×

Modular symbols for reductive groups and \(p\)-adic Rankin-Selberg convolutions over number fields. (English) Zbl 1288.11049

Let \(\pi\) and \(\sigma\) be irreducible cuspidal automorphic representations of \(\mathrm{GL}_n\) and \(\mathrm{GL}_{n-1}\), which occur in cohomology and are further assumed to be ordinary at \(p\). The paper proves the existence of a \(p\)-adic measure \(\mu\) such that for some complex constant \(\Omega\): \[ \int\chi\, d\mu=\Omega \cdot g(\chi) \cdot L(\tfrac12,(\pi\otimes\chi)\times\sigma). \] Here \(\chi\) is any Hecke character of finite order with a nontrivial \(p\)-power conductor, \(g(\chi)\) is a Gauss sum, \(L(\cdot, \pi_1\times\pi_2)\) is the Rankin-Selberg \(L\)-function constructed by Jacquet, Piatetski-Shapiro and Shalika. It is also proved in the case \(n=2,3\) that \(\Omega\neq 0\). The author expects that the nonvanishing of \(\Omega\) can be established in general by the method of H. Kasten and C.-G. Schmidt [Int. J. Number Theory 9, No. 1, 205–256 (2013; Zbl 1305.11039)].
As a corollary, the paper also gives the construction of a \(p-\)adic \(L-\)function of the symmetric cube of a modular elliptic curve. The author also points out an oversight in a prior work of A. Ash and D. Ginzburg [Invent. Math. 116, No. 1–3, 27–73 (1994; Zbl 0807.11029)], where the assumption on the Hecke character \(\chi\) was overlooked.

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
22E55 Representations of Lie and linear algebraic groups over global fields and adèle rings

Software:

SageMath
PDFBibTeX XMLCite
Full Text: DOI arXiv

References:

[1] DOI: 10.1007/BF01231556 · Zbl 0807.11029 · doi:10.1007/BF01231556
[2] Borel A., Ann. Sci. E’c. Norm. Sup. 7 (4) pp 235– (1974)
[3] Borel A., Progr. Math. 14 pp 21– (1981)
[4] DOI: 10.2307/1970210 · Zbl 0107.14804 · doi:10.2307/1970210
[5] DOI: 10.1007/BF02566134 · Zbl 0274.22011 · doi:10.1007/BF02566134
[6] DOI: 10.1007/BF02684375 · Zbl 0145.17402 · doi:10.1007/BF02684375
[7] DOI: 10.1090/S0894-0347-01-00370-8 · Zbl 0982.11033 · doi:10.1090/S0894-0347-01-00370-8
[8] Clozel L., Perspect. Math. 10 pp 77– (1990)
[9] DOI: 10.1007/BF02698889 · Zbl 0814.11033 · doi:10.1007/BF02698889
[10] Deligne P., Proc. Symp. Pure Math. 33 (2) pp 313– (1979)
[11] DOI: 10.2307/2374726 · Zbl 0776.11027 · doi:10.2307/2374726
[12] Gelbart S., Ann. Sci. E’c. Norm. Sup. 11 (4) pp 471– (1978)
[13] DOI: 10.1007/BF01098922 · doi:10.1007/BF01098922
[14] DOI: 10.1007/BF02684396 · Zbl 0228.20015 · doi:10.1007/BF02684396
[15] DOI: 10.2307/1971270 · Zbl 0401.10037 · doi:10.2307/1971270
[16] DOI: 10.2307/1971112 · doi:10.2307/1971112
[17] DOI: 10.1007/BF01450798 · Zbl 0443.22013 · doi:10.1007/BF01450798
[18] DOI: 10.2307/2374264 · Zbl 0525.22018 · doi:10.2307/2374264
[19] DOI: 10.2307/2374103 · Zbl 0473.12008 · doi:10.2307/2374103
[20] DOI: 10.2307/2374050 · Zbl 0491.10020 · doi:10.2307/2374050
[21] Kazhdan D., Math. 512 pp 97– (2000)
[22] DOI: 10.2307/3062134 · Zbl 1040.11036 · doi:10.2307/3062134
[23] Langlands R. P., Ann. Math. Stud. pp 96– (1980)
[24] DOI: 10.1070/RM1976v031n01ABEH001444 · Zbl 0348.12016 · doi:10.1070/RM1976v031n01ABEH001444
[25] Mazur B., Sem. Bourbaki 414 pp 1– (1972)
[26] DOI: 10.1007/BF01389997 · Zbl 0281.14016 · doi:10.1007/BF01389997
[27] DOI: 10.2307/2007099 · Zbl 0065.01404 · doi:10.2307/2007099
[28] DOI: 10.2307/1970307 · Zbl 0119.27801 · doi:10.2307/1970307
[29] DOI: 10.1007/978-0-8176-4639-4_10 · doi:10.1007/978-0-8176-4639-4_10
[30] DOI: 10.1007/BF01388636 · Zbl 0565.14006 · doi:10.1007/BF01388636
[31] DOI: 10.1007/BF01389047 · Zbl 0677.10020 · doi:10.1007/BF01389047
[32] Satake I., Publ. Math. Inst. Hautes Et. Sci. 18 pp 1– (1963)
[33] DOI: 10.1007/BF01232425 · Zbl 0810.11036 · doi:10.1007/BF01232425
[34] DOI: 10.1007/PL00005884 · Zbl 1004.11024 · doi:10.1007/PL00005884
[35] DOI: 10.2307/1971071 · Zbl 0316.12010 · doi:10.2307/1971071
[36] DOI: 10.1002/cpa.3160290618 · Zbl 0348.10015 · doi:10.1002/cpa.3160290618
[37] DOI: 10.1007/BF01391466 · Zbl 0363.10019 · doi:10.1007/BF01391466
[38] DOI: 10.1215/S0012-7094-78-04529-5 · Zbl 0394.10015 · doi:10.1215/S0012-7094-78-04529-5
[39] Shintani T., on GLn over P-adic fields, Proc. Japan Acad. 52 pp 180– (1976) · Zbl 0387.43002
[40] DOI: 10.2307/1970221 · Zbl 0222.12018 · doi:10.2307/1970221
[41] DOI: 10.2307/2118560 · Zbl 0823.11030 · doi:10.2307/2118560
[42] DOI: 10.2307/2118559 · Zbl 0823.11029 · doi:10.2307/2118559
[43] DOI: 10.1215/S0012-7094-94-07505-4 · Zbl 0823.11018 · doi:10.1215/S0012-7094-94-07505-4
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.