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Transfer of Siegel Cusp Forms of Degree 2
Ameya Pitale, University of Oklahoma, Norman, Oklahoma, Abhishek Saha, University of Bristol, United Kingdom, and Ralf Schmidt, University of Oklahoma, Norman, Oklahoma
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Memoirs of the American Mathematical Society
2014; 107 pp; softcover
Volume: 232
ISBN-10: 0-8218-9856-6
ISBN-13: 978-0-8218-9856-7
List Price: US$75 Individual Members: US$45
Institutional Members: US\$60
Order Code: MEMO/232/1090

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 $$\textrm{GSp}_4\times\textrm{GL}_2$$.

• Global $$L$$-functions for $$\textrm{GSp}_4\times\textrm{GL}_2$$
• Holomorphy of global $$L$$-functions for $$\textrm{GSp}_4\times\textrm{GL}_2$$