Academia.eduAcademia.edu

Outline

Title
Abstract
References
All Topics
Mathematics
Number Theory

Theory of the Siegel Modular Variety

Profile image of Jae-Hyun YangJae-Hyun Yang

2007

https://doi.org/10.1007/978-93-86279-46-0_16
visibility

…

description

52 pages

link

1 file

Abstract

In this paper, we discuss the theory of the Siegel modular variety in the aspects of arithmetic and geometry. This article covers the theory of Siegel modular forms, the Hecke theory, a lifting of elliptic cusp forms, geometric properties of the Siegel modular variety, (hypothetical) motives attached to Siegel modular forms and a cohomology of the Siegel modular variety.

Related papers

Siegel modular forms and representations
Ralf Schmidt

manuscripta mathematica, 2001

This paper explicitly describes the procedure of associating an automorphic representation of PGSp(2n, A) with a Siegel modular form of degree n for the full modular group n = Sp(2n, Z), generalizing the well-known procedure for n = 1. This will show that the so-called "standard" and "spinor" L-functions associated with such forms are obtained as Langlands L-functions. The theory of Euler products, developed by Langlands, applied to a Levi subgroup of the exceptional group of type F 4 , is then used to establish meromorphic continuation for the spinor L-function when n = 3.

View PDFchevron_right
Pullback formula for vector valued Siegel modular forms and its applications
noritomo kozima

arXiv (Cornell University), 2021

View PDFchevron_right
Galois representations for holomorphic Siegel modular forms
Andrei Jorza

Mathematische Annalen, 2013

We prove local-global compatibility (up to a quadratic twist) of Galois representations associated to holomorphic Hilbert-Siegel modular forms in many cases (induced from Borel or Klingen parabolic), and as a corollary we obtain a conjecture of Skinner and Urban. For Siegel modular forms, when the local representation is an irreducible principal series we get local-global compatibility without a twist. We achieve this by proving a version of rigidity (strong multiplicity one) for GSp(4) using, on the one hand the doubling method to compute the standard L-function, and on the other hand the explicit classification of the irreducible local representations of GSp(4) over p-adic fields; then we use the existence of a globally generic Hilbert-Siegel modular form weakly equivalent to the original and we refer to Sorensen (Mathematica 15:623-670, 2010) for local-global compatibility in that case.

View PDFchevron_right
The Trace of Hecke operators on the space of classical holomorphic Siegel modular forms of genus two
Rainer Weissauer

Eprint Arxiv 0909 1744, 2009

We prove multiplicity one for vector valued holomorphic Siegel modular forms of weights greater or equal to 3 and the full Siegel modular group and give a trace formula for the action of the Hecke operators T(p) in the regular cases.

View PDFchevron_right
Relating Siegel cusp forms to Siegel–Maaß forms
Antareep Mandal

Research in Number Theory

In this paper we generalize a well-known isomorphism between the space of cusp forms of weight k for a Fuchsian subgroup of the first kind $$\Gamma \subset \mathrm {SL}_{2}({\mathbb {R}})$$ Γ ⊂ SL 2 ( R ) and the space of certain Maaß forms of weight k for $$\Gamma $$ Γ to an isomorphism between the space of Siegel cusp forms of weight k for a subgroup $$\Gamma \subset \mathrm {Sp}_{n}({\mathbb {R}})$$ Γ ⊂ Sp n ( R ) , which is commensurable with the Siegel modular group $$\mathrm {Sp}_{n}({\mathbb {Z}})$$ Sp n ( Z ) , and a suitable space of Siegel–Maaß forms of weight k for $$\Gamma $$ Γ .

View PDFchevron_right
A note on Fourier-Jacobi coefficients of Siegel modular forms
Narasimha Kumar

Archiv der Mathematik, 2013

Let F be a Siegel cusp form of weight k and genus n > 1 with Fourier-Jacobi coefficients f m. In this article, we estimate the growth of the Petersson norms of f m , where m runs over an arithmetic progression. This result sharpens a recent result of Kohnen in [5].

View PDFchevron_right
On the transcendence degree of the differential field generated by Siegel modular forms
Wadim Zudilin

Journal für die reine und angewandte Mathematik (Crelles Journal), 2000

It is a classical fact that the elliptic modular function λ = (ϑ 10 /ϑ 00 ) 4 satisfies an algebraic differential equation of order 3 (this goes back to Jacobi's Fundamenta nova), and none of lower order (cf.

