Test vectors and central \(L\)-values for \(\mathrm{GL}(2)\). (English) Zbl 1425.11104
Summary: We determine local test vectors for Waldspurger functionals for \(\mathrm{GL}_2\), in the case where both the representation of \(\mathrm{GL}_2\) and the character of the degree two extension are ramified, with certain restrictions. We use this to obtain an explicit version of Waldspurger’s formula relating twisted central \(L\)-values of automorphic representations on \(\mathrm{GL}_2\) with certain toric period integrals. As a consequence, we generalize an average value formula of Feigon and Whitehouse, and obtain some nonvanishing results.
MSC:
| 11F67 | Special values of automorphic \(L\)-series, periods of automorphic forms, cohomology, modular symbols |
| 11F66 | Langlands \(L\)-functions; one variable Dirichlet series and functional equations |
| 11F41 | Automorphic forms on \(\mbox{GL}(2)\); Hilbert and Hilbert-Siegel modular groups and their modular and automorphic forms; Hilbert modular surfaces |
| 11F70 | Representation-theoretic methods; automorphic representations over local and global fields |
| 22E50 | Representations of Lie and linear algebraic groups over local fields |
References:
| [1] | 10.1006/jabr.1998.7542 · Zbl 0913.22013 · doi:10.1006/jabr.1998.7542 |
| [2] | 10.1007/3-540-31511-X · doi:10.1007/3-540-31511-X |
| [3] | 10.2140/ant.2014.8.2523 · Zbl 1311.11054 · doi:10.2140/ant.2014.8.2523 |
| [4] | 10.1215/00127094-2009-041 · Zbl 1241.11057 · doi:10.1215/00127094-2009-041 |
| [5] | 10.1515/crll.1993.438.187 · Zbl 0770.11025 · doi:10.1515/crll.1993.438.187 |
| [6] | ; Gelbart, Automorphic forms on adele groups. Automorphic forms on adele groups. Annals of Mathematics Studies, 83 (1975) · Zbl 0329.10018 |
| [7] | 10.1007/978-3-0348-0351-9 · Zbl 1285.11073 · doi:10.1007/978-3-0348-0351-9 |
| [8] | ; Gross, Number theory. Number theory. CMS Conf. Proc., 7, 115 (1987) |
| [9] | 10.2307/2374689 · Zbl 0675.12011 · doi:10.2307/2374689 |
| [10] | 10.1007/BF01445212 · Zbl 0768.22004 · doi:10.1007/BF01445212 |
| [11] | 10.1007/s00229-014-0696-4 · Zbl 1392.11043 · doi:10.1007/s00229-014-0696-4 |
| [12] | 10.1215/S0012-7094-02-11522-1 · Zbl 1042.11030 · doi:10.1215/S0012-7094-02-11522-1 |
| [13] | 10.5802/aif.1258 · Zbl 0725.11025 · doi:10.5802/aif.1258 |
| [14] | 10.1215/0023608X-2010-014 · Zbl 1271.11057 · doi:10.1215/0023608X-2010-014 |
| [15] | ; Hida, Ann. Sci. École Norm. Sup. (4), 26, 189 (1993) · Zbl 0778.11061 |
| [16] | ; Hsieh, Doc. Math., 19, 709 (2014) · Zbl 1314.11034 |
| [17] | ; Jacquet, Compositio Math., 63, 315 (1987) · Zbl 0633.10029 |
| [18] | ; Jacquet, Bull. Soc. Math. France, 129, 33 (2001) · Zbl 1069.11017 |
| [19] | 10.1007/BF01450798 · Zbl 0443.22013 · doi:10.1007/BF01450798 |
| [20] | 10.1090/S0002-9947-09-04923-X · Zbl 1230.11061 · doi:10.1090/S0002-9947-09-04923-X |
| [21] | 10.2307/2041005 · Zbl 0375.22005 · doi:10.2307/2041005 |
| [22] | 10.2307/2373875 · Zbl 0417.22012 · doi:10.2307/2373875 |
| [23] | 10.1093/imrn/rnn127 · Zbl 1193.11046 · doi:10.1093/imrn/rnn127 |
| [24] | 10.1007/978-1-4614-1260-1_20 · Zbl 1276.11057 · doi:10.1007/978-1-4614-1260-1_20 |
| [25] | 10.1007/s00208-009-0447-0 · Zbl 1221.11127 · doi:10.1007/s00208-009-0447-0 |
| [26] | 10.1016/j.jnt.2013.01.001 · Zbl 1288.11050 · doi:10.1016/j.jnt.2013.01.001 |
| [27] | 10.2140/pjm.2011.250.365 · Zbl 1264.11038 · doi:10.2140/pjm.2011.250.365 |
| [28] | 10.1016/j.jnt.2009.01.017 · Zbl 1258.11059 · doi:10.1016/j.jnt.2009.01.017 |
| [29] | 10.1112/S0010437X06002259 · Zbl 1144.11041 · doi:10.1112/S0010437X06002259 |
| [30] | 10.4310/PAMQ.2005.v1.n4.a1 · Zbl 1143.11029 · doi:10.4310/PAMQ.2005.v1.n4.a1 |
| [31] | ; Saito, Compositio Math., 85, 99 (1993) · Zbl 0795.22009 |
| [32] | ; Schmidt, J. Ramanujan Math. Soc., 17, 115 (2002) · Zbl 0997.11040 |
| [33] | ; Shimura, Duke Math. J., 45, 637 (1978) · Zbl 0394.10015 |
| [34] | ; Sugano, J. Fac. Sci. Univ. Tokyo Sect. IA Math., 31, 521 (1985) |
| [35] | 10.4064/aa8021-4-2016 · Zbl 1393.11043 · doi:10.4064/aa8021-4-2016 |
| [36] | 10.1007/BF01393255 · Zbl 0385.12006 · doi:10.1007/BF01393255 |
| [37] | 10.2307/2374441 · Zbl 0532.12015 · doi:10.2307/2374441 |
| [38] | ; Waldspurger, Compositio Math., 54, 173 (1985) · Zbl 0567.10021 |
| [39] | 10.1155/IMRN/2006/26150 · Zbl 1141.11028 · doi:10.1155/IMRN/2006/26150 |
| [40] | 10.4310/AJM.2001.v5.n2.a1 · Zbl 1111.11030 · doi:10.4310/AJM.2001.v5.n2.a1 |
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