eBook ISBN:  9781470418939 
Product Code:  MEMO/232/1090.E 
List Price:  $75.00 
MAA Member Price:  $67.50 
AMS Member Price:  $45.00 
eBook ISBN:  9781470418939 
Product Code:  MEMO/232/1090.E 
List Price:  $75.00 
MAA Member Price:  $67.50 
AMS Member Price:  $45.00 

Book DetailsMemoirs of the American Mathematical SocietyVolume: 232; 2014; 107 ppMSC: Primary 11
Let \(\pi\) be the automorphic representation of \(\textrm{GSp}_4(\mathbb{A})\) generated by a full level cuspidal Siegel eigenform that is not a SaitoKurokawa 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 PiatetskiShapiro then implies that such representations \(\pi\) have a functorial lifting to a cuspidal representation of \(\textrm{GL}_4(\mathbb{A})\). Combined with the exteriorsquare 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

Chapters

Introduction

Notation

1. Distinguished vectors in local representations

2. Global $L$functions for $\textup {GSp}_4\times \textup {GL}_2$

3. The pullback formula

4. Holomorphy of global $L$functions for $\textup {GSp}_4 \times \textup {GL}_2$

5. Applications


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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 SaitoKurokawa 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 PiatetskiShapiro then implies that such representations \(\pi\) have a functorial lifting to a cuspidal representation of \(\textrm{GL}_4(\mathbb{A})\). Combined with the exteriorsquare 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 \(\textrm{GSp}_4\times\textrm{GL}_2\).

Chapters

Introduction

Notation

1. Distinguished vectors in local representations

2. Global $L$functions for $\textup {GSp}_4\times \textup {GL}_2$

3. The pullback formula

4. Holomorphy of global $L$functions for $\textup {GSp}_4 \times \textup {GL}_2$

5. Applications