View PDFchevron_right
Siegel modular forms and finite symplectic groups
Francesco Dalla Piazza

Advances in Theoretical and Mathematical Physics, 2009

The finite symplectic group Sp(2g) over the field of two elements has a natural representation on the vector space of Siegel modular forms of given weight for the principal congruence subgroup of level two. In this paper we decompose this representation, for various (small) values of the genus and the level, into irreducible representations. As a consequence we obtain uniqueness results for certain modular forms related to the superstring measure, a better understanding of certain modular forms in genus three studied by D'Hoker and Phong as well as a new construction of Miyawaki's cusp form of weight twelve in genus three.

View PDFchevron_right
Mathieu moonshine and Siegel Modular Forms
Suresh Govindarajan

arXiv: High Energy Physics - Theory, 2020

A second-quantized version of Mathieu moonshine leads to product formulae for functions that are potentially genus-two Siegel Modular Forms analogous to the Igusa Cusp Form. The modularity of these functions do not follow in an obvious manner. For some conjugacy classes, but not all, they match known modular forms. In this paper,we prove that the product formulae for all conjugacy classes of $M_{24}$ are indeed Siegel Modular Forms.

View PDFchevron_right
On Fundamental Fourier Coefficients of Siegel Modular Forms
Soumya Das

Journal of the Institute of Mathematics of Jussieu, 2021

We prove that if F is a nonzero (possibly noncuspidal) vector-valued Siegel modular form of any degree, then it has infinitely many nonzero Fourier coefficients which are indexed by half-integral matrices having odd, square-free (and thus fundamental) discriminant. The proof uses an induction argument in the setting of vector-valued modular forms. Further, as an application of a variant of our result and complementing the work of A. Pollack, we show how to obtain an unconditional proof of the functional equation of the spinor L-function of a holomorphic cuspidal Siegel eigenform of degree $3$ and level $1$ .

View PDFchevron_right

References (169)

