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Equidistribution theorems for holomorphic Siegel modular forms for \(GSp_4\); Hecke fields and n-level density

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Abstract

This paper is a continuation of Kim et al. (J Inst Math Jussieu, 2018). We supplement four results on a family of holomorphic Siegel cusp forms for \(GSp_4/\mathbb {Q}\). First, we improve the result on Hecke fields. Namely, we prove that the degree of Hecke fields is unbounded on the subspace of genuine forms which do not come from functorial lift of smaller subgroups of \(GSp_4\). Second, we prove simultaneous vertical Sato–Tate theorem. Namely, we prove simultaneous equidistribution of Hecke eigenvalues at finitely many primes. Third, we compute the n-level density of degree 4 spinor L-functions, and thus we can distinguish the symmetry type depending on the root numbers. This is conditional on certain conjecture on root numbers. Fourth, we consider equidistribution of paramodular forms. In this case, we can prove the conjecture on root numbers. Main tools are the equidistribution theorem in our previous work and Shin–Templier’s (Compos Math 150(12):2003–2053, 2014) work.

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References

  1. Bushnell, C.J., Henniart, G., Kutzko, P.: Local Rankin–Selberg convolutions for \({\rm GL}_n\): explicit conductor formula. J. Am. Math. Soc. 11(3), 703–730 (1998)

    MATH  Google Scholar 

  2. Calegari, F., Gee, T.: Irreducibility of automorphic Galois representations. Annales de l’Institut Fourier 63(5), 1881–1912 (2013)

    MathSciNet  MATH  Google Scholar 

  3. Cho, P.J., Kim, H.H.: \(n\)-level densities of Artin \(L\)-functions. IMRN 2015, 7861–7883 (2015)

    MathSciNet  MATH  Google Scholar 

  4. Cohen, H., Diaz, F., Diaz, Y., Olivier, M.: Enumerating quartic dihedral extensions of \({\mathbb{Q}}\). Comput. Math 133, 65–93 (2002)

    MathSciNet  MATH  Google Scholar 

  5. Emerton, M.: A local-global compatibility conjecture in the \(p\)-adic Langlands programme for \(GL_2/{\mathbb{Q}}\). Pure Appl. Math. Q. 2(2), 279–393 (2006) (Special Issue: In honor of John H. Coates. Part 2)

  6. Gan, W.-T., Takeda, S.: The local Langlands conjecture for \(GSp(4)\). Ann. Math. (2) 173(3), 1841–1882 (2011)

    MathSciNet  MATH  Google Scholar 

  7. Gee, T., Taïbi, O.: Arthur’s multiplicity formula for \(GSp_4\) and restriction to \(Sp_4\). arXiv:1807.03988

  8. Harris, M., Soudry, D., Taylor, R.: \(l\)-adic representations associated to modular forms over imaginary quadratic fields. I. Lifting to \(GSp_4({\mathbb{Q}})\). Invent. Math. 112(2), 377–411 (1993)

    MathSciNet  MATH  Google Scholar 

  9. Ibukiyama, T., Kitayama, H.: Dimensional formulas of paramodular forms of squarefree level and comparison with inner twist. J. Math. Soc. Japan 69, 597–671 (2017)

    MathSciNet  MATH  Google Scholar 

  10. Jorza, A.: Galois representations for holomorphic Siegel modular forms. Math. Ann. 355, 381–400 (2013)

    MathSciNet  MATH  Google Scholar 

  11. Katz, N., Sarnak, P.: Random Matrices, Frobenius Eigenvalues, and Monodromy. American Mathematical Society Colloquium Publications, vol. 45. American Mathematical Society, Providence, RI (1999)

  12. Kim, H.: An Application of Exterior Square Functoriality of \(GL_4\); Asai Lift, Number theory, CRM Proc. Lecture Notes, vol. 36, pp. 197–202. Amer. Math. Soc., Providence (2004)

