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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

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
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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 \(\textrm{GSp}_4\times\textrm{GL}_2\).

Table of Contents

  • Introduction
  • Notation
  • Distinguished vectors in local representations
  • Global \(L\)-functions for \(\textrm{GSp}_4\times\textrm{GL}_2\)
  • The pullback formula
  • Holomorphy of global \(L\)-functions for \(\textrm{GSp}_4\times\textrm{GL}_2\)
  • Applications
  • Bibliography
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