  1. A. N. Andrianov, Modular descent and the Saito-Kurokawa lift, Invent. Math. 289, Springer-Verlag (1987).
  2. A. N. Andrianov, Quadratic forms and Hecke operators, Grundlehren der Mathematik 53 (1979), 267- 280.
  3. A. N. Andrianov and V. L. Kalinin, On the anlaytic properties of standard zeta functions of Siegel modular forms, Math. USSR. Sb. 35 (1979), 1-17.
  4. A. N. Andrianov and V. G. Zhuravlev, Modular forms and Hecke operators, Translations of Mathemat- ical Monographs 145, AMS, Providence, RI (1995).
  5. A. Ash, D. Mumford, M. Rapoport and Y. Tai, Smooth compactification of locally symmetric varieties, Math. Sci. Press (53 Jordan Rd., Brookline, MA. 02146, USA), 1975.
  6. W. Baily, Satake's compactification of V * n , Amer. J. Math. 80 (1958), 348-364.
  7. W. Baily and A. Borel, Compactification of arithmetic quotients of bounded symmetric domains, Ann. Math. 84 (1966), 442-528.
  8. R. Berndt, Zur Arithmetik der elliptischen Funktionenkörper höherer Stufe, J. reine angew. Math., 326(1981), 79-94.
  9. R. Berndt, Meromorphic Funktionen auf Mumfords Kompaktifizierung der universellen elliptischen Kurve N -ter Stufe, J. reine angew. Math., 326(1981), 95-103.
  10. R. Berndt, Shimuras Reziprozitätsgesetz für den Körper der arithmetischen elliptischen Funktionen beliebiger Stufe , J. reine angew. Math., 343(1983), 123-145.
  11. R. Berndt, Die Jacobigruppe und die Wärmeleitungsgleichung , Math. Z., 191(1986), 351-361.
  12. R. Berndt, The Continuous Part of L 2 (Γ J \G J ) for the Jacobi Group, Abh. Math. Sem. Univ. Hamburg., 60(1990), 225-248.
  13. R. Berndt and S. Böcherer, Jacobi Forms and Discrete Series Representations of the Jacobi Group, Math. Z., 204(1990), 13-44.
  14. R. Berndt, On Automorphic Forms for the Jacobi Group , Jb. d. Dt. Math.-Verein., 97(1995), 1-18.
  15. R. Berndt and R. Schmidt, Elements of the Representation Theory of the Jacobi Group, Birkhäuser, 1998.
  16. F. A. Bogomolov and P. I. Katsylo, Rationality of some quotient varieties, Math. USSR Sbornik., 54(1986), 571-576.
  17. R. Borcherds, E. Freitag and R. Weissauer, A Siegel cusp form of degree 12 and weight 12, J. reine angew. Math. 494 (1998), 141-153.
  18. S. Breulmann and M. Kuss, On a conjecture of Duke-Imamoǧlu, Proc. Ameri. Math. Soc. 128 (2000), 1595-1604.
  19. D. Bump and Y. J. Choie, Derivatives of modular forms of negative weight, Pure Appl. Math. Q. 2 (2006), no. 1, 111-133.
  20. R. Charney and R. Lee, Cohomology of the Satake compactification , Topology 22(1983), 389-423.
  21. H. Clemens, Double Solids, Adv. Math. 47(1983), 107-230.
  22. H. Clemens and P. Griffiths, , Ann. of Math. 95(1972), 281-356.
  23. C. Consani and C. Faber, On the cusp forms motives in genus 1 and level 1, arXiv:math.AG/0504418.
  24. P. Deligne, Formes modulaires et représentations ℓ-adiques: Sém. Bourbaki 1968/9, no. 355; Lecture Notes in Math. 179, Springer-Verlag, Berlin-Heidelberg-New York (1971), 139-172.
  25. P. Deligne, Valeurs de fonctions L, et périods d'intégrals, Proc. Symp. Pure Math., Amer. Math. Soc., Providence, RI, 33, Part 2 (1979), 313-346.
  26. R. Donagi, The unirationality of A5, Ann. of Math., 119(1984), 269-307.
  27. W. Duke and Ö. Imamoǧlu, A converse theorem and the Saito-Kurokawa lift, IMRN 310 (1996), 347- 355.
  28. W. Duke and Ö. Imamoǧlu, Siegel modular forms of small weight, Math. Ann. 310 (1998), 73-82.
  29. N. Dummigan, Period ratios of modular forms, Math. Ann. 318 (2000), 621-636.
  30. M. Eichler and D. Zagier, The Theory of Jacobi Forms, Progress in Mathematics 55, Birkhäuser, Boston, Basel and Stuttgart, 1985.
  31. G. Faltings and C.-L. Chai, Degeneration of abelian varieties, Ergebnisse der Math. 22, Springer-Verlag, Berlin-Heidelberg-New York (1990).
  32. A. J. Feingold and I. B. Frenkel, A Hyperbolic Kac-Moody Algebra and the Theory of Siegel Modular Forms of genus 2, Math. Ann., 263(1983), 87-144.
  33. E. Freitag, Die Kodairadimension von Körpern automorpher Funktionen, J. reine angew. Math. 296 (1977), 162-170.
  34. E. Freitag, Stabile Modulformen, Math. Ann. 230 (1977), 162-170.
  35. E. Freitag, Siegelsche Modulfunktionen, Grundlehren de mathematischen Wissenschaften 55, Springer- Verlag, Berlin-Heidelberg-New York (1983).
