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Arithmetic distribution of tempered components of cuspidal representations of \({{\,\textrm{GL}\,}}(3)\)

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Abstract

Let \(\pi =\otimes _v'\pi _v\) be an arbitrary unitary cuspidal representation of \(\textrm{GL}(3)\) over a number field F. We show, for certain ray classes \(\mathcal {C},\) that

$$\begin{aligned} \sum _{\begin{array}{c} \mathfrak {p}^2\in \mathcal {C}\\ |a_{\pi }(\mathfrak {p})|<1 \end{array}}\frac{\log N(\mathfrak {p})}{N(\mathfrak {p})^{1/2}}=+\infty , \end{aligned}$$

which implies that there are infinitely many unramified places v in \(\mathcal {C}\) such that \(\pi _v\)’s are tempered with Hecke eigenvalues lying inside the open unit disk. Furthermore, when \(F=\mathbb {Q},\) we consider the problem on the least prime p in an arithmetic progression such that \(\pi _p\) satisfies the Ramanujan conjecture. An effective upper bound of Linnik type for such a prime p is proved.

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References

  1. Allen, P.B., Calegari, F., Caraiani, A., Gee, T., Helm, D., Hung, B.V.L., Newton, J., Scholze, P., Taylor, R., Thorne, J.A.: Potential automorphy over cm fields. arXiv:1812.09999 (arXiv preprint) (2018)

  2. Blomer, V., Brumley, F.: On the Ramanujan conjecture over number fields. Ann. Math. (2) 174(1), 581–605 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  3. Blomer, V., Brumley, F.: The role of the Ramanujan conjecture in analytic number theory. Bull. Am. Math. Soc. 50(2), 267–320 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  4. Brylinski, J.-L., Labesse, J.-P.: Cohomologie d’intersection et fonctions \( L \) de certaines variétés de shimura. Ann. Sci. Ecol. Norm. Sup. 17, 361–412 (1984)

    Article  MATH  Google Scholar 

  5. Blasius, D.: Hilbert modular forms and the Ramanujan conjecture. In: Noncommutative Geometry and Number Theory, pp. 35–56. Springer, Berlin (2006)

    Chapter  MATH  Google Scholar 

  6. Brumley, F.: Effective multiplicity one on gl n and narrow zero-free regions for Rankin–Selberg l-functions. Am. J. Math. 128(6), 1455–1474 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  7. Chenevier, G., Harris, M.: Construction of automorphic galois representations, II. Camb. J. Math. 1(1), 53–73 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  8. Clozel, L.: Représentations galoisiennes associées aux représentations automorphes autoduales de \({\rm GL}(n)\). Inst. Hautes Études Sci. Publ. Math. 73, 97–145 (1991)

    Article  MATH  Google Scholar 

  9. Deligne, P.: La conjecture de weil. I. Publ. Math. Inst. Hautes Études Sci. 43(1), 273–307 (1974)

    Article  MathSciNet  MATH  Google Scholar 

  10. Deligne, P., Serre, J.-P.: Formes modulaires de poids \(1\). Ann. Sci. Ecole Norm. Sup. 7, 507–530 (1974)

    Article  MathSciNet  MATH  Google Scholar 

  11. Fröhlich, A.: Artin-root numbers and normal integral bases for quaternion fields. Inventiones Math. 17(2), 143–166 (1972)

    Article  MathSciNet  MATH  Google Scholar 

  12. Gelbart, S., Jacquet, H.: A relation between automorphic forms on \({\rm GL}(2)\) and \({\rm GL}(3)\). Proc. Nat. Acad. Sci. USA 73(10), 3348–3350 (1976)

    Article  MathSciNet  MATH  Google Scholar 

  13. Heath-Brown, D.R.: Zero-free regions for Dirichlet \(L\)-functions, and the least prime in an arithmetic progression. Proc. Lond. Math. Soc. (3) 64(2), 265–338 (1992)

    Article  MathSciNet  MATH  Google Scholar 

  14. Humphries, P., Brumley, F.: Standard zero-free regions for Rankin–Selberg l-functions via sieve theory. Math. Z. 20, 1–18 (2018)

