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On congruences of Jacobi forms

Author: Olav K. Richter
Journal: Proc. Amer. Math. Soc. 136 (2008), 2729-2734
MSC (2000): Primary 11F50; Secondary 11F60
Published electronically: April 15, 2008
MathSciNet review: 2399034
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Abstract | References | Similar Articles | Additional Information

Abstract: We consider congruences and filtrations of Jacobi forms. More specifically, we extend Tate's theory of theta cycles to Jacobi forms, which allows us to prove a criterion for an analog of Atkin's $ U$-operator applied to a Jacobi form to be nonzero modulo a prime.

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  • 1. S. Ahlgren and K. Ono, Arithmetic of singular moduli and class polynomials, Compos. Math. 141 (2005), no. 2, 293-312. MR 2134268 (2006a:11058)
  • 2. M. Chida and M. Kaneko, On ordinary primes for modular forms and the theta operator, Proc. Amer. Math. Soc. 135 (2007), no. 4, 1001-1005. MR 2262900
  • 3. Y. Choie and W. Eholzer, Jacobi forms and generalized RC-algebras, Rocky Mountain J. Math. 31 (2001), no. 4, 1265-1275. MR 1895295 (2003d:11072)
  • 4. M. Eichler and D. Zagier, The theory of Jacobi forms, Birkhäuser, Boston, 1985. MR 781735 (86j:11043)
  • 5. N. Elkies, K. Ono, and T. Yang, Reduction of CM elliptic curves and modular function congruences, Internat. Math. Res. Notices 2005, no. 44, 2695-2707. MR 2181309 (2006k:11076)
  • 6. P. Guerzhoy, On $ {U}(p)$-congruences, Proc. Amer. Math. Soc. 135 (2007), no. 9, 2743-2746. MR 2317947
  • 7. N. Jochnowitz, A study of the local components of the Hecke algebra mod $ l$, Trans. Amer. Math. Soc. 270 (1982), no. 1, 253-267. MR 642340 (83e:10033a)
  • 8. M. Kaneko and D. Zagier, A generalized Jacobi theta function and quasimodular forms, in: The moduli space of curves (Texel Island, 1994), Progr. Math. 129, pp. 165-172, Birkhäuser, Boston, 1995. MR 1363056 (96m:11030)
  • 9. T. Kawai and K. Yoshioka, String partition functions and infinite products, Adv. Theor. Math. Phys. 4 (2000), no. 2, 397-485. MR 1838446 (2002g:11054)
  • 10. K. Ono, The web of modularity: Arithmetic of the coefficients of modular forms and $ q$-series, CBMS Regional Conference Series in Mathematics, vol. 102, Published for the Conference Board of the Mathematical Sciences, Washington, DC, by the Amer. Math. Soc., Providence, RI, 2004. MR 2020489 (2005c:11053)
  • 11. S. Ramanujan, On certain arithmetical functions, Trans. Camb. Phil. Soc. 22 (1916), 159-184 (Collected Papers, No. 18).
  • 12. O. Richter, The action of the heat operator on Jacobi forms, to appear in the Proceedings of the AMS.
  • 13. J-P. Serre, Formes modulaires et fonctions zeta $ p$-adiques, in: Modular functions of one variable III, Lecture Notes in Math. 350, pp. 191-268, Springer, 1973. MR 0404145 (53:7949a)
  • 14. A. Sofer, $ p$-adic aspects of Jacobi forms, J. Number Theory 63 (1997), no. 2, 191-202. MR 1443756 (98b:11058)
  • 15. H. P. F. Swinnerton-Dyer, On $ l$-adic representations and congruences for coefficients of modular forms, in: Modular functions of one variable III, Lecture Notes in Math. 350, pp. 1-55, Springer, 1973. MR 0406931 (53:10717a)

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Additional Information

Olav K. Richter
Affiliation: Department of Mathematics, University of North Texas, Denton, Texas 76203

Received by editor(s): June 25, 2007
Published electronically: April 15, 2008
Communicated by: Ken Ono
Article copyright: © Copyright 2008 American Mathematical Society

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