Transfer of Siegel Cusp Forms of Degree 2

Let $\pi$ be the automorphic representation of $\textrm{GSp}_4(\mathbb{A})$ generated by a full level cuspidal Siegel eigenform that is not a Saito-Kurokawa lift, and $\tau$ be an arbitrary cuspidal, automorphic representation of $\textrm{GL}_2(\mathbb{A})$. Using Furusawa's integral representation for $\textrm{GSp}_4\times\textrm{GL}_2$ combined with a pullback formula involving the unitary group $\textrm{GU}(3,3)$, the authors prove that the $L$-functions $L(s,\pi\times\tau)$ are 'nice'. The converse theorem of Cogdell and Piatetski-Shapiro then implies that such representations $\pi$ have a functorial lifting to a cuspidal representation of $\textrm{GL}_4(\mathbb{A})$. Combined with the exterior-square lifting of Kim, this also leads to a functorial lifting of $\pi$ to a cuspidal representation of $\textrm{GL}_5(\mathbb{A})$. As an application, the authors obtain analytic properties of various $L$-functions related to full level Siegel cusp forms. They also obtain special value results for $\textrm{GSp}_4\times\textrm{GL}_1$ and $\textrm{GSp}_4\times\textrm{GL}_2$.