    MATH  Google Scholar 

  15. Henniart, G.: Une preuve simple des conjectures de Langlands pour \({\rm GL}(n)\) sur un corps \(p\)-adique. Invent. Math. 139(2), 439–455 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  16. Harris, M., Taylor, R.: The Geometry and Cohomology of Some Simple Shimura Varieties, Volume 151 of Annals of Mathematics Studies. Princeton University Press, Princeton (2001). (With an appendix by Vladimir G. Berkovich)

    Google Scholar 

  17. Iwaniec, H., Kowalski, E.: Analytic Number Theory. American Mathematical Society Colloquium Publications, vol. 53. American Mathematical Society, Providence (2004)

    Google Scholar 

  18. Kim, H.H., Shahidi, F.: Cuspidality of symmetric powers with applications. Duke Math. J. 112(1), 177–197 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  19. Kim, H., Sarnak, P.: Refined estimates towards the Ramanujan and Selberg conjectures. J. Am. Math. Soc 16(1), 175–181 (2003)

    Google Scholar 

  20. Li, X.: Upper bounds on \(L\)-functions at the edge of the critical strip. Int. Math. Res. Not. 4, 727–755 (2010)

    MathSciNet  MATH  Google Scholar 

  21. Linnik, U.V.: On the least prime in an arithmetic progression. I. The basic theorem. Rec. Math. [Mat. Sbornik] N.S. 15(57), 139–178 (1944)

    MathSciNet  MATH  Google Scholar 

  22. Linnik, U.V.: On the least prime in an arithmetic progression. II. The Deuring–Heilbronn phenomenon. Rec. Math. [Mat. Sbornik] N.S. 15(57), 347–368 (1944)

    MathSciNet  MATH  Google Scholar 

  23. Liu, J., Yan, Q., Jie, W.: Two Linnik-type problems for automorphic \(L\)-functions. Math. Proc. Camb. Philos. Soc. 151(2), 219–227 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  24. Luo, W., Rudnick, Z., Sarnak, P.: On the generalized ramanujan conjecture for gl (n). In Proceedings of Symposia in Pure Mathematics, vol 66. American Mathematical Society, Providence, pp 301–310 (1999)

  25. Yan, Q.: Linnik-type problems for automorphic \(L\)-functions. J. Number Theory 130(3), 786–802 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  26. Ramakrishnan, D.: On the coefficients of cusp forms. Math. Res. Lett. 4(2–3), 295–307 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  27. Ramakrishnan, D.: Existence of Ramanujan primes for GL(3). In: Contributions to Automorphic Forms, Geometry, and Number Theory, pp. 711–717. Johns Hopkins University Press, Baltimore (2004)

    Google Scholar 

  28. Ramakrishnan, D., Yang, L.: A constraint for twist equivalence of cusp forms on gl \((n) \). Funct. Approx. Comment. Math. 20, 20 (2021)

    MathSciNet  Google Scholar 

  29. Shin, S.W.: Galois representations arising from some compact shimura varieties. Ann. Math. 20, 1645–1741 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  30. Walji, N.: On the size of Satake parameters for unitary cuspidal automorphic representations for \({\rm GL}(4)\). J. Number Theory 133(10), 3470–3484 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  31. Winnie Li, W.-C.: The Ramanujan conjecture and its applications. Philos. Trans. R. Soc. A 378(2163), 20180441 (2020)

    Article  MathSciNet  MATH  Google Scholar 

  32. Xylouris, T.: Über die Nullstellen der Dirichletschen L-Funktionen und die kleinste Primzahl in einer arithmetischen Progression, volume 404 of Bonner Mathematische Schriften [Bonn Mathematical Publications]. Universität Bonn, Mathematisches Institut, Bonn, 2011. Dissertation for the degree of Doctor of Mathematics and Natural Sciences at the University of Bonn, Bonn (2011)

  33. Yang, L.: Holomorphy of adjoint \(L\)-functions for \({\rm GL}(n)\): \(n\le 4\). Math. Ann. 20, 1–61 (2021)

    Google Scholar 

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Acknowledgements

I am very grateful to Dinakar Ramakrishnan for pleasant discussions. I would like to thank Kimball Martin for his helpful suggestions. Sincere thanks are also due to the anonymous referee for his/her careful reading and valuable comments.

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    Liyang Yang

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Yang, L. Arithmetic distribution of tempered components of cuspidal representations of \({{\,\textrm{GL}\,}}(3)\). Math. Z. 303, 66 (2023). https://doi.org/10.1007/s00209-023-03213-w

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