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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
Published electronically: January 6, 2011
MathSciNet review: 2775370
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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.


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  • 1. James Arthur, Unipotent automorphic representations: conjectures, Astérisque 171-172 (1989), 13–71. Orbites unipotentes et représentations, II. MR 1021499
  • 2. Cristina M. Ballantine, Ramanujan type buildings, Canad. J. Math. 52 (2000), no. 6, 1121–1148. MR 1794299, 10.4153/CJM-2000-047-4
  • 3. D. Barbasch, D. Ciubotaru, Unitary functorial correspondences for $ p$-adic groups, preprint, arXiv:0909.5241.
  • 4. Armand Borel, Admissible representations of a semi-simple group over a local field with vectors fixed under an Iwahori subgroup, Invent. Math. 35 (1976), 233–259. MR 0444849
  • 5. Armand Borel, Some finiteness properties of adele groups over number fields, Inst. Hautes Études Sci. Publ. Math. 16 (1963), 5–30. MR 0202718
  • 6. P. Cartier, Representations of 𝑝-adic groups: a survey, Automorphic forms, representations and 𝐿-functions (Proc. Sympos. Pure Math., Oregon State Univ., Corvallis, Ore., 1977) Proc. Sympos. Pure Math., XXXIII, Amer. Math. Soc., Providence, R.I., 1979, pp. 111–155. MR 546593
  • 7. W. Casselman, Introduction to admissible representations of p-adic groups, http://www.math. ubc.ca/$ \sim$cass/research.html
  • 8. Laurent Clozel, Michael Harris, and Richard Taylor, Automorphy for some 𝑙-adic lifts of automorphic mod 𝑙 Galois representations, Publ. Math. Inst. Hautes Études Sci. 108 (2008), 1–181. With Appendix A, summarizing unpublished work of Russ Mann, and Appendix B by Marie-France Vignéras. MR 2470687, 10.1007/s10240-008-0016-1
  • 9. Keqin Feng and Wen-Ch’ing Winnie Li, Spectra of hypergraphs and applications, J. Number Theory 60 (1996), no. 1, 1–22. MR 1405722, 10.1006/jnth.1996.0109
  • 10. Ki-ichiro Hashimoto, Zeta functions of finite graphs and representations of 𝑝-adic groups, Automorphic forms and geometry of arithmetic varieties, Adv. Stud. Pure Math., vol. 15, Academic Press, Boston, MA, 1989, pp. 211–280. MR 1040609
  • 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. Yasutaka Ihara, Discrete subgroups of 𝑃𝐿(2,𝑘_{℘}), Algebraic Groups and Discontinuous Subgroups (Proc. Sympos. Pure Math., Boulder, Colo., 1965) Amer. Math. Soc., Providence, R.I., 1966, pp. 272–278. MR 0205952
  • 13. Martin Kneser, Strong approximation, Algebraic Groups and Discontinuous Subgroups (Proc. Sympos. Pure Math., Boulder, Colo., 1965) Amer. Math. Soc., Providence, R.I., 1966, pp. 187–196. MR 0213361
  • 14. Max-Albert Knus, Alexander Merkurjev, Markus Rost, and Jean-Pierre Tignol, The book of involutions, American Mathematical Society Colloquium Publications, vol. 44, American Mathematical Society, Providence, RI, 1998. With a preface in French by J. Tits. MR 1632779
  • 15. A. Lubotzky, R. Phillips, and P. Sarnak, Ramanujan graphs, Combinatorica 8 (1988), no. 3, 261–277. MR 963118, 10.1007/BF02126799
  • 16. Alexander Lubotzky, Discrete groups, expanding graphs and invariant measures, Progress in Mathematics, vol. 125, Birkhäuser Verlag, Basel, 1994. With an appendix by Jonathan D. Rogawski. MR 1308046
  • 17. George Lusztig, Affine Hecke algebras and their graded version, J. Amer. Math. Soc. 2 (1989), no. 3, 599–635. MR 991016, 10.1090/S0894-0347-1989-0991016-9
  • 18. G. A. Margulis, Explicit group-theoretic constructions of combinatorial schemes and their applications in the construction of expanders and concentrators, Problemy Peredachi Informatsii 24 (1988), no. 1, 51–60 (Russian); English transl., Problems Inform. Transmission 24 (1988), no. 1, 39–46. MR 939574
  • 19. Jonathan D. Rogawski, Automorphic representations of unitary groups in three variables, Annals of Mathematics Studies, vol. 123, Princeton University Press, Princeton, NJ, 1990. MR 1081540
  • 20. Winfried Scharlau, Zur Existenz von Involutionen auf einfachen Algebren. II, Math. Z. 176 (1981), no. 3, 399–404 (German). MR 610220, 10.1007/BF01214616
  • 21. Atle Selberg, On discontinuous groups in higher-dimensional symmetric spaces, Contributions to function theory (internat. Colloq. Function Theory, Bombay, 1960) Tata Institute of Fundamental Research, Bombay, 1960, pp. 147–164. MR 0130324
  • 22. Patrick Solé, Ramanujan hypergraphs and Ramanujan geometries, Emerging applications of number theory (Minneapolis, MN, 1996) IMA Vol. Math. Appl., vol. 109, Springer, New York, 1999, pp. 583–590. MR 1691550, 10.1007/978-1-4612-1544-8_24
  • 23. J. Tits, Reductive groups over local fields, Automorphic forms, representations and 𝐿-functions (Proc. Sympos. Pure Math., Oregon State Univ., Corvallis, Ore., 1977) Proc. Sympos. Pure Math., XXXIII, Amer. Math. Soc., Providence, R.I., 1979, pp. 29–69. MR 546588

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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: http://dx.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.