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Transfer of Siegel cusp forms of degree $2$
About this Title
Ameya Pitale, Abhishek Saha and Ralf Schmidt
Publication: Memoirs of the American Mathematical Society
Publication Year:
2014; Volume 232, Number 1090
ISBNs: 978-0-8218-9856-7 (print); 978-1-4704-1893-9 (online)
DOI: https://doi.org/10.1090/memo/1090
Published electronically: February 19, 2014
MSC: Primary 11F70, 11F46, 11F67
Table of Contents
Chapters
- Introduction
- Notation
- 1. Distinguished vectors in local representations
- 2. Global $L$-functions for $\mathrm {GSp}_4\times \mathrm {GL}_2$
- 3. The pullback formula
- 4. Holomorphy of global $L$-functions for $\mathrm {GSp}_4 \times \mathrm {GL}_2$
- 5. Applications
Abstract
Let $\pi$ be the automorphic representation of $\mathrm {GSp}_4(\mathbb {A})$ generated by a full level cuspidal Siegel eigenform that is not a Saito-Kurokawa lift, and $\tau$ be an arbitrary cuspidal, automorphic representation of $\mathrm {GL}_2(\mathbb {A})$. Using Furusawa’s integral representation for $\mathrm {GSp}_4\times \mathrm {GL}_2$ combined with a pullback formula involving the unitary group $\mathrm {GU}(3,3)$, we 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 $\mathrm {GL}_4(\mathbb {A})$. Combined with the exterior-square lifting of Kim, this also leads to a functorial lifting of $\pi$ to a cuspidal representation of $\mathrm {GL}_5(\mathbb {A})$. As an application, we obtain analytic properties of various $L$-functions related to full level Siegel cusp forms. We also obtain special value results for $\mathrm {GSp}_4\times \mathrm {GL}_1$ and $\mathrm {GSp}_4\times \mathrm {GL}_2$.- A. N. Andrianov, Euler products that correspond to Siegel’s modular forms of genus $2$, Uspehi Mat. Nauk 29 (1974), no. 3 (177), 43–110 (Russian). MR 0432552
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