DOI QR코드

DOI QR Code

THE REPRESENTABILITY OF MODULAR FORMS BY A CERTAIN THETA SERIES AND ITS APPLICATIONS

  • Jun, Sung-Tae (Division of Mathematics and Computer sicience, Konkuk University) ;
  • Kim, In-Suk (Department of Mathematics education Wonkwang University)
  • Received : 2008.04.03
  • Accepted : 2008.05.13
  • Published : 2008.06.25

Abstract

The primitive orders in a quaternion algebra play a central role of the theory of Hecke operators. In this paper, we study theta series generated by Brandt matrices and its applications to almost Ramanujan graphs.

Keywords

References

  1. A. Atkin and J. Lehner Hecke operators on $\Gamma_0$(N), Math. Ann., 185 (1970), pp. 134-160. https://doi.org/10.1007/BF01359701
  2. F. Bien, Constructions of telephone networks by group representations, Notices Amer. Math. Soc., 36 (1989), pp. 5-22.
  3. J. Brzezinski, On embedding numbers into quaternion orders, Comment. Math. Helvetici, 66 (1991), pp. 302-318. https://doi.org/10.1007/BF02566649
  4. F.R.K. Chung, Diameters and eigenvalues., J. Amer. Math. Soc., 2 (1989), pp. 187-196. https://doi.org/10.1090/S0894-0347-1989-0965008-X
  5. G. Davidoff, P. Sarnak, A. Valette, Elementary number theory, Group theory and Ramanujan graphs, London Math. Soc. Student Text, 65.
  6. M. Eichler, "Uber die Idealklassenzahl tota definiter Quaternionen-Algebren, J. Mat. Zeit, 43 (1937) pp. 102-109.
  7. M. Eichler, The basis problem for modular forms and the traces of Hecke operators, Springer-Verlag, Lecture Notes in Math., 320 (1972), pp. 75-151.
  8. E. Hecke, Analytische Arithmetik der positiven quadratischen Formen, Math. Werke, pp. 789-918, 1959.
  9. H. Hijikata, Explicit formula of the traces of the Hecke operators for $\Gamma_0$(N), J. Math. Soc. Japan, 26 (1974), pp. 56-82. https://doi.org/10.2969/jmsj/02610056
  10. H. Hijikata, A. Pizer and T. Shemanske, Orders in Quaternion Algebras, J. Reine angew Math., 394 (1989), pp.59-106.
  11. H. Hijikata, A. Pizer and T. Shemanske, The basis problem for modular forms on $\Gamma_0$(N), Memoirs of AMS., 32 No. 418, (1989).
  12. Y. Ihara, Descrete subgroups of PL($2,\,k_p$), Proc. Sympos. in Pure Math., IX (1966), pp. 272-278.
  13. A. Lbotzky, R. Phillips, P. Sarnak, Explicit expanders and Ramanujan conjectures, Proc. of Eighteenth Annual ACM Sympos. on the theory of computing, 18 (1986), pp. 240-246.
  14. S. Jun, On the Representability of Modular forms by a certain Theta series, J. Korean Math. Soc., 34, no.4 (1997), pp. 809-824.
  15. S. Jun, Optimal embeddings in a quatenion algebra over a dyadic local field, preprint.
  16. A. Pizer, An Algorithm for computing modular forms on $\Gamma_0$(N), J. Algebra, 64 (1980), pp. 177-241.
  17. A. Pizer, Ramanujan graphs and Hecke operators, Bull. Amer. Math. Soc., 23, no.1 (1990), pp. 127-137. https://doi.org/10.1090/S0273-0979-1990-15918-X
  18. A. Pizer, Theta series and Modular forms of level $p^2M$, Comp. Math., 40 (1980), pp. 177-241.