Rankin-Cohen Brackets and Invariant Theory
Y. Choie
, B. Mourrain2
and P. Solé3
2INRIA Sophia Route des Lucioles, 06 903 Sophia-Antipolis, France P. SOL É
DOI: 10.1023/A:1008722316223
Abstract
Using maps due to Ozeki and Broué-Enguehard between graded spaces of invariants for certain finite groups and the algebra of modular forms of even weight we equip these invariants spaces with a differential operator which gives them the structure of a Rankin-Cohen algebra. A direct interpretation of the Rankin-Cohen bracket in terms of transvectant for the group SL(2, C) is given.
Pages: 5–13
Keywords: Rankin-Cohen brackets; ozeki and broué-enguehard maps; invariants; codes
Full Text: PDF
References
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2. M. Broué and M. Enguehard, “Polyn\hat omes des poids de certains codes et fonctions theta de certains réseaux,” Ann. Sc. ENS. 5 (1972), 157-181.
3. Y. Choie and W. Eholzer, “Rankin-Cohen operators of Jacobi and Siegel forms,” Journal of Number Theory 68(2) (1998), 160-177.
4. H. Cohen, “Sums involving the values at negative integers of L-functions of quadratic characters,” Math. Ann. 217 (1975), 271-285.
5. J.H. Conway and N.J.A. Sloane, Sphere Packings Lattices and Groups, 3rd ed., Springer Verlag, New York, 1999.
6. W. Ebeling, Lattices and Codes, Vieweg & John, Braunschweig, 1994.
7. P.J. Olver, Equivalence, Invariants and Symmetry, Cambridge, 1995.
8. P.J. Olver, Classical Invariant Theory, Cambridge, 1999.
9. P.J. Olver and J.A. Sanders, Transvectants, Modular forms and the Heisenberg Algebra, Preprint, 1999.
10. M. Ozeki, “On the notion of Jacobi polynomials for codes,” Math. Proc. Cambridge Phil. Soc. 121 (1997), 15-30.
11. R.A. Rankin, “The construction of automorphic forms from the derivatives of given forms,” Michigan Math. Jour. 4 (1957), 181-186.
12. P. Solé, “Codes and modular forms,” proceeding of KIAS number theory conference, submitted.
13. B. Sturmfels, Algorithms in Invariant Theory, Springer, 1993.
14. A. Unterberger and J. Unterberger, “Algebras of symbols and modular forms,” Jour. d'Analyse Math. 68 (1996), 121-143.
15. D. Zagier, “Modular forms and differential operators,” Proc. Indian Acad. Sci. (Math. Sci.) 104(1) (1994), 57-75.