## Relations between cusp forms sharing Hecke eigenvalues

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- 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
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## 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.ย 1, 191โ207. 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.ย 11, 2298โ2367. MR**4190046**, DOI 10.1112/S0010437X20007496 - Stephen Gelbart and Hervรฉ Jacquet,
*A relation between automorphic representations of $\textrm {GL}(2)$ and $\textrm {GL}(3)$*, Ann. Sci. รcole Norm. Sup. (4)**11**(1978), no.ย 4, 471โ542. 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 - Hervรฉ 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.ย 143โ226. MR**1044830** - H. Jacquet and J. A. Shalika,
*On Euler products and the classification of automorphic representations. I*, Amer. J. Math.**103**(1981), no.ย 3, 499โ558. 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.ย 4, 777โ815. 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.ย 9-10, 1085โ1089 (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.ย 1, 139โ183. 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.ย 3, 495โ502. 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.ย 2, 645โ662. 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.ย 3, 837โ893. 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.ย 1, 177โ197. 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.ย 237โ256. 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.ย 5, 777โ785. MR**3286086**, DOI 10.1007/s13226-014-0088-1 - Freydoon Shahidi,
*On certain $L$-functions*, Amer. J. Math.**103**(1981), no.ย 2, 297โ355. 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.ย 1-2, 267โ270. MR**3299852**, DOI 10.1007/s00209-014-1366-7

## Bibliographic Information

**Dipendra Prasad**- Affiliation: Indian Institute of Technology Bombay, Powai, Mumbai-400076, India; and St Petersburg State University, St Petersburg, Russia
- MR Author ID: 291342
- Email: prasad.dipendra@gmail.com
**Ravi Raghunathan**- Affiliation: Indian Institute of Technology Bombay, Powai, Mumbai-400076, India
- MR Author ID: 601543
- Email: ravir@math.iitb.ac.in
- Received by editor(s): December 7, 2021
- Received by editor(s) in revised form: July 16, 2022
- Published electronically: October 7, 2022
- Additional Notes: The first author was supported by the Science and Engineering Research Board of the Department of Science and Technology, India through the JC Bose National Fellowship of the Govt. of India, project number JBR/2020/000006. His work was also supported by a grant of the Government of the Russian Federation for the state support of scientific research carried out under the agreement 14.W03.31.0030 dated 15.02.2018.
- © Copyright 2022 American Mathematical Society
- Journal: Represent. Theory
**26**(2022), 1063-1079 - MSC (2020): Primary 11F70; Secondary 22E55
- DOI: https://doi.org/10.1090/ert/626
- MathSciNet review: 4493873