  13. Khare, C., Wintenberger, J.-P.: Serre’s modularity conjecture. I. Invent. Math. 178(3), 485–504 (2009)

    MathSciNet  MATH  Google Scholar 

  14. Kisin, M.: The Fontaine–Mazur conjecture for \(GL_2\). J. Am. Math. Soc. 22(3), 641–690 (2009)

    MathSciNet  MATH  Google Scholar 

  15. Kim, H.-H., Shahidi, F.: Functorial products for \(GL_2\times GL_3\) and the symmetric cube for \(GL_2\). With an appendix by Colin J. Bushnell and Guy Henniart. Ann. Math (2) 155(3), 837–893 (2002)

    MathSciNet  Google Scholar 

  16. Kim, H.H., Wakatsuki, S., Yamauchi, T.: An equidistribution theorem for holomorphic siegel modular forms for GSP4 and its applications. J. Inst. Math. Jussieu (2018). https://doi.org/10.1017/S147474801800004X

    MATH  Google Scholar 

  17. Kim, H.H., Wakatsuki, S., Yamauchi, T.: Equidistribution theorems of holomorphic Siegel cusp forms of general degree (in preparation)

  18. Kret, A., Shin, S.W.: Galois representations for general symplectic groups (2016). arXiv:1609.04223

  19. Lapid, E.: On the root number of representations of orthogonal type. Comput. Math. 140(2), 274–286 (2004)

    MathSciNet  MATH  Google Scholar 

  20. Laumon, G.: Sur la cohomologie a supports compacts des varietes de Shimura pour \(GSp(4)_{\mathbb{Q}}\). Comput. Math. 105(3), 267–359 (1997)

    MathSciNet  MATH  Google Scholar 

  21. Laumon, G.: Fonctions zetas des varietes de Siegel de dimension trois, Formes automorphes. II. Le cas du groupe GSp(4). Astérisque 302, 1–66 (2005)

    Google Scholar 

  22. Liu, T., Yu, J.-K.: On automorphy of certain Galois representations of \(GO_4\)-type. With an appendix by Liang Xiao. J. Number Theory 161, 49–74 (2016)

    MathSciNet  Google Scholar 

  23. Mazur, B.: Modular curves and the Eisenstein ideal. Inst. Hautes Études Sci. Publ. Math. 47, 33–186 (1977)

    MathSciNet  MATH  Google Scholar 

  24. Murty, R., Sinha, K.: Effective equidistribution of eigenvalues of Hecke operators. J. Number Theory 129(3), 681–714 (2009)

    MathSciNet  MATH  Google Scholar 

  25. Roberts, B., Schmidt, R.: Local Newforms for \(GSp(4)\). Lecture Notes in Mathematics, vol. 1918. Springer, Berlin (2007)

    Google Scholar 

  26. Royer, E.: Facteurs \({\mathbb{Q}}\)-simples de \(J_0(N)\) de grande dimension et de grand rang. Bull. Soc. Math. France 128(2), 219–248 (2000)

    MathSciNet  MATH  Google Scholar 

  27. Rubinstein, M.: Low-lying zeros of \(L\)-functions and random matrix theory. Duke Math. J. 109(1), 147–181 (2001)

    MathSciNet  MATH  Google Scholar 

  28. Schmidt, R.: Iwahori-spherical representations of GSp(2) and Siegel modular forms of degree 2 with square-free level. J. Math. Soc. Japan 57, 259–293 (2005)

    MathSciNet  MATH  Google Scholar 

  29. Schmidt, R.: On classical Saito–Kurokawa liftings. J. Reine Angew. Math. 604, 211–236 (2007)

    MathSciNet  MATH  Google Scholar 

  30. Schmidt, R.: Archimedean aspects of Siegel modular forms of degree 2. Rocky Mountain J. Math. 47(7), 2381–2422 (2017)