  36. E. Freitag, Holomorphic tensors on subvarieties of the Siegel modular variety, Birkhäuser, Prog. Math. Boston, 46(1984), 93-113.
  37. G. Frobenius, Über die Beziehungen zwischen 28 Doppeltangenten einer eben Curve vierte Ordnung , J. reine angew. Math., 99(1886), 285-314.
  38. G. van der Geer, Note on abelian schemes of level three, Math. Ann., 278(1987), 401-408.
  39. G. van der Geer, Hilbert Modular Surfaces, Springer-Verlag, Berlin-Heidelberg-New York (1987).
  40. G. van der Geer, Siegel Modular Forms, arXiv:math.AG/0605346.
  41. S. Gelbart and H. Jacquet, A relation between automorphic representations of GL(2) and GL(3), Ann. Sci. École Norm Sup. 11 (1978), 471-542.
  42. M. Goresky and W. Pardon, Chern classes of automorphic vector bundles, Invent. Math. 147 (2002), 561-612.
  43. M. Goresky and Y.-S. Tai, Toroidal and reductive Borel-Serre compactifications of locally symmetric spaces, Amer. J. Math. 121 (1999), no. 5, 1095-1151.
  44. D. Grenier, An analogue of Siegel's φ-operator for automorphic forms for GL(n, Z), Trans. Amer. Math. Soc. 331, No. 1 (1992), 463-477.
  45. V. A. Gritsenko, The action of modular operators on the Fourier-Jacobi coefficients of modular forms, Math. USSR Sbornik, 74(1984), 237-268.
  46. S. Grushevsky and D. Lehavi, Ag is of general type, arXiv:math.AG/0512530.
  47. Harish-Chandra, Representations of a semisimple Lie group on a Banach space. I., Trans. Amer. Math. Soc. 75 (1953), 185-243.
  48. Harish-Chandra, The characters of semisimple Lie groups, Trans. Amer. Math. Soc. 83 (1956), 98-163.
  49. J. Harris and D. Mumford, On the Kodaira dimension of the moduli space of curves, Invent. Math., 67(1982), 23-88.
  50. M. Harris, Arithmetic vector bundles and automorphic forms on Shimura varieties I, Invent. Math. 82 (1985), 151-189.
  51. M. Harris, Arithmetic vector bundles and automorphic forms on Shimura varieties II, Compositio Math. 60 (1986), no. 3, 323-378.
  52. M. Harris and S. Zucker, Boundary cohomology of Shimura varieties I, Ann. Sci. École Norm Sup. 27 (1994), 249-344.
  53. M. Harris and S. Zucker, Boundary cohomology of Shimura varieties II, Invent. Math. 116 (1994), 243-308.
  54. S. Helgason, Differential operators on homogeneous spaces, Acta Math. 102 (1959), 239-299.
  55. S. Helgason, Groups and geometric analysis, Academic Press, New York (1984).
  56. H. Hida, p-adic Automorphic Forms on Shimura Varieties, Springer-Verlag, New York (2004).
  57. F. Hirzebruch, Automorphe Formen und der Satz von Riemann-Roch, Symposium Internacional de Topologia, Unesco (1958).
  58. J. W. Hoffman and S. H. Weintraub, Cohomology of the Siegel modular group of degree 2 and level 4 , Mem. Amer. Math. Soc. 631 (1998), 59-75.
  59. J. W. Hoffman and S. H. Weintraub, Cohomology of the boundary of the Siegel modular varieties of degree 2, with applications, Fund. Math. 178 (2003), 1-47.
  60. R. Howe, Perspectives on invariant theory: Schur duality, multiplicity-free actions and beyond, The Schur lectures (1992) (Tel Aviv), Israel Math. Conf. Proceedings, vol. 8 (1995), 1-182.
  61. K. Hulek and G. K. Sankaran, The Geometry of Siegel modular varieties, Advanced Studies in Pure Mathematics, Higher Dimensional Birational Geometry 35 (2002), 89-156 or ArXiv:math.AG/9810153.
  62. J. Igusa, On Siegel modular forms of genus two, Amer. J. Math. 84 (1962), 175-200.
  63. J. Igusa, On Siegel modular forms of genus two II, Amer. J. Math. 86 (1964), 392-412.
  64. J. Igusa, On the grades ring of theta constants, Amer. J. Math. 86 (1964), 219-246.
  65. J. Igusa, On the grades ring of theta constants II, Amer. J. Math. 88 (1966), 221-236.
  66. J. Igusa, Modular forms and projective invariants, Amer. J. Math. 89 (1967), 817-855.
  67. J. Igusa, Theta Functions, Springer-Verlag, Berlin-Heidelberg-New York (1971).
  68. T. Ikeda, On the lifting of elliptic cusp forms to Siegel cusp forms of degree 2n, Ann. Math. 154 (2001), 641-681.
  69. T. Ikeda, Pullback of the lifting of elliptic cusp forms and Miyawaki's conjecture, Duke Math. J. 131 (2006), no. 3, 469-497.
  70. C. G. J. Jacobi, Fundamenta nova theoriae functionum ellipticum, Königsberg, (1829).
