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Grobner, Harald; Sachdeva, Gunja

Relations of rationality for special values of Rankin-Selberg \(L\)-functions of \(\mathrm{GL}_n \times \mathrm{GL}_m\) over CM-fields. (English) Zbl 1469.11135

Pac. J. Math. 308, No. 2, 281-305 (2020).
Summary: We establish an “automorphic version” of Deligne’s conjecture for motivic \(L\)-functions in the case of Rankin-Selberg \(L\)-functions \(L(s,\Pi\times\Pi')\) of \( \operatorname{GL}_n\times\operatorname{GL}_m\) over arbitrary CM-fields \(F\). Our main results are of two different kinds: Firstly, for arbitrary integers \(1\leq m< n\) and suitable pairs \((\Pi,\Pi')\) of cohomological automorphic representations, we relate critical values of \(L(s,\Pi\times\Pi')\) with a product of Whittaker periods attached to \(\Pi\) and \(\Pi'\), Blasius’s CM-periods of Hecke-characters and certain nonzero values of standard \(L\)-functions. Secondly, these relations lead to quite broad generalizations of fundamental rationality-results of J. L. Waldspurger [Compos. Math. 54, 173–242 (1985; Zbl 0567.10021)], G. Harder and A. Raghuram [Eisenstein cohomology for \(\mathrm{GL}_N\) and the special values of Rankin-Selberg \(L\)-functions. Princeton, NJ: Princeton University Press (2020; Zbl 1466.11001)], and others.

MSC:

11F67 Special values of automorphic \(L\)-series, periods of automorphic forms, cohomology, modular symbols
11F70 Representation-theoretic methods; automorphic representations over local and global fields
11G18 Arithmetic aspects of modular and Shimura varieties
11R39 Langlands-Weil conjectures, nonabelian class field theory
22E55 Representations of Lie and linear algebraic groups over global fields and adèle rings

Keywords:

periods; rationality; special values; \(L\)-function; Rankin-Selberg; \(\mathrm{GL}(n)\)

Citations:

Zbl 0567.10021; Zbl 1466.11001
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Full Text: DOI

References:

[1] ; Arthur, Simple algebras, base change, and the advanced theory of the trace formula. Ann. of Math. Stud., 120 (1989) · Zbl 0682.10022
[2] ; Bellaïche, Families of Galois representations and Selmer groups. Astérisque, 324 (2009) · Zbl 1192.11035
[3] 10.2307/1971386 · Zbl 0608.10029 · doi:10.2307/1971386
[4] 10.1090/pspum/033.2/546608 · doi:10.1090/pspum/033.2/546608
[5] 10.1090/pspum/033.1/546598 · doi:10.1090/pspum/033.1/546598
[6] ; Borel, Continuous cohomology, discrete subgroups, and representations of reductive groups. Ann. of Math. Stud., 94 (1980) · Zbl 0443.22010
[7] 10.1007/BF02699257 · Zbl 0739.11020 · doi:10.1007/BF02699257
[8] 10.1090/pspum/033.2/546622 · doi:10.1090/pspum/033.2/546622
[9] 10.1215/S0012-7094-79-04626-X · Zbl 0427.22010 · doi:10.1215/S0012-7094-79-04626-X
[10] 10.1090/S0894-0347-04-00455-2 · Zbl 1057.11029 · doi:10.1090/S0894-0347-04-00455-2
[11] 10.1007/978-0-387-79852-3 · Zbl 1173.22001 · doi:10.1007/978-0-387-79852-3
[12] 10.1007/s00605-018-1188-5 · Zbl 1451.11040 · doi:10.1007/s00605-018-1188-5
[13] 10.1017/S1474748014000462 · Zbl 1423.11096 · doi:10.1017/S1474748014000462
[14] 10.1142/S1793042114500110 · Zbl 1309.11044 · doi:10.1142/S1793042114500110
[15] 10.1353/ajm.2014.0021 · Zbl 1297.11048 · doi:10.1353/ajm.2014.0021
[16] 10.2118/178924-pa · doi:10.2118/178924-pa
[17] ; Harder, Seminar on number theory. Progr. Math., 38, 73 (1983)
[18] 10.2307/j.ctvhrd1b0 · doi:10.2307/j.ctvhrd1b0
[19] 10.2307/2152780 · Zbl 0779.11023 · doi:10.2307/2152780
[20] 10.1515/crll.1997.483.75 · Zbl 0859.11032 · doi:10.1515/crll.1997.483.75
[21] 10.2307/2944321 · Zbl 0731.11031 · doi:10.2307/2944321
[22] 10.24033/bsmf.2622 · Zbl 1332.22020 · doi:10.24033/bsmf.2622
[23] 10.1215/S0012-7094-94-07417-6 · Zbl 0838.11036 · doi:10.1215/S0012-7094-94-07417-6
[24] 10.1090/tran/7527 · Zbl 1443.11076 · doi:10.1090/tran/7527
[25] 10.4171/jems/955 · Zbl 1452.11060 · doi:10.4171/jems/955
[26] 10.1090/pspum/055.2/1265560 · doi:10.1090/pspum/055.2/1265560
[27] ; Kurchanov, Izv. Akad. Nauk SSSR Ser. Mat., 42, 588 (1978) · Zbl 0393.10030
[28] ; Kurchanov, Izv. Akad. Nauk SSSR Ser. Mat., 43, 67 (1979) · Zbl 0416.10024
[29] ; Labesse, On the stabilization of the trace formula. Stab. Trace Formula Shimura Var. Arith. Appl., 1, 429 (2011) · Zbl 1255.11027
[30] 10.1515/forum-2014-0043 · Zbl 1417.11082 · doi:10.1515/forum-2014-0043
[31] 10.1007/BF01389047 · Zbl 0677.10020 · doi:10.1007/BF01389047
[32] 10.1016/j.jnt.2019.10.013 · Zbl 1460.11072 · doi:10.1016/j.jnt.2019.10.013
[33] 10.2307/2374219 · Zbl 0467.12013 · doi:10.2307/2374219
[34] ; Waldspurger, Compos. Math., 54, 173 (1985) · Zbl 0567.10021
[35] ; Weil, Proceedings of the International Symposium on Algebraic Number Theory, 1 (1956) · Zbl 0071.26501
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