an arbitrary cuspidal, automorphic representation of $\GL_2(\A)$. The converse theorem of Cogdell and Piatetski-Shapiro then implies that such representations $\pi$ have a functorial lifting to a cuspidal representation of $\GL_4(\A)$. Combined with the exterior-square lifting of Kim, this also leads to a functorial lifting of $\pi$ to a cuspidal representation of $\GL_5(\A)$. As an application, we obtain analytic properties of various $L$-functions related to full level Siegel cusp forms. We also obtain special value results for $\GSp_4\times\GL_1$ and $\GSp_4\times\GL_2$.

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