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JACOBI-LIKE FORMS, PSEUDODIFFERENTIAL OPERATORS, AND GROUP COHOMOLOGY

  • MIN HO LEE (a1)

Abstract

Pseudodifferential operators are formal Laurent series in the formal inverse −1 of the derivative operator whose coefficients are holomorphic functions on the Poincaré upper half-plane. Given a discrete subgroup Γ of SL(2,ℝ), automorphic pseudodifferential operators for Γ are pseudodifferential operators that are Γ-invariant, and they are closely linked to Jacobi-like forms and modular forms for Γ. We construct linear maps from the space of automorphic pseudodifferential operators and from the space of Jacobi-like forms for Γ to the cohomology space of the group Γ, and prove that these maps are compatible with the respective Hecke operator actions.

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References

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[1]Andrianov, A., Quadratic Forms and Hecke Operators (Springer, Heidelberg, 1987).
[2]Bayer, P. and Neukirch, J., ‘On automorphic forms and Hodge theory’, Math. Ann. 257 (1981), 135155.
[3]Choie, Y., ‘Pseudodifferential operators and Hecke operators’, Appl. Math. Lett. 11 (1998), 2934.
[4]Cohen, P. B., Manin, Y. and Zagier, D., ‘Automorphic pseudodifferential operators’, in: Algebraic Aspects of Nonlinear Systems (Birkhäuser, Boston, 1997), pp. 1747.
[5]Dickey, L., Soliton Equations and Hamiltonian Systems (World Scientific, Singapore, 1991).
[6]Dong, C. and Mason, G., Transformation Laws for Theta Functions, CRM Proceedings Lecture Notes, 30 (American Mathematical Society, Providence, RI, 2001), pp. 1526.
[7]Eichler, M. and Zagier, D., The Theory of Jacobi Forms, Progress in Mathematics, 55 (Birkhäuser, Boston, 1985).
[8]Hida, H., Elementary Theory of L-functions and Eisenstein Series (Cambridge University Press, Cambridge, 1993).
[9]Kuga, M., Fiber varieties over a symmetric space whose fibers are abelian varieties I, II, (University of Chicago, Chicago, 1963/64).
[10]Kuga, M., Group Cohomology and Hecke Operators. II. Hilbert Modular Surface Case, Advanced Studies in Pure Mathematics, 7 (North-Holland, Amsterdam, 1985), pp. 113148.
[11]Kuga, M., Parry, W. and Sah, C. H., Group Cohomology and Hecke Operators, Progress in Mathematics, 14 (Birkhäuser, Boston, 1981), pp. 223266.
[12]Lee, M. H., ‘Hilbert modular pseudodifferential operators’, Proc. Amer. Math. Soc. 129 (2001), 31513160.
[13]Lee, M. H., ‘Jacobi-like forms, differential equations, and Hecke operators’, Complex Var. Theory Appl. 50 (2005), 10951104.
[14]Lee, M. H. and Myung, H. C., ‘Hecke operators on Jacobi-like forms’, Canad. Math. Bull. 44 (2001), 282291.
[15]Martin, F. and Royer, E., Formes Modulaires et Transcendance (eds. S. Fischler, E. Gaudron and S. Khémira) (Soc. Math. de France, 2005), pp. 1117.
[16]Miyake, T., Modular Forms (Springer, Heidelberg, 1989).
[17]Rhie, Y. H. and Whaples, G., ‘Hecke operators in cohomology of groups’, J. Math. Soc. Japan 22 (1970), 431442.
[18]Shimura, G., Introduction to the Arithmetic Theory Automorphic Functions (Princeton University Press, Princeton, NJ, 1971).
[19]Steinert, B., On Annihilation of Torsion in the Cohomology of Boundary Strata of Siegel Modular Varieties, Bonner Mathematische Schriften, 276 (Universität Bonn, Bonn, 1995).
[20]Zagier, D., ‘Modular forms and differential operators’, Proc. Indian Acad. Sci. Math. Sci. 104 (1994), 5775.
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JACOBI-LIKE FORMS, PSEUDODIFFERENTIAL OPERATORS, AND GROUP COHOMOLOGY

  • MIN HO LEE (a1)

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