Base change and triple product 𝐿-series

Hsieh, Ming-Lun; Yamana, Shunsuke

doi:10.1090/ert/602

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

Shih-Yu Chen, Gamma factors for the Asai cube representation, Math. Z. 297 (2021), no. 1-2, 747–773. MR 4204712, DOI 10.1007/s00209-020-02531-7
Shih-Yu Chen, Yao Cheng, and Isao Ishikawa, Gamma factors for the Asai representation of $\textrm {GL}_2$, J. Number Theory 209 (2020), 83–146. MR 4053061, DOI 10.1016/j.jnt.2019.08.008
Yuval Z. Flicker and Dmitrii Zinoviev, On poles of twisted tensor $L$-functions, Proc. Japan Acad. Ser. A Math. Sci. 71 (1995), no. 6, 114–116. MR 1344660
Wee Teck Gan and Atsushi Ichino, On endoscopy and the refined Gross-Prasad conjecture for $(\rm SO_5,SO_4)$, J. Inst. Math. Jussieu 10 (2011), no. 2, 235–324. MR 2787690, DOI 10.1017/S1474748010000198
Wee Teck Gan, Yannan Qiu, and Shuichiro Takeda, The regularized Siegel-Weil formula (the second term identity) and the Rallis inner product formula, Invent. Math. 198 (2014), no. 3, 739–831. MR 3279536, DOI 10.1007/s00222-014-0509-0
Michael Harris and Stephen S. Kudla, The central critical value of a triple product $L$-function, Ann. of Math. (2) 133 (1991), no. 3, 605–672. MR 1109355, DOI 10.2307/2944321
Atsushi Ichino, Trilinear forms and the central values of triple product $L$-functions, Duke Math. J. 145 (2008), no. 2, 281–307. MR 2449948, DOI 10.1215/00127094-2008-052
Tamotsu Ikeda, On the functional equations of the triple $L$-functions, J. Math. Kyoto Univ. 29 (1989), no. 2, 175–219. MR 1015866, DOI 10.1215/kjm/1250520261
Tamotsu Ikeda, On the location of poles of the triple $L$-functions, Compositio Math. 83 (1992), no. 2, 187–237. MR 1174424
Tamotsu Ikeda, On the functional equation of the Siegel series, J. Number Theory 172 (2017), 44–62. MR 3573143, DOI 10.1016/j.jnt.2016.08.002
Stephen S. Kudla and Stephen Rallis, Ramified degenerate principal series representations for $\textrm {Sp}(n)$, Israel J. Math. 78 (1992), no. 2-3, 209–256. MR 1194967, DOI 10.1007/BF02808058
Stephen S. Kudla and Stephen Rallis, A regularized Siegel-Weil formula: the first term identity, Ann. of Math. (2) 140 (1994), no. 1, 1–80. MR 1289491, DOI 10.2307/2118540
Henry H. Kim and Freydoon Shahidi, Functorial products for $\textrm {GL}_2\times \textrm {GL}_3$ and the symmetric cube for $\textrm {GL}_2$, Ann. of Math. (2) 155 (2002), no. 3, 837–893. With an appendix by Colin J. Bushnell and Guy Henniart. MR 1923967, DOI 10.2307/3062134
Shin-ichi Kato and Keiji Takano, Square integrability of representations on $p$-adic symmetric spaces, J. Funct. Anal. 258 (2010), no. 5, 1427–1451. MR 2566307, DOI 10.1016/j.jfa.2009.10.026
Hengfei Lu, A new proof to the period problems of $\textrm {GL}(2)$, J. Number Theory 180 (2017), 1–25. MR 3679785, DOI 10.1016/j.jnt.2017.03.010
Vladimir Platonov and Andrei Rapinchuk, Algebraic groups and number theory, Pure and Applied Mathematics, vol. 139, Academic Press, Inc., Boston, MA, 1994. Translated from the 1991 Russian original by Rachel Rowen. MR 1278263
Dipendra Prasad, Trilinear forms for representations of $\textrm {GL}(2)$ and local $\epsilon$-factors, Compositio Math. 75 (1990), no. 1, 1–46. MR 1059954
I. Piatetski-Shapiro and Stephen Rallis, Rankin triple $L$ functions, Compositio Math. 64 (1987), no. 1, 31–115. MR 911357
Dinakar Ramakrishnan, Modularity of the Rankin-Selberg $L$-series, and multiplicity one for $\textrm {SL}(2)$, Ann. of Math. (2) 152 (2000), no. 1, 45–111. MR 1792292, DOI 10.2307/2661379
Hideo Shimizu, Theta series and automorphic forms on $\textrm {GL}_{2}$, J. Math. Soc. Japan 24 (1972), 638–683. MR 333081, DOI 10.2969/jmsj/02440638
Goro Shimura, Arithmetic and analytic theories of quadratic forms and Clifford groups, Mathematical Surveys and Monographs, vol. 109, American Mathematical Society, Providence, RI, 2004. MR 2027702, DOI 10.1090/surv/109
J.-L. Waldspurger, Sur les valeurs de certaines fonctions $L$ automorphes en leur centre de symétrie, Compositio Math. 54 (1985), no. 2, 173–242 (French). MR 783511
André Weil, Sur la formule de Siegel dans la théorie des groupes classiques, Acta Math. 113 (1965), 1–87 (French). MR 223373, DOI 10.1007/BF02391774
Xinyi Yuan, Shou-Wu Zhang, and Wei Zhang, The Gross-Zagier formula on Shimura curves, Annals of Mathematics Studies, vol. 184, Princeton University Press, Princeton, NJ, 2013. MR 3237437
Wei Zhang, Automorphic period and the central value of Rankin-Selberg L-function, J. Amer. Math. Soc. 27 (2014), no. 2, 541–612. MR 3164988, DOI 10.1090/S0894-0347-2014-00784-0