  71. H. Klingen, Zum Darstellungssatz für Siegelsche Modulformen, Math. Zeitschr. 102 (1967), 30-43.
  72. H. Klingen, Introductory lectures on Siegel modular forms, Cambridge Studies in advanced mathematics 20, Cambridge University Press (1990).
  73. A. W. Knapp, Representation Theory of Semisimple Groups, Princeton University Press, Princeton, New Jersey (1986).
  74. W. Kohnen, Modular forms of half-integral weight on Γ0(4), Math. Ann. 248 (1980), 249-266.
  75. W. Kohnen, Lifting modular forms of half-integral weight to Siegel modular forms of even degree, Math. Ann. 322 (2003), 787-809.
  76. W. Kohnen and H. Kojima, A Maass space in higher genus, Compositio Math. 141 (2005), 313-322.
  77. W. Kohnen and D. Zagier, Modular forms with rational periods, in: Modular forms (Durham, 1983), 197-249, Ellis Horwood Ser. Math. Appl. (1984).
  78. J. Kollar and F. O. Schreyer, The moduli of curves is stably rational for g ≤ 6 , Duke Math. J., 51(1984), 239-242.
  79. A. Korányi and J. Wolf, Generalized Cayley transformations of bounded symmetric domains, Amer. J. Math. 87 (1965), 899-939.
  80. J. Kramer, A geometrical approach to the theory of Jacobi forms, Compositio Math., 79(1991), 1-19.
  81. J. Kramer, An arithmetic theory of Jacobi forms in higher dimensions , J. reine angew. Math., 458(1995), 157-182.
  82. S. Kudla and S. Rallis, A regularized Siegel-Weil formula : the first term identity, Ann. Math. 140 (1994), 1-80.
  83. N. Kurokawa, Examples of eigenvalues of Hecke operators on Siegel cusp forms of degree two, Invent. Math. 49 (1978), 149-165.
  84. N. V. Kuznetsov, A new class of identities for the Fourier coefficients of modular forms, Acta Arith. (1975), 505-519.
  85. J.P. Labesse and J. Schwermer, On liftings and cusp cohomology of arithmetic groups, Invent. Math. 83 (1986), 283-401.
  86. R. Langlands, Base change for GL(2), Ann. of Math. Syud., 96, Princeton Univ. Press, Princeton (1980).
  87. G. Laumon, Sur la cohomologie à supports compacts des variétés de Shimura GSp(4) Q , Compositio Math. 105 (1997), 267-359.
  88. R. Lee and S. H. Weintraub, Cohomology of a Siegel modular variety of degree 2, Contemp. Math. 36, Amer. Math. Soc. (1985), 433-488.
  89. J.-S. Li and J. Schwermer, On the Eisenstein cohomology of arithmetic groups, Duke Math. J. 123 (2004), 141-169.
  90. G. Lion and M. Vergne, The Weil representation, Maslov index and Theta series, Progress in Mathe- matics, 6, Birkhäuser, Boston, Basel and Stuttgart, 1980.
  91. H. Maass, Die Differentialgleichungen in der Theorie der Siegelschen Modulfunktionen, Math. Ann. 126 (1953), 44-68.
  92. H. Maass, Siegel modular forms and Dirichlet series, Lecture Notes in Math. 216, Springer-Verlag, Berlin-Heidelberg-New York (1971).
  93. H. Maass, Über eine Spezialschar von Modulformen zweiten Grades I, Invent. Math. 52 (1979), 95-104.
  94. H. Maass, Über eine Spezialschar von Modulformen zweiten Grades II, Invent. Math. 53 (1979), 249-253.
  95. H. Maass, Über eine Spezialschar von Modulformen zweiten Grades III, Invent. Math. 53 (1979), 255- 265.
  96. R. MacPherson, Chern classes for singular varieties, Ann. of Math. 100 (1974), 423-432.
  97. J. S. Milne, Canonical models of (mixed) Shimura varieties and automorphic vector bundles, in: Auto- morphic forms, Shimura varieties and L-functions, edited by L. Clozel and J. S. Milne, Academic Press (1990), 283-414.
  98. H. Minkowski, Gesammelte Abhandlungen: Chelsea, New York (1967).
  99. S. Mizumoto, Poles and residues of standard L-functions attached to Siegel modular forms, Math. Ann. 289 (1991), 589-612.
  100. D. Mumford, On the equations defining Abelian varieties I, II, III, Invent. math. 1 (1966), 287-354; 3 (1967), 75-135; 3 (1967), 215-244.
  101. D. Mumford, Abelian Varieties, Oxford University Press (1970): Reprinted (1985).
  102. D. Mumford, Hirzebruch's Proportionality Theorem in the Non-Compact Case, Invent. Math. 42 (1977), 239-272.
  103. D. Mumford, On the Kodaira dimension of the Siegel modular variety, Lecture Notes in Math. 997, Springer-Verlag, Berlin (1983), 348-375.
  104. D. Mumford, Tata Lectures on Theta I, Progress in Mathematics, 28, Birkhäuser, Boston, Basel and Stuttgart, 1983.
