A local converse theorem for ${\mathrm {U}}_{2r+1}$
HTML articles powered by AMS MathViewer
- by Qing Zhang PDF
- Trans. Amer. Math. Soc. 371 (2019), 5631-5654 Request permission
Abstract:
Let $E/F$ be a quadratic extension of $p$-adic fields, and let ${\mathrm {U}}_{2r+1}$ be the unitary group associated with $E/F$. We prove the following local converse theorem for ${\mathrm {U}}_{2r+1}$: given two irreducible generic supercuspidal representations $\pi ,\pi _0$ of ${\mathrm {U}}_{2r+1}$ with the same central character, if $\gamma (s,\pi \times \tau ,\psi )=\gamma (s,\pi _0\times \tau ,\psi )$ for all irreducible generic representations $\tau$ of ${\mathrm {GL}}_n(E)$ and for all $n$ with $1\le n\le r$, then $\pi \cong \pi _0$. The proof depends on analysis of the local integrals which defines local gamma factors and uses certain properties of partial Bessel functions developed recently by Cogdell, Shahidi, and Tsai.References
- Avraham Aizenbud, Dmitry Gourevitch, Stephen Rallis, and GΓ©rard Schiffmann, Multiplicity one theorems, Ann. of Math. (2) 172 (2010), no.Β 2, 1407β1434. MR 2680495, DOI 10.4007/annals.2010.172.1413
- Ehud Moshe Baruch, Local factors attached to representations of p-adic groups and strong multiplicity one, ProQuest LLC, Ann Arbor, MI, 1995. Thesis (Ph.D.)βYale University. MR 2692992
- Ehud Moshe Baruch, On the gamma factors attached to representations of $\textrm {U}(2,1)$ over a $p$-adic field, Israel J. Math. 102 (1997), 317β345. MR 1489111, DOI 10.1007/BF02773805
- Asher Ben-Artzi and David Soudry, $L$-functions for $\textrm {U}_m\times R_{E/F}\textrm {GL}_n\ (n\leq [{m\over 2}])$, Automorphic forms and $L$-functions I. Global aspects, Contemp. Math., vol. 488, Amer. Math. Soc., Providence, RI, 2009, pp.Β 13β59. MR 2522026, DOI 10.1090/conm/488/09563
- W. Casselman, Introduction to the theory of admissible representations of $p$-adic groups, online notes, available at https://www.math.ubc.ca/ cass/research/pdf/p-adic-book.pdf
- J. Chai, Bessel functions and local converse conjecture of Jacquet, Jour. Eur. Math. Soc. (to appear).
- Jiang-Ping Jeff Chen, The $n\times (n-2)$ local converse theorem for $\textrm {GL}(n)$ over a $p$-adic field, J. Number Theory 120 (2006), no.Β 2, 193β205. MR 2257542, DOI 10.1016/j.jnt.2005.12.001
- J. W. Cogdell, I. I. Piatetski-Shapiro, and F. Shahidi, Partial Bessel functions for quasi-split groups, Automorphic representations, $L$-functions and applications: progress and prospects, Ohio State Univ. Math. Res. Inst. Publ., vol. 11, de Gruyter, Berlin, 2005, pp.Β 95β128. MR 2192821, DOI 10.1515/9783110892703.95
- J. W. Cogdell, I. I. Piatetski-Shapiro, and F. Shahidi, Stability of $\gamma$-factors for quasi-split groups, J. Inst. Math. Jussieu 7 (2008), no.Β 1, 27β66. MR 2398146, DOI 10.1017/S1474748007000163
- J. W. Cogdell, F. Shahidi, and T.-L. Tsai, Local Langlands correspondence for $\textrm {GL}_n$ and the exterior and symmetric square $\varepsilon$-factors, Duke Math. J. 166 (2017), no.Β 11, 2053β2132. MR 3694565, DOI 10.1215/00127094-2017-0001
- Wee Teck Gan, Benedict H. Gross, and Dipendra Prasad, Symplectic local root numbers, central critical $L$ values, and restriction problems in the representation theory of classical groups, AstΓ©risque 346 (2012), 1β109 (English, with English and French summaries). Sur les conjectures de Gross et Prasad. I. MR 3202556
- Stephen Gelbart and Ilya Piatetski-Shapiro, Automorphic forms and $L$-functions for the unitary group, Lie group representations, II (College Park, Md., 1982/1983) Lecture Notes in Math., vol. 1041, Springer, Berlin, 1984, pp.Β 141β184. MR 748507, DOI 10.1007/BFb0073147
- HervΓ© Jacquet, Germs for Kloosterman integrals, a review, Advances in the theory of automorphic forms and their $L$-functions, Contemp. Math., vol. 664, Amer. Math. Soc., Providence, RI, 2016, pp.Β 173β185. MR 3502982, DOI 10.1090/conm/664/13048
- H. Jacquet, B. Liu, On the local converse theorem for $p$-adic ${\mathrm {GL}}_n$, Amer. Jour. Math. (to appear).
