DOI QR코드

DOI QR Code

Problems in the Geometry of the Siegel-Jacobi Space

  • Received : 2023.09.05
  • Accepted : 2023.10.26
  • Published : 2024.09.30

Abstract

The Siegel-Jacobi space is a non-symmetric homogeneous space which is very important geometrically and arithmetically. In this short paper, we propose the basic problems in the geometry of the Siegel-Jacobi space.

Keywords

References

  1. A. Ash, D. Mumford, M. Rapoport and Y.-S. Tai, Smooth Compactifications of Locally Symmetric Varieties, Cambridge Math. Lib., Cambridge University Press, Cambridge, 2010, x+230 pp. 
  2. E. Balslev, Spectral theory of the Laplacian on the modular Jacobi group manifold, preprint, Aarhus University (2012). 
  3. J. I. Burgos Gil, J. Kramer and U. Kuhn, The Singularities of the invariant metric on the Jacobi line bundle, London Math. Soc. Lecture Note Ser. 427, Cambridge University Press, Cambridge, 2016, 45-77. 
  4. M. Eichler and D. Zagier, The Theory of Jacobi Forms, Progr. Math. 55, Birkhauser Boston, Inc., Boston, MA, 1985, v+148 pp. 
  5. Harish-Chandra, Representations of a semisimple Lie group on a Banach space. I, Trans. Amer. Math. Soc., 75(1953), 185-243.  https://doi.org/10.1090/S0002-9947-1953-0056610-2
  6. Harish-Chandra, The characters of semisimple Lie groups, Trans. Amer. Math. Soc., 83(1956), 98-163.  https://doi.org/10.1090/S0002-9947-1956-0080875-7
  7. S. Helgason, Groups and geometric analysis, Pure Appl. Math. 113, Academic Press, Inc., Orlando, FL, 1984, xix+654 pp. 
  8. G. Khan and J. Zhang, A hall of statistical mirrors, Asian J. Math., 26(6)(2022), 809-846.  https://doi.org/10.4310/AJM.2022.v26.n6.a3
  9. J. Kramer, A geometrical approach to the theory of Jacobi forms, Compositio Math., 79(1)(1991), 1-19. 
  10. J. Kramer, A geometrical approach to Jacobi forms, revisited, 2014, Oberwolfach reports from the workshop on modular forms held April 22-May 2, 2014, organized by J. H. Bruinier, A. Ichino, T. Ikeda and O. Imamoglu, Report No. 22/2014. DOI: 10.4171/OWR/2014/22. 
  11. J. Kramer and A.-M. von Pippich, Snapshots of modern mathematics from Oberwolfach: Special Values of Zeta Functions and Areas of Triangles, Notices Amer. Math. Soc., 63(8)(2016), 917-922.  https://doi.org/10.1090/noti1413
  12. H. Maass, Die Differentialgleichungen in der Theorie der Siegelschen Modulfunktionen, Math. Ann., 126(1953), 44-68.  https://doi.org/10.1007/BF01343149
  13. H. Maass, Siegel's modular forms and Dirichlet series, Lecture Notes in Math. 216, Springer-Verlag, Berlin-New York, 1971, v+328 pp. 
  14. D. B. Mumford, Hirzebruch's Proportionality Theorem in the Non-Compact Case, Invent. Math., 42(1977), 239-272.  https://doi.org/10.1007/BF01389790
  15. R. Pink, Arithmetical compactification of mixed Shimura varieties, Bonner Math. Schriften, 209 [Bonn Mathematical Publications] Universitat Bonn, Mathematisches Institut, Bonn, 1990, xviii+340 pp. 
  16. C. L. Siegel, Symplectic Geometry, Amer. J. Math. 65(1943), 1-86; Academic Press, New York and London (1964); Gesammelte Abhandlungen, no. 41, vol. II, Springer-Verlag (1966), 274-359.  https://doi.org/10.2307/2371774
  17. C. L. Siegel, Topics in complex function theory. Vol. III: Abelian Functions and Modular Functions of Several Variables, Wiley Classics Lib. Wiley-Intersci. Publ., John Wiley & Sons, Inc., New York, 1989, x+244 pp. 
  18. J.-H. Yang, Invariant metrics and Laplacians on Siegel-Jacobi space, J. Number Theory, 127(1)(2007), 83-102.  https://doi.org/10.1016/j.jnt.2006.12.014
  19. J.-H. Yang, Invariant differential operators on Siegel-Jacobi space and Maass-Jacobi forms, Proceedings of the International Conference (October 10-12, 2012) on Geometry, number theory and representation theory, 37-63. KM Kyung Moon Sa, Seoul (2013). ISBN:978-89-88615-35-5. 
  20. J.-H. Yang, Geometry and Arithmetic on the Siegel-Jacobi Space, Geometry and Analysis on Manifolds, In Memory of Professor Shoshichi Kobayashi (edited by T. Ochiai, A. Weinstein et al), Progr. Math. 308, Birkhauser, Springer International Publishing AG Switzerland (2015), 275-325. 
  21. J.-H. Yang, Y.-H. Yong, S.-N. Huh, J.-H. Shin and G.-H. Min, Sectional Curvatures of the Siegel-Jacobi Space, Bull. Korean Math. Soc., 50(3)(2013), 787-799.  https://doi.org/10.4134/BKMS.2013.50.3.787
  22. J. Yang and L. Yin, Differential Operators for Siegel-Jacobi forms, Sci. China Math., 59(6)(2016), 1029-1050.  https://doi.org/10.1007/s11425-015-5111-4
  23. C. Ziegler, Jacobi Forms of Higher Degree, Abh. Math. Sem. Univ. Hamburg, 59(1989), 191-224. https://doi.org/10.1007/BF02942329