  105. A. Murase, L-functions attached to Jacobi forms of degree n. Part I : The Basic Identity, J. reine angew. Math., 401(1989), 122-156.
  106. A. Murase, L-functions attached to Jacobi forms of degree n. Part II : Functional Equation, Math. Ann., 290 (1991), 247-276.
  107. A. Murase and T. Sugano, Whittaker-Shintani Functions on the Symplectic Group of Fourier-Jacobi Type, Compositio Math., 79(1991), 321-349.
  108. Y. Namikawa, Toroidal compactification of Siegel spaces, Lecture Notes in Math. 812, Springer-Verlag, Berlin-Heidelberg-New York (1980).
  109. T. Oda and J. Schwermer, Mixed Hodge structures and automorphic forms for Siegel modular varieties of degree two, Math. Ann. 286 (1990), 481-509.
  110. I. Piateski-Sharpiro, Automorphic Functions and the Geometry of Classical Domains, Gordan-Breach, New York (1966).
  111. C. Poor, Schottky's form and the hyperelliptic locus, Proc. Amer. Math. Soc. 124 (1996), 1987-1991.
  112. B. Riemann, Zur Theorie der Abel'schen Funktionen für den p = 3, Math. Werke, Teubener, Leipzig, (1876), 456-476.
  113. B. Runge, Theta functions and Siegel-Jacobi functions, Acta Math., 175(1995), 165-196.
  114. N. C. Ryan, Computing the Satake p-parameters of Siegel modular forms, arXiv:math.NT/0411393.
  115. L. Saper, L2-cohomology of arithmetic varieties, Ann. Math.(2) 132 (1990), no. 1, 1-69.
  116. L. Saper, On the cohomology of locally symmetric spaces and of their compactifications, Current de- velopments in mathematics, 2002, 219-289, Int. Press, Somerville, MA, 2003.
  117. L. Saper, Geometric rationality of equal-rank Satake compactifications, Math. Res. Lett. 11 (2004), n0. 5-6, 653-671.
  118. I. Satake, On the compactification of the Siegel space, J. Indian Math. Soc. 20 (1956), 259-281.
  119. I. Satake, Theory of spherical functions on reductive algebraic groups over p-adic fields, Publ. Math. IHES, Nr. 20 (1963).
  120. I. Satake, Fock Representations and Theta Functions, Ann. Math. Studies, 66(1971), 393-405.
  121. I. Satake, Algebraic Structures of Symmetric Domains, Kano Memorial Lectures 4, Iwanami Shoton, Publishers and Princeton University Press (1980).
  122. T. Satoh, On certain vector valued Siegel modular forms of degree two, Math. Ann. 274 (1986), 335- 352.
  123. A. J. Scholl, Motives for modular forms, Invent. Math. 100 (1990), 419-430.
  124. J. Schwermer, On arithmetic quotients of the Siegel upper half space of degree two, Compositio Math., 58(1986), 233-258.
  125. J. Schwermer, On Euler products and residual Eisenstein cohomology classes for Siegel modular vari- eties, Forum Math., 7(1995), 1-28.
  126. G. Shimura, Introduction to the arithmetic theory of automorphic functions, Kano Memorial Lectures 1, Iwanami Shoton, Publishers and Princeton University Press (1971).
  127. G. Shimura, On modular correspondence for Sp(n, Z) and their congruence relations, Proc. Acad. Sci. USA 49 (1963), 824-828.
  128. G. Shimura, On modular forms of half integral weight , Ann. of Math., 97(1973), 440-481.
  129. G. Shimura, On certain reciprocity laws for theta functions and modular forms , Acta Math., 141(1979), 35-71.
  130. G. Shimura, Invariant differential operators on hermitian symmetric spaces, Ann. of Math., 132(1990), 237-272.
  131. C. L. Siegel, Symplectic Geometry, Amer. J. Math. 65 (1943), 1-86; Academic Press, New York and London (1964);
  132. Gesammelte Abhandlungen, no. 41, vol. II, Springer-Verlag (1966), 274-359.
  133. C. L. Siegel, Gesammelte Abhandlungen I-IV, Springer-Verlag(I-III: 1966; IV: 1979).
  134. C. L. Siegel, Topics in Complex Function Theory : Abelian Functions and Modular Functions of Several Variables, vol. III, Wiley-Interscience, 1973.
  135. Y.-S. Tai, On the Kodaira Dimension of the Moduli Space of Abelian Varieties, Invent. Math. 68 (1982), 425-439.
  136. R. Taylor, On the ℓ-adic cohomology of Siegel threefolds, Invent. Math. 114 (1993), 289-310.
  137. B. Totaro, Chern numbers for singular varieties and elliptic homology, Ann. Math.(2) 151 (2000), no. 2, 757-791.
  138. R. Tsushima, A formula for the dimension of spaces of Siegel cusp forms of degree three, Amer. J. Math. 102 (1980), 937-977.
  139. S. Tsuyumine, On Siegel modular forms of degree three, Amer. J. Math. 108 (1986), 755-862 : Appen- dum. Amer. J. Math. 108 (1986), 1001-1003.