- HervΓ© Jacquet and Joseph Shalika, A lemma on highly ramified $\epsilon$-factors, Math. Ann. 271 (1985), no.Β 3, 319β332. MR 787183, DOI 10.1007/BF01456070
- Dihua Jiang, On local $\gamma$-factors, Arithmetic geometry and number theory, Ser. Number Theory Appl., vol. 1, World Sci. Publ., Hackensack, NJ, 2006, pp.Β 1β28. MR 2258071, DOI 10.1142/9789812773531_{0}001
- Dihua Jiang and David Soudry, The local converse theorem for $\textrm {SO}(2n+1)$ and applications, Ann. of Math. (2) 157 (2003), no.Β 3, 743β806. MR 1983781, DOI 10.4007/annals.2003.157.743
- Dihua Jiang and David Soudry, Appendix: On the local descent from $\textrm {GL}(n)$ to classical groups [appendix to MR2931222], Amer. J. Math. 134 (2012), no.Β 3, 767β772. MR 2931223, DOI 10.1353/ajm.2012.0023
- Eyal Kaplan, Complementary results on the Rankin-Selberg gamma factors of classical groups, J. Number Theory 146 (2015), 390β447. MR 3267119, DOI 10.1016/j.jnt.2013.12.002
- K. Morimoto, On the irreducibility of global descents for even unitary groups and its applications, Trans. Amer. Math. Soc. (to appear).
- David Soudry and Yaacov Tanay, On local descent for unitary groups, J. Number Theory 146 (2015), 557β626. MR 3267124, DOI 10.1016/j.jnt.2014.03.003
- Boaz Tamir, On $L$-functions and intertwining operators for unitary groups, Israel J. Math. 73 (1991), no.Β 2, 161β188. MR 1135210, DOI 10.1007/BF02772947
- Qing Zhang, Stability of Rankin-Selberg gamma factors for $\textrm {Sp}(2n)$, $\widetilde \textrm {Sp}(2n)$ and $\textrm {U}(n,n)$, Int. J. Number Theory 13 (2017), no.Β 9, 2393β2432. MR 3704368, DOI 10.1142/S1793042117501317
- Q. Zhang, A local converse theorem for ${\mathrm {Sp}}_{2r}$, Math. Ann. (to appear), arXiv:1705.01692.
Additional Information
- Qing Zhang
- Affiliation: Department of Mathematics and Statistics, University of Calgary, Calgary, Canada
- Email: qingzhang0@gmail.com
- Received by editor(s): May 26, 2017
- Received by editor(s) in revised form: November 13, 2017
- Published electronically: August 31, 2018
- © Copyright 2018 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 371 (2019), 5631-5654
- MSC (2010): Primary 22E50; Secondary 11F70
- DOI: https://doi.org/10.1090/tran/7469
- MathSciNet review: 3937305