    MathSciNet  MATH  Google Scholar 

  31. Schmidt, R.: Packet structure and paramodular forms. Trans. Am. Math. Soc. 370(5), 3085–3112 (2018)

    MathSciNet  MATH  Google Scholar 

  32. Serre, J.-P.: Répartition asymptotique des valeurs propres de l’opérateur de Hecke \(T_p\). J. Am. Math. Soc 10(1), 75–102 (1997)

    MATH  Google Scholar 

  33. Shimizu, H.: On discontinuous groups operating on the product of the upper half planes. Ann. Math (2) 77, 33–71 (1963)

    MathSciNet  MATH  Google Scholar 

  34. Shimura, G.: Class fields over real quadratic fields and Hecke operators. Ann. Math. (2) 95, 130–190 (1972)

    MathSciNet  MATH  Google Scholar 

  35. Shin, S.W., Templier, N.: Sato–Tate theorem for families and low-lying zeros of automorphic \(L\)-functions. Invent. Math. 203(1), 1–177 (2016)

    MathSciNet  MATH  Google Scholar 

  36. Shin, S.W., Templier, N.: On fields of rationality for automorphic representations. Compos. Math. 150(12), 2003–2053 (2014)

    MathSciNet  MATH  Google Scholar 

  37. Sorensen, C.M.: Galois representations attached to Hilbert–Siegel modular forms. Doc. Math. 15, 623–670 (2010)

    MathSciNet  MATH  Google Scholar 

  38. Taylor, R.: On congruences of modular forms. Thesis (1988)

  39. Wakatsuki, S.: Multiplicity formulas for discrete series representations in \(L^2(\Gamma \backslash {\rm Sp}(2,{\mathbb{R}}))\). J. Number Theory 133, 3394–3425 (2013)

    MathSciNet  MATH  Google Scholar 

  40. Weissauer, R.: Existence of Whittaker Models Related to Four Dimensional Symplectic Galois Representations, Modular Forms on Schiermonnikoog, pp. 285–310. Cambridge Univ. Press, Cambridge (2008)

    MATH  Google Scholar 

  41. Weissauer, R.: Endoscopy for \(GSp(4)\) and the cohomology of Siegel modular threefolds. Lecture Notes in Mathematics, vol. 1968. Springer, Berlin (2009)

  42. Yamauchi, T.: On \({\mathbb{Q}}\)-simple factors of Jacobian varieties of modular curves. Yokohama Math. J. 53(2), 149–160 (2007)

    MathSciNet  MATH  Google Scholar 

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Acknowledgements

We would like to thank K. Morimoto, R. Schmidt and S-W. Shin for helpful discussions. We thank the referee and P. Michel for their help.

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Authors and Affiliations

  1. Department of Mathematics, University of Toronto, Toronto, ON, M5S 2E4, Canada

    Henry H. Kim

  2. Korea Institute for Advanced Study, Seoul, South Korea

    Henry H. Kim

  3. Faculty of Mathematics and Physics, Institute of Science and Engineering, Kanazawa University, Kakumamachi, Kanazawa, Ishikawa, 920-1192, Japan

    Satoshi Wakatsuki

  4. Mathematical Institute, Tohoku University, 6-3, Aoba, Aramaki, Aoba-Ku, Sendai, 980-8578, Japan

    Takuya Yamauchi

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  1. Henry H. Kim
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  2. Satoshi Wakatsuki
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Correspondence to Henry H. Kim.

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Henry H. Kim is partially supported by NSERC #482564. Satoshi Wakatsuki is partially supported by JSPS Grant-in-Aid for Scientific Research (Nos. 26800006, 25247001, 15K04795). Takuya Yamauchi is partially supported by JSPS Grant-in-Aid for Scientific Research (C) No. 15K04787.

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Kim, H.H., Wakatsuki, S. & Yamauchi, T. Equidistribution theorems for holomorphic Siegel modular forms for \(GSp_4\); Hecke fields and n-level density. Math. Z. 295, 917–943 (2020). https://doi.org/10.1007/s00209-019-02378-7

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