Relations between cusp forms sharing Hecke eigenvalues

Prasad, Dipendra; Raghunathan, Ravi

doi:10.1090/ert/626

Relations between cusp forms sharing Hecke eigenvalues
HTML articles powered by AMS MathViewer

by Dipendra Prasad and Ravi Raghunathan

Represent. Theory 26 (2022), 1063-1079

DOI: https://doi.org/10.1090/ert/626

Published electronically: October 7, 2022

PDF | Request permission

Abstract:

In this paper we consider the question of when the set of Hecke eigenvalues of a cusp form on $GL_n(\mathbb {A}_F)$ is contained in the set of Hecke eigenvalues of a cusp form on $GL_m(\mathbb {A}_F)$ for $n \leq m$. This question is closely related to a question about finite dimensional representations of an abstract group, which also we consider in this work.

References

Frank Calegari, The Artin conjecture for some $S_5$-extensions, Math. Ann. 356 (2013), no. 1, 191–207. MR 3038126, DOI 10.1007/s00208-012-0839-4
Wee Teck Gan, Benedict H. Gross, and Dipendra Prasad, Branching laws for classical groups: the non-tempered case, Compos. Math. 156 (2020), no. 11, 2298–2367. MR 4190046, DOI 10.1112/S0010437X20007496
Stephen Gelbart and Hervé Jacquet, A relation between automorphic representations of $\textrm {GL}(2)$ and $\textrm {GL}(3)$, Ann. Sci. École Norm. Sup. (4) 11 (1978), no. 4, 471–542. MR 533066, DOI 10.24033/asens.1355
James E. Humphreys, Introduction to Lie algebras and representation theory, Graduate Texts in Mathematics, Vol. 9, Springer-Verlag, New York-Berlin, 1972. MR 0323842, DOI 10.1007/978-1-4612-6398-2
Hervé Jacquet and Joseph Shalika, Exterior square $L$-functions, Automorphic forms, Shimura varieties, and $L$-functions, Vol. II (Ann Arbor, MI, 1988) Perspect. Math., vol. 11, Academic Press, Boston, MA, 1990, pp. 143–226. MR 1044830
H. Jacquet and J. A. Shalika, On Euler products and the classification of automorphic representations. I, Amer. J. Math. 103 (1981), no. 3, 499–558. MR 618323, DOI 10.2307/2374103
H. Jacquet and J. A. Shalika, On Euler products and the classification of automorphic forms. II, Amer. J. Math. 103 (1981), no. 4, 777–815. MR 623137, DOI 10.2307/2374050
Chandrashekhar B. Khare and Michael Larsen, Abelian varieties with isogenous reductions, C. R. Math. Acad. Sci. Paris 358 (2020), no. 9-10, 1085–1089 (English, with English and French summaries). MR 4196779, DOI 10.5802/crmath.129
Henry H. Kim, Functoriality for the exterior square of $\textrm {GL}_4$ and the symmetric fourth of $\textrm {GL}_2$, J. Amer. Math. Soc. 16 (2003), no. 1, 139–183. With appendix 1 by Dinakar Ramakrishnan and appendix 2 by Kim and Peter Sarnak. MR 1937203, DOI 10.1090/S0894-0347-02-00410-1
Henry H. Kim, An example of non-normal quintic automorphic induction and modularity of symmetric powers of cusp forms of icosahedral type, Invent. Math. 156 (2004), no. 3, 495–502. MR 2061327, DOI 10.1007/s00222-003-0340-5
Henry H. Kim and Freydoon Shahidi, Symmetric cube $L$-functions for $\rm GL_2$ are entire, Ann. of Math. (2) 150 (1999), no. 2, 645–662. MR 1726704, DOI 10.2307/121091
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
Henry H. Kim and Freydoon Shahidi, Cuspidality of symmetric powers with applications, Duke Math. J. 112 (2002), no. 1, 177–197. MR 1890650, DOI 10.1215/S0012-9074-02-11215-0
Dinakar Ramakrishnan, Remarks on the symmetric powers of cusp forms on $\rm GL(2)$, Automorphic forms and $L$-functions I. Global aspects, Contemp. Math., vol. 488, Amer. Math. Soc., Providence, RI, 2009, pp. 237–256. MR 2522032, DOI 10.1090/conm/488/09570
Dinakar Ramakrishnan, An exercise concerning the selfdual cusp forms on $\textrm {GL}(3)$, Indian J. Pure Appl. Math. 45 (2014), no. 5, 777–785. MR 3286086, DOI 10.1007/s13226-014-0088-1
Freydoon Shahidi, On certain $L$-functions, Amer. J. Math. 103 (1981), no. 2, 297–355. MR 610479, DOI 10.2307/2374219
Shunsuke Yamana, On poles of the exterior cube $L$-functions for $\rm {GL}_6$, Math. Z. 279 (2015), no. 1-2, 267–270. MR 3299852, DOI 10.1007/s00209-014-1366-7