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)

 
 

 

Ramanujan bigraphs associated with $ SU(3)$ over a $ p$-adic field


Authors: Cristina Ballantine and Dan Ciubotaru
Journal: Proc. Amer. Math. Soc. 139 (2011), 1939-1953
MSC (2010): Primary 11F70, 22E50
DOI: https://doi.org/10.1090/S0002-9939-2011-10856-6
Published electronically: January 6, 2011
MathSciNet review: 2775370
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: We use the representation theory of the quasisplit form $ G$ of $ SU(3)$ over a $ p$-adic field to investigate whether certain quotients of the Bruhat-Tits tree associated to this form are Ramanujan bigraphs. We show that a quotient of the tree associated with $ G$ (which is a biregular bigraph) is Ramanujan if and only if $ G$ satisfies a Ramanujan type conjecture. This result is analogous to the seminal case of $ PGL_2(\mathbb{Q}_p)$ considered by Lubotzky, Phillips, and Sarnak. As a consequence, the classification of the automorphic spectrum of the unitary group in three variables by Rogawski implies the existence of certain infinite families of Ramanujan bigraphs.


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

  • 1. James Arthur, Unipotent automorphic representations: conjectures, Astérisque 171-172 (1989), 13-71. MR 1021499 (91f:22030)
  • 2. C. Ballantine, Ramanujan type buildings, Canad. J. Math. 52 (2000), no. 6, 1121-1148. MR 1794299 (2001j:11027)
  • 3. D. Barbasch, D. Ciubotaru, Unitary functorial correspondences for $ p$-adic groups, preprint, arXiv:0909.5241.
  • 4. A. Borel, Admissible representations of a semi-simple group over a local field with vectors fixed under an Iwahori subgroup, Invent. Math. 35 (1976), 233-259. MR 0444849 (56:3196)
  • 5. A. Borel, Some finiteness properties of adele groups over number fields, Publ. Math. IHES 16 (1963), 5-30. MR 0202718 (34:2578)
  • 6. P. Cartier, Representations of $ \mathfrak{p}$-Adic Groups: A Survey, Proc. Symp. Pure Math., 33, Vol. 1, Amer. Math. Soc., 1979, 111-155. MR 546593 (81e:22029)
  • 7. W. Casselman, Introduction to admissible representations of p-adic groups, http://www.math. ubc.ca/$ \sim$cass/research.html
  • 8. L. Clozel, M. Harris, R. Taylor, Automorphy for some $ l$-adic lifts of automorphic mod $ l$ Galois representations, Publ. Math. Inst. Hautes Études Sci. 108 (2008), 1-181. MR 2470687 (2010j:11082)
  • 9. K. Feng, W.-C.W. Li, Spectra of hypergraphs and applications, J. Number Theory 60, no. 1 (1996), 1-22. MR 1405722 (97f:05128)
  • 10. K. Hashimoto, Zeta Functions of Finite Graphs and Representations of $ p$-Adic Groups, Advanced Studies in Pure Mathematics, 15, Academic Press, Boston, MA, 1989, Automorphic Forms and Geometry of Arithmetic Varieties, 211-280. MR 1040609 (91i:11057)
  • 11. K. Hashimoto and A. Hori, Selberg-Ihara's Zeta Function for $ p$-adic Discrete Subgroups, Advanced Studies in Pure Mathematics 15, Academic Press, Boston, MA, 1989, Automorphic Forms and Geometry of Arithmetic Varieties, 171-210.
  • 12. Y. Ihara, Discrete subgroups of $ PL(2,k_p)$, Proc. Symp. Pure Math. IX, Amer. Math. Soc., 1966, 272-278. MR 0205952 (34:5777)
  • 13. M. Kneser, Strong approximation, Proc. Symp. Pure Math. IX, Amer. Math. Soc., 1966, 187-196. MR 0213361 (35:4225)
  • 14. M.-A. Knus, A. Merkurjev, M. Rost, J.-P. Tignol, The book of involutions, Amer. Math. Soc. Coll. Publ., 44, Amer. Math. Soc., 1998. MR 1632779 (2000a:16031)
  • 15. A. Lubotzky, R. Phillips, P. Sarnak, Ramanujan graphs, Combinatorica 8 (1988), 261-277. MR 963118 (89m:05099)
  • 16. A. Lubotzky, Discrete Groups, Expanding Graphs and Invariant Measures, Progress in Math., 125, Birkhäuser Verlag, 1994. MR 1308046 (96g:22018)
  • 17. G. Lusztig, Affine Hecke algebras and their graded version, J. Amer. Math. Soc. 2 (1989), no. 3, 599-635. MR 991016 (90e:16049)
  • 18. G.A. Margulis, Explicit group-theoretic constructions of combinatorial schemes and their applications in the construction of expanders and concentrators, Problems of Information Transmission 24(1) (1988), 39-46. MR 939574 (89f:68054)
  • 19. J. Rogawski, Automorphic Representations of Unitary Groups in Three Variables, Princeton University Press, Princeton, NJ, 1990. MR 1081540 (91k:22037)
  • 20. W. Scharlau, Zur Existenz von Involutionen auf einfachen Algebren. II, Math. Z. 176 (1981), 399-404. MR 610220 (82k:12013)
  • 21. A. Selberg, On discontinuous groups in higher-dimensional symmetric spaces, Contributions to function theory, International colloquia on function theory (Bombay, 1960), Tata Institute of Fundamental Research, Bombay, 1960, 147-164. MR 0130324 (24:A188)
  • 22. P. Solé, Ramanujan Hypergraphs and Ramanujan Geometries, Emerging Applications of Number Theory (eds. D.A. Hejhal, J. Friedman, M.C. Gutzwiller, A.M. Odlyzko), Springer, New York, 1999, 583-590. MR 1691550 (2000c:05108)
  • 23. J. Tits, Reductive groups over local fields, Automorphic forms, representations and $ L$-functions (Proc. Sympos. Pure Math., Oregon State Univ., Corvallis, Ore., 1977), Part 1, 29-69, Proc. Sympos. Pure Math., XXXIII, Amer. Math. Soc., Providence, RI, 1979. MR 546588 (80h:20064)

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC (2010): 11F70, 22E50

Retrieve articles in all journals with MSC (2010): 11F70, 22E50


Additional Information

Cristina Ballantine
Affiliation: Department of Mathematics and Computer Science, College of the Holy Cross, Worcester, Massachusetts 01610
Email: cballant@holycross.edu

Dan Ciubotaru
Affiliation: Department of Mathematics, University of Utah, Salt Lake City, Utah 84112
Email: ciubo@math.utah.edu

DOI: https://doi.org/10.1090/S0002-9939-2011-10856-6
Received by editor(s): June 1, 2010
Published electronically: January 6, 2011
Additional Notes: The second author was supported in part by NSA-AMS 081022.
Communicated by: Matthew A. Papanikolas
Article copyright: © Copyright 2011 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

American Mathematical Society