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)

 
 

 

Hermitian Jacobi forms and $ U(p)$ congruences


Authors: Olav K. Richter and Jayantha Senadheera
Journal: Proc. Amer. Math. Soc. 143 (2015), 4199-4210
MSC (2010): Primary 11F50; Secondary 11F33
DOI: https://doi.org/10.1090/S0002-9939-2015-12562-2
Published electronically: March 18, 2015
MathSciNet review: 3373920
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: We introduce a new space of Hermitian Jacobi forms, and we determine its structure. Moreover, we characterize $ U(p)$ congruences of Hermitian Jacobi forms, and we discuss an explicit example.


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

  • [1] Scott Ahlgren and Ken Ono, Arithmetic of singular moduli and class polynomials, Compos. Math. 141 (2005), no. 2, 293-312. MR 2134268 (2006a:11058), https://doi.org/10.1112/S0010437X04001198
  • [2] Anatolii N. Andrianov, Modular descent and the Saito-Kurokawa conjecture, Invent. Math. 53 (1979), no. 3, 267-280. MR 549402 (80k:10024), https://doi.org/10.1007/BF01389767
  • [3] YoungJu Choie, Jacobi forms and the heat operator, Math. Z. 225 (1997), no. 1, 95-101. MR 1451334 (98c:11042), https://doi.org/10.1007/PL00004603
  • [4] YoungJu Choie, Jacobi forms and the heat operator. II, Illinois J. Math. 42 (1998), no. 2, 179-186. MR 1612731 (99d:11049)
  • [5] Soumya Das, Note on Hermitian Jacobi forms, Tsukuba J. Math. 34 (2010), no. 1, 59-78. MR 2723724 (2012b:11077)
  • [6] Soumya Das, Some aspects of Hermitian Jacobi forms, Arch. Math. (Basel) 95 (2010), no. 5, 423-437. MR 2738862 (2012a:11055), https://doi.org/10.1007/s00013-010-0176-3
  • [7] T. Dern, Hermitesche Modulformen zweiten Grades, Ph.D. thesis, RWTH Aachen University, Germany, 2001.
  • [8] Martin Eichler and Don Zagier, The theory of Jacobi forms, Progress in Mathematics, vol. 55, Birkhäuser Boston, Inc., Boston, MA, 1985. MR 781735 (86j:11043)
  • [9] Noam Elkies, Ken Ono, and Tonghai Yang, Reduction of CM elliptic curves and modular function congruences, Int. Math. Res. Not. 44 (2005), 2695-2707. MR 2181309 (2006k:11076), https://doi.org/10.1155/IMRN.2005.2695
  • [10] Klaus Haverkamp, Hermitesche Jacobiformen, Schriftenreihe des Mathematischen Instituts der Universität Münster. 3. Serie, Vol. 15, Schriftenreihe Math. Inst. Univ. Münster 3. Ser., vol. 15, Univ. Münster, Münster, 1995, pp. 105 (German). MR 1335247 (96c:11053)
  • [11] Klaus K. Haverkamp, Hermitian Jacobi forms, Results Math. 29 (1996), no. 1-2, 78-89. MR 1377681 (97d:11077), https://doi.org/10.1007/BF03322207
  • [12] Jun-ichi Igusa, On the graded ring of theta-constants, Amer. J. Math. 86 (1964), 219-246. MR 0164967 (29 #2258)
  • [13] Naomi 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), https://doi.org/10.2307/1999771
  • [14] Haesuk Kim, Differential operators on Hermitian Jacobi forms, Arch. Math. (Basel) 79 (2002), no. 3, 208-215. MR 1933379 (2003g:11047), https://doi.org/10.1007/s00013-002-8306-1
  • [15] Hans Maass, Über eine Spezialschar von Modulformen zweiten Grades, Invent. Math. 52 (1979), no. 1, 95-104 (German). MR 532746 (80f:10031), https://doi.org/10.1007/BF01389857
  • [16] Hans Maass, Über eine Spezialschar von Modulformen zweiten Grades. II, Invent. Math. 53 (1979), no. 3, 249-253 (German). MR 549400 (81a:10037), https://doi.org/10.1007/BF01389765
