## Base change and triple product $L$-series

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- by Ming-Lun Hsieh and Shunsuke Yamana
- Represent. Theory
**26**(2022), 402-431 - DOI: https://doi.org/10.1090/ert/602
- Published electronically: March 25, 2022
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## Abstract:

Let $\pi _i$ be an irreducible cuspidal automorphic representation of $\mathrm {GL}_2$ with central character $\omega _i$. When $\omega _1\omega _2\omega _3$ is trivial, Atsushi Ichino proved a formula for the central value $L(\frac {1}{2}, \pi _1\times \pi _2\times \pi _3)$ of the triple product $L$-series in terms of global trilinear forms. We will extend this formula to the case when $\omega _1\omega _2\omega _3$ is a quadratic character, giving a non-vanishing criterion of a local trilinear form in terms of the central value of the gamma factor.## References

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## Bibliographic Information

**Ming-Lun Hsieh**- Affiliation: Institute of Mathematics, Academia Sinica, Taipei 10617, Taiwan
- MR Author ID: 904717
- ORCID: 0000-0002-7329-5167
- Email: mlhsieh@math.sinica.edu.tw
**Shunsuke Yamana**- Affiliation: Osaka City University, Advanced Mathematical Institute, 3-3-138 Sugimoto, Sumiyoshi-ku, Osaka 558-8585, Japan
- MR Author ID: 871756
- Email: yamana@sci.osaka-cu.ac.jp
- Received by editor(s): March 17, 2020
- Received by editor(s) in revised form: March 23, 2021
- Published electronically: March 25, 2022
- Additional Notes: The first author was partially supported by a MOST grants 108-2628-M-001-009-MY4 and 110-2628-M-001-004-. The second author was partially supported by JSPS Grant-in-Aid for Scientific Research (C)18K03210 and (B)19H01778. This work was partially supported by Osaka City University Advanced Mathematical Institute (MEXT Joint Usage/Research Center on Mathematics and Theoretical Physics).

The second author is the corresponding author - © Copyright 2022 American Mathematical Society
- Journal: Represent. Theory
**26**(2022), 402-431 - MSC (2020): Primary 11F67, 11F70, 11F27
- DOI: https://doi.org/10.1090/ert/602
- MathSciNet review: 4400091