  140. T. Veenstra, Siegel modular forms, L-functions and Satake parameters, Journal of Number Theory 87 (2001), 15-30.
  141. W. Wang, On the Smooth Compactification of Siegel Spaces , J. Diff. Geometry, 38(1993), 351-386.
  142. W. Wang, On the moduli space of principally polarized abelian varieties, Contemporary Math., 150(1993), 361-365.
  143. H. Weber, Theorie der Abel'schen Funktionen von Geschlecht 3, Berlin, (1876).
  144. R. Weissauer, Vektorwertige Modulformen kleinen Gewichts, J. Reine Angew. Math. 343 (1983), 184- 202.
  145. R. Weissauer, Stabile Modulformen und Eisensteinreihnen, Lecture Notes in Math. 1219, Springer- Verlag, Berlin (1986).
  146. R. Weissauer, Untervarietäten der Siegelschen Modulmannigfatigkeiten von allgemeinen Typ, Math. Ann., 343(1983), 209-220.
  147. R. Weissauer, Differentialformen zu Untergruppen der Siegelschen Modulgruppe zweiten Grades, J. reine angew Math., 391(1988), 100-156.
  148. H. Weyl, The classical groups: Their invariants and representations, Princeton Univ. Press, Princeton, New Jersey, second edition (1946).
  149. E. Witt, Eine Identität zwischen Modulformen zweiten Grades, Math. Sem. Hansisch Univ. 14 (1941), 323-337.
  150. J.-H. Yang, The Siegel-Jacobi Operator, Abh. Math. Sem. Univ. Hamburg 63 (1993), 135-146.
  151. J.-H. Yang, Vanishing theorems on Jacobi forms of higher degree, J. Korean Math. Soc., 30(1)(1993), 185-198.
  152. J.-H. Yang, Remarks on Jacobi forms of higher degree, Proc. of the 1993 Workshop on Automorphic Forms and Related Topics, edited by Jin-Woo Son and Jae-Hyun Yang, the Pyungsan Institute for Mathematical Sciences, (1993), 33-58.
  153. J.-H. Yang, Singular Jacobi Forms, Trans. Amer. Math. Soc. 347 (6) (1995), 2041-2049.
  154. J.-H. Yang, Construction of vector valued modular forms from Jacobi forms, Canadian J. of Math. 47 (6) (1995), 1329-1339 or arXiv:math.NT/0612502.
  155. J.-H. Yang, Kac-Moody algebras, the monstrous moonshine, Jacobi forms and infinite products, Pro- ceedings of the 1995 Symposium on Number theory, geometry and related topics, the Pyungsan Institute for Mathematical Sciences (1996), 13-82 or arXiv:math.NT/0612474.
  156. J.-H. Yang, A geometrical theory of Jacobi forms of higher degree, Proceedings of Symposium on Hodge Theory and Algebraic Geometry ( edited by Tadao Oda ), Sendai, Japan (1996), 125-147 or Kyungpook Math. J. 40 (2) (2000), 209-237 or arXiv:math.NT/0602267.
  157. J.-H. Yang, The Method of Orbits for Real Lie Groups, Kyungpook Math. J. 42 (2) (2002), 199-272 or arXiv:math.RT/0602056.
  158. J.-H. Yang, A note on a fundamental domain for Siegel-Jacobi space, Houston Journal of Mathematics, 32 (3) (2006), 701-712.
  159. J.-H. Yang, Invariant metrics and Laplacians on Siegel-Jacobi space, arXiv:math.NT/0507215 v1 or Journal of Number Theory (2007), doi:10.1016/j.jnt.2006.014.
  160. J.-H. Yang, Invariant metrics and Laplacians on Siegel-Jacobi disk, arXiv:math.NT/0507217 v1 or revised version (2006).
  161. H. Yoshida, On the zeta functions of Shimura varieties and periods of Hilbert modular forms, Duke Math. J. 75 (1994), 121-191.
  162. H. Yoshida, Motives and Siegel modular forms, Amer. J. Math. 123 (2001), no. 6, 1171-1197.
  163. H. Yoshida, Hiroyuki Yoshida's letter to Shalika : April 16, 2001.
  164. H. Yoshida, Motives and Siegel modular forms, Proceedings of Japanese-German Seminar: Explicit structure of modular forms and zeta functions, edited by T. Ibukiyama and W. Kohnen (2002), 197-115.
  165. D. Zagier, Sur la conjecture de Saito-Kurokawa (d'après H. Maass): Seminaire Delange-Pisot-Poitou, Paris, 1979-80, Progress in Mathematics 12, Birkhäuser, Boston, Basel and Stuttgart (1981), 371-394.
  166. C. Ziegler, Jacobi Forms of Higher Degree, Abh. Math. Sem. Hamburg 59 (1989), 191-224.
  167. S. Zucker, On the reductive Borel-Serre compactification: L p -cohomology of arithmetic groups (for large p), Amer. J. Math. 123 (2001), no. 5, 951-984.
  168. S. Zucker, On the reductive Borel-Serre compactification. III: Mixed Hodge structures, Asian J. Math. 8 (2004), no. 4, 881-911.
  169. Department of Mathematics, Inha University, Incheon 402-751, Korea E-mail address: jhyang@inha.ac.kr

