# Transfer of Siegel Cusp Forms of Degree 2

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*Ameya Pitale; Abhishek Saha; Ralf Schmidt*

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

# Table of Contents

## Transfer of Siegel Cusp Forms of Degree 2

- Cover Cover11 free
- Title page i2 free
- Introduction 18 free
- Notation 1724 free
- Chapter 1. Distinguished vectors in local representations 2128
- Chapter 2. Global 𝐿-functions for 𝐺𝑆𝑝₄×𝐺𝐿₂ 4552
- Chapter 3. The pullback formula 6370
- Chapter 4. Holomorphy of global 𝐿-functions for 𝐺𝑆𝑝₄×𝐺𝐿₂ 8188
- Chapter 5. Applications 8996
- Bibliography 103110
- Back Cover Back Cover1120