  • [17] Hans Maass, Über eine Spezialschar von Modulformen zweiten Grades. III, Invent. Math. 53 (1979), no. 3, 255-265 (German). MR 549401 (81a:10038), https://doi.org/10.1007/BF01389766
  • [18] Ken 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 American Mathematical Society, Providence, RI, 2004. MR 2020489 (2005c:11053)
  • [19] S. Ramanujan, On certain arithmetical functions, Trans. Camb.Phil. Soc. 22 (1916), 159-184, (Collected Papers, No. 18).
  • [20] M. Raum and O. Richter, The structure of Siegel modular forms mod $ p$ and $ {U}(p)$ congruences, to appear in Mathematical Research Letters.
  • [21] Olav K. Richter, On congruences of Jacobi forms, Proc. Amer. Math. Soc. 136 (2008), no. 8, 2729-2734. MR 2399034 (2009i:11060), https://doi.org/10.1090/S0002-9939-08-09274-5
  • [22] Olav K. Richter, The action of the heat operator on Jacobi forms, Proc. Amer. Math. Soc. 137 (2009), no. 3, 869-875. MR 2457425 (2010i:11064), https://doi.org/10.1090/S0002-9939-08-09566-X
  • [23] Ryuji Sasaki, Hermitian Jacobi forms of index one, Tsukuba J. Math. 31 (2007), no. 2, 301-325. MR 2371175 (2008j:11055)
  • [24] J. Senadheera, Hermitian Jacobi forms and congruences, Ph.D. thesis, University of North Texas, 2014.
  • [25] Jean-Pierre Serre, Formes modulaires et fonctions zêta $ p$-adiques, Modular functions of one variable, III (Proc. Internat. Summer School, Univ. Antwerp, 1972) Springer, Berlin, 1973, pp. 191-268. Lecture Notes in Math., Vol. 350 (French). MR 0404145 (53 #7949a)
  • [26] Adriana Sofer, $ p$-adic aspects of Jacobi forms, J. Number Theory 63 (1997), no. 2, 191-202. MR 1443756 (98b:11058), https://doi.org/10.1006/jnth.1997.2095
  • [27] W.A. Stein et al., Sage Mathematics Software (Version 5.7), The Sage Development Team, 2013, http://www.sagemath.org.
  • [28] H. P. F. Swinnerton-Dyer, On $ l$-adic representations and congruences for coefficients of modular forms, Modular functions of one variable, III (Proc. Internat. Summer School, Univ. Antwerp, 1972) Springer, Berlin, 1973, pp. 1-55. Lecture Notes in Math., Vol. 350. MR 0406931 (53 #10717a)
  • [29] D. Zagier, Sur la conjecture de Saito-Kurokawa (d'après H. Maass), Seminar on Number Theory, Paris 1979-80, Progr. Math., vol. 12, Birkhäuser, Boston, Mass., 1981, pp. 371-394 (French). MR 633910 (83b:10031)

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC (2010): 11F50, 11F33

Retrieve articles in all journals with MSC (2010): 11F50, 11F33


Additional Information

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

Jayantha Senadheera
Affiliation: Department of Mathematics, University of North Texas, Denton, Texas 76203
Address at time of publication: Department of Mathematics and Computer Science, Faculty of Natural Sciences, The Open University of Sri Lanka, Nawala 10250, Sri Lanka
Email: jayantha.senadheera@gmail.com

DOI: https://doi.org/10.1090/S0002-9939-2015-12562-2
Received by editor(s): June 11, 2014
Published electronically: March 18, 2015
Additional Notes: The first author was partially supported by Simons Foundation Grant $#200765$
Communicated by: Ken Ono
Article copyright: © Copyright 2015 American Mathematical Society

American Mathematical Society