Related papers

Siegel varieties and p-adic Siegel modular forms
Jacques Tilouine

Doc. Math, 2006

In this paper, we present a conjecture concerning the classicality of a genus two overconvergent Siegel cusp eigenform whose associated Galois representation happens to be geometric, and more precisely, given by the Tate module of an abelian surface. This conjecture is inspired by the Fontaine-Mazur conjecture. It generalizes known results in the genus one case, due to Kisin, Buzzard-Taylor and Buzzard. The main difference in the genus two case is the complexity of the arithmetic geometry involved. This is why most of the paper consists in recalling (mostly with proofs) old and new results on the bad reduction of parahoric type Siegel varieties, with some consequences on their rigid geometry. Our conjecture would imply, in certain cases, a conjecture posed by H. Yoshida in 1980 on the modularity of abelian surfaces defined over the rationals.

View PDFchevron_right
The geometry of Siegel modular varieties
K. Hulek

1998

This is a survey article about Siegel modular varieties over the complex numbers. It is written mostly from the point of view of moduli of abelian varieties, especially surfaces. We cover compactification of Siegel modular varieties; classification of the compactified varieties by Kodaira dimension, etc.; moduli of abelian surfaces and especially applications of the lifting of Jacobi forms to modular

View PDFchevron_right
On The Notion Of $p$-old Siegel Modular Forms
Arash Rastegar

2004

We introduce a higher dimensional Atkin-Lehner theory for Siegel-parahoric and Borel congruence subgroups of $GSp(2g)$. Old Siegel forms are induced by geometric correspondences on Siegel moduli spaces which commute with almost all local Hecke algebras. Any satisfactory local definition of $p$-old Siegel forms which guarantees that, any new eigenform of almost all local Hecke algebras is an eigenform of all local Hecke algebras prime to the level, will imply that our correspondences introduce a geometric formulation for the notion of $p$-old Siegel forms inside all Siegel forms of Siegel parahoric level $pn$. We also prove an injection result for Siegel modular forms in finite characteristics.

View PDFchevron_right
Siegel modular forms of genus $2$ attached to elliptic curves
Dinakar Ramakrishnan

Mathematical Research Letters, 2007

View PDFchevron_right
On the images of the Galois representations attached to genus 2 Siegel modular forms
ariel weiss

Journal für die reine und angewandte Mathematik (Crelles Journal), 2002

We address the problem of the determination of the images of the Galois representations attached to genus 2 Siegel cusp forms of level 1 having multiplicity one. These representations are symplectic. We prove that the images are as large as possible for almost every prime, if the Siegel cusp form is not a Maass spezialform and verifies two easy to check conditions.

View PDFchevron_right

Related topics

  • Number Theory
  • Algebraic Geometry
    • 580 California St., Suite 400
    San Francisco, CA, 94104
    © 2025 Academia. All rights reserved