Journal of the Korean Mathematical Society (대한수학회지)

Korean Mathematical Society (대한수학회)
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STABLE AUTOMORPHIC FORMS FOR THE GENERAL LINEAR GROUP

  • Jae-Hyun Yang (Yang Institute for Advanced Study, Department of Mathematics Inha University)
  • Received : 2023.03.06
  • Accepted : 2023.11.03
  • Published : 2024.01.01
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Abstract

In this paper, we introduce the notion of the stability of automorphic forms for the general linear group and relate the stability of automorphic forms to the moduli space of real tori and the Jacobian real locus.

Keywords

References

  1. W. L. Baily, Satake's compactification of Vn, Amer. J. Math. 80 (1958), 348-364. https://doi.org/10.2307/2372789
  2. D. Bump, Automorphic forms on GL(3, R), Lecture Notes in Mathematics, 1083, Springer, Berlin, 1984. https://doi.org/10.1007/BFb0100147
  3. G. Codogni, Hyperelliptic Schottky problem and stable modular forms, Doc. Math. 21 (2016), 445-466. https://doi.org/10.4171/dm/538
  4. G. Codogni and N. I. Shepherd-Barron, The non-existence of stable Schottky forms, Compos. Math. 150 (2014), no. 4, 679-690. https://doi.org/10.1112/S0010437X13007586
  5. A. Comessatti, Sulle variet'a abeliane reali. I, II., Ann. Mat. Pura Appl. 2 (1925), no. 1, 67-106; 4 (1926), 27-72. https://doi.org/10.1007/BF02418645
  6. E. Freitag, Stabile Modulformen, Math. Ann. 230 (1977), no. 3, 197-211. https://doi.org/10.1007/BF01367576
  7. E. Freitag, Siegelsche Modulfunktionen, Grundlehren der mathematischen Wissenschaften, 254, Springer, Berlin, 1983. https://doi.org/10.1007/978-3-642-68649-8
  8. D. Goldfeld, Automorphic forms and L-functions for the group GL(n, ℝ), Cambridge Studies in Advanced Mathematics, 99, Cambridge Univ. Press, Cambridge, 2006. https://doi.org/10.1017/CBO9780511542923
  9. M. Goresky and Y.-S. Tai, The moduli space of real abelian varieties with level structure, Compositio Math. 139 (2003), no. 1, 1-27. https://doi.org/10.1023/B:COMP.0000005079.56232.e3
  10. D. Grenier, Fundamental domains for the general linear group, Pacific J. Math. 132 (1988), no. 2, 293-317. http://projecteuclid.org/euclid.pjm/1102689682 102689682
  11. D. Grenier, An analogue of Siegel's ϕ-operator for automorphic forms for GLn(ℤ), Trans. Amer. Math. Soc. 333 (1992), no. 1, 463-477. https://doi.org/10.2307/2154119
  12. D. Grenier, On the shape of fundamental domains in GL(n, ℝ)/O(n), Pacific J. Math. 160 (1993), no. 1, 53-66. http://projecteuclid.org/euclid.pjm/1102624564 102624564
  13. K. Imai and A. Terras, The Fourier expansion of Eisenstein series for GL(3, ℤ), Trans. Amer. Math. Soc. 273 (1982), no. 2, 679-694. https://doi.org/10.2307/1999935
  14. H. Maass, Die Bestimmung der Dirichletreihen mit Grossencharakteren zu den Modulformen n-ten Grades, J. Indian Math. Soc. (N.S.) 19 (1955), 1-23.
  15. H. Maass, Siegel's modular forms and Dirichlet series, Lecture Notes in Mathematics, Vol. 216, Springer, Berlin, 1971.
  16. H. Minkowski, Gesammelte Abhandlungen, Chelsea, New York, 1967.
  17. D. Mumford, Abelian Varieties, Oxford University Press, 1970: Reprinted 1985.
  18. A. Selberg, Harmonic analysis and discontinuous groups in weakly symmetric Riemannian spaces with applications to Dirichlet series, J. Indian Math. Soc. (N.S.) 20 (1956), 47-87; Collected Papers, Volume I, Springer-Verlag (1989), 423-463.
  19. A. Selberg, Discontinuous groups and harmonic analysis, Proceedings of ICM, Stockholm (1962), 177-189; Collected Papers, Volume I, Springer-Verlag (1989), 493-505.
  20. M. Seppala and R. Silhol, Moduli spaces for real algebraic curves and real abelian varieties, Math. Z. 201 (1989), no. 2, 151-165. https://doi.org/10.1007/BF01160673
  21. C. L. Siegel, The volume of the fundamental domain for some infinite groups, Trans. Amer. Math. Soc. 39 (1936), no. 2, 209-218. https://doi.org/10.2307/1989745
  22. R. Silhol, Real abelian varieties and the theory of Comessatti, Math. Z. 181 (1982), no. 3, 345-364. https://doi.org/10.1007/BF01161982
  23. R. Silhol, Real Algebraic Surfaces, Lecture Notes in Mathematics, 1392, Springer, Berlin, 1989. https://doi.org/10.1007/BFb0088815
  24. R. Silhol, Compactifications of moduli spaces in real algebraic geometry, Invent. Math. 107 (1992), no. 1, 151-202. https://doi.org/10.1007/BF01231886
  25. A. Terras, Harmonic Analysis on Symmetric Spaces and Applications. II, Springer, Berlin, 1988. https://doi.org/10.1007/978-1-4612-3820-1
  26. J.-H. Yang, Polarized real tori, J. Korean Math. Soc. 52 (2015), no. 2, 269-331. https://doi.org/10.4134/JKMS.2015.52.2.269