Advertisement

Zeta Functions of Heisenberg Graphs over Finite Rings

  • Michelle DeDeo
  • María Martínez
  • Archie Medrano
  • Marvin Minei
  • Harold Stark
  • Audrey Terras
Chapter
Part of the Developments in Mathematics book series (DEVM, volume 13)

Abstract

We investigate Ihara-Selberg zeta functions of Cayley graphs for the Heisenberg group over finite rings ℤ/p n ℤ, where p is a prime. In order to do this, we must compute the Galois group of the covering obtained by reducing coordinates in ℤ/pn+1ℤ modulo pn+1. The Ihara-Selberg zeta functions of the Heisenberg graph mod pn+1 factor as a product of Artin L-functions corresponding to the irreducible representations of the Galois group of the covering. Emphasis is on graphs of degree four. These zeta functions are compared with zeta functions of finite torus graphs which are Cayley graphs for the abelian groups (ℤ/p n ℤ) r .

Keywords

Span Tree Zeta Function Adjacency Matrix Heisenberg Group Galois Group 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. DeDeo, M., Martinez, M., Medrano, A., Minei, M., Stark, H., and Terras, A. (2004). Spectra of Heisenberg graphs over finite rings. In Feng, W., Hu, S., and Lin, X., editors, Discrete and Continuous Dynamical Systems, 2003 Supplement Volume, pages 213–222. Proc. of 4th Internatl. Conf. on Dynamical Systems and Differential Equations.Google Scholar
  2. Diaconis, P. and Saloff-Coste, L. (1994). Moderate growth and random walk on finite groups. Geom. Funct. Anal., 4:1–36.CrossRefMathSciNetGoogle Scholar
  3. Godsil, C. D. (1993). Algebraic Combinatorics. Chapman and Hall, New York.Google Scholar
  4. Hashimoto, K. (1990). On Zeta and L-functions of finite graphs. Intl. J. Math., 1(4):381–396.CrossRefzbMATHMathSciNetGoogle Scholar
  5. Hofstadter, D. R. (1976). Energy levels and wave functions of Bloch electrons in rational and irrational magnetic fields. Physical Review B, 14:2239–2249.CrossRefGoogle Scholar
  6. Katz, N. and Sarnak, P. (1999). Zeros of zeta functions and symmetry. Bull. Amer. Math. Soc., 36(1):1–26.CrossRefMathSciNetGoogle Scholar
  7. Kemperman, J. H. B. (1961). The Passage Problem for a Stationary Markov Chain. Univ. of Chicago Press, Chicago, IL.Google Scholar
  8. Kotani, M. and Sunada, T. (2000). Spectral geometry of crystal lattices. Preprint.Google Scholar
  9. Lin, X.-S. and Wang, Z. (2001). Random walk on knot diagrams, colored Jones polynomial and Ihara-Selberg zeta function. In Gillman, J., Menasco, W., and Lin, X.-S., editors, Knots, Braids, and Mapping Class Groups-Papers dedicated to Joan S. Birman (New York, 1998), volume 24 of AMS/IP Studies in Adv. Math., pages 107–121. Amer. Math. Soc., Providence, RI.Google Scholar
  10. Lubotzky, A., Phillips, R., and Sarnak, P. (1988). Ramanujan graphs. Combinatorica, 8:261–277.CrossRefMathSciNetGoogle Scholar
  11. Myers, P. (1995). Euclidean and Heisenberg Graphs: Spectral Properties and Applications. PhD thesis, Univ. of California, San Diego.Google Scholar
  12. Rosen, M. (2002). Number Theory in Function Fields, volume 210 of Graduate Texts in Mathematics. Springer-Verlag, New York.Google Scholar
  13. Sarnak, P. (1995). Arithmetic quantum chaos. In The Schur lectures (1992) (Tel Aviv), volume 8 of Israel Math. Soc. Conf. Proc., pages 183–236. Bar-Ilan Univ., Ramat-Gan, Israel.Google Scholar
  14. Stark, H. M. and Terras, A. (1996). Zeta functions of finite graphs and coverings. Adv. in Math., 121:124–165.CrossRefMathSciNetGoogle Scholar
  15. Stark, H. M. and Terras, A. (2000). Zeta functions of finite graphs and coverings. II. Adv. in Math., 154(1):132–195.CrossRefMathSciNetGoogle Scholar
  16. Terras, A. (1999). Fourier Analysis on Finite Groups and Applications. Cambridge Univ. Press, Cambridge, UK.Google Scholar
  17. Terras, A. (2000). Statistics of graph spectra for some finite matrix groups: finite quantum chaos. In Dunkl, C., Ismail, M., and Wong, R., editors, Special Functions (Hong Kong, 1999), pages 351–374. World Scientific, Singapore.Google Scholar
  18. Terras, A. (2002). Finite quantum chaos. Amer. Math. Monthly, 109(2):121–139.CrossRefzbMATHMathSciNetGoogle Scholar
  19. Zack, M. (1990). Measuring randomness and evaluating random number generators using the finite Heisenberg group. In Limit Theorems in Probability and Statistics (Pécs, 1989), volume 57 of Colloq. Math. Soc. János Bolyai, pages 537–544. North-Holland, Amsterdam.Google Scholar

Copyright information

© Springer Science+Business Media, Inc. 2005

Authors and Affiliations

  • Michelle DeDeo
    • 1
  • María Martínez
    • 1
  • Archie Medrano
    • 1
  • Marvin Minei
    • 1
  • Harold Stark
    • 1
  • Audrey Terras
    • 1
  1. 1.Department of MathematicsUniversity of CaliforniaSan Diego, La Jolla

Personalised recommendations