Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 
 

 

The action of the heat operator on Jacobi forms


Author: Olav K. Richter
Journal: Proc. Amer. Math. Soc. 137 (2009), 869-875
MSC (2000): Primary 11F50; Secondary 11F60
DOI: https://doi.org/10.1090/S0002-9939-08-09566-X
Published electronically: September 15, 2008
MathSciNet review: 2457425
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: We investigate the action of the heat operator on Jacobi forms. In particular, we present two explicit characterizations of this action on Jacobi forms of index $ 1$. Furthermore, we study congruences and filtrations of Jacobi forms. As an application, we determine when an analog of Atkin's $ U$-operator applied to a Jacobi form is nonzero modulo a prime.


References [Enhancements On Off] (What's this?)

  • 1. Ahlgren, S., and Ono, K.
    Arithmetic of singular moduli and class polynomials.
    Compos. Math. 141, no. 2 (2005), 293-312. MR 2134268 (2006a:11058)
  • 2. Atkinson, J.
    Divisors of modular forms on $ \Gamma_0(4)$.
    J. Number Theory 112, no. 1 (2005), 189-204. MR 2131143 (2005m:11084)
  • 3. Bruinier, J., Ono, K., and Kohnen, W.
    The arithmetic of the values of modular functions and the divisors of modular forms.
    Compos. Math. 140, no. 3 (2004), 552-566. MR 2041768 (2005h:11083)
  • 4. Choie, Y.
    Jacobi forms and the heat operator.
    Math. Z. 225 (1997), 95-101. MR 1451334 (98c:11042)
  • 5. Choie, Y.
    Jacobi forms and the heat operator II.
    Illinois J. Math. 42, no. 2 (1998), 179-186. MR 1612731 (99d:11049)
  • 6. Choie, Y., and Kohnen, W.
    Special values of elliptic functions at points of the divisors of Jacobi forms.
    Proc. Amer. Math. Soc. 131, no. 11 (2003), 3309-3317. MR 1990618 (2004e:11042)
  • 7. Eichler, M., and Zagier, D.
    The theory of Jacobi forms.
    Birkhäuser, Boston, 1985. MR 781735 (86j:11043)
  • 8. Elkies, N., Ono, K., and Yang, T.
    Reduction of CM elliptic curves and modular function congruences.
    Internat. Math. Res. Notices 2005, no. 44, 2695-2707. MR 2181309 (2006k:11076)
  • 9. Guerzhoy, P.
    An approach to the $ p$-adic theory of Jacobi forms.
    Internat. Math. Res. Notices 1994, no. 1, 31-39. MR 1255251 (94m:11057)
  • 10. Guerzhoy, P.
    On $ {U}(p)$-congruences.
    Proc. Amer. Math. Soc. 135, no. 9 (2007), 2743-2746. MR 2317947
  • 11. Kaneko, M., and Zagier, D.
    A generalized Jacobi theta function and quasimodular forms, The moduli space of curves (Texel Island, 1994).
    Progr. Math. 129. Birkhäuser, 1995, pp. 165-172. MR 1363056 (96m:11030)
  • 12. Kawai, T., and Yoshioka, K.
    String partition functions and infinite products.
    Adv. Theor. Math. Phys. 4, no. 2 (2000), 397-485. MR 1838446 (2002g:11054)
  • 13. Ono, K.
    The web of modularity: Arithmetic of the coefficients of modular forms and $ q$-series, vol. 102 of CBMS Regional Conference Series in Mathematics.
    Published for the Conference Board of the Mathematical Sciences, Washington, DC, by the Amer. Math. Soc., Providence, RI, 2004. MR 2020489 (2005c:11053)
  • 14. Ramanujan, S.
    On certain arithmetical functions.
    Trans. Camb. Phil. Soc. 22 (1916), 159-184
    (Collected Papers, No. 18).
  • 15. Serre, J-P.
    Formes modulaires et fonctions zeta $ p$-adiques, Modular functions of one variable, III.
    Lecture Notes in Math., 350. Springer, 1973, pp. 191-268. MR 0404145 (53:7949a)
  • 16. Sofer, A.
    $ p$-adic aspects of Jacobi forms.
    J. Number Theory 63, no. 2 (1997), 191-202. MR 1443756 (98b:11058)
  • 17. Swinnerton-Dyer, H. P. F.
    On $ l$-adic representations and congruences for coefficients of modular forms, Modular functions of one variable, III.
    Lecture Notes in Math., 350. Springer, 1973, pp. 1-55. MR 0406931 (53:10717a)

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC (2000): 11F50, 11F60

Retrieve articles in all journals with MSC (2000): 11F50, 11F60


Additional Information

Olav K. Richter
Affiliation: Department of Mathematics, University of North Texas, Denton, Texas 76203
Email: richter@unt.edu

DOI: https://doi.org/10.1090/S0002-9939-08-09566-X
Received by editor(s): March 5, 2008
Published electronically: September 15, 2008
Communicated by: Ken Ono
Article copyright: © Copyright 2008 American Mathematical Society

American Mathematical Society