Hostname: page-component-7c8c6479df-7qhmt Total loading time: 0 Render date: 2024-03-28T13:01:16.623Z Has data issue: false hasContentIssue false

Ramanujan Type Buildings

Published online by Cambridge University Press:  20 November 2018

Cristina M. Ballantine*
Affiliation:
Department of Mathematics, University of Toronto, Toronto, Ontario, M5A 3G3 Department of Mathematics, Bowdoin College, Brunswick, Maine 04011, USA email: cballant@bowdoin.edu
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

We will construct a finite union of finite quotients of the affine building of the group $\text{G}{{\text{L}}_{3}}$ over the field of $p$-adic numbers ${{\mathbb{Q}}_{p}}$. We will view this object as a hypergraph and estimate the spectrum of its underlying graph.

Type
Research Article
Copyright
Copyright © Canadian Mathematical Society 2000

References

[1] Arthur, J., Unipotent automorphic representations: conjectures. Asterisque 171–172(1989), 1371.Google Scholar
[2] Arthur, J. and Gelbart, S., Lectures on automorphic L-functions. London Math. Soc. Lecture Note Ser. 153(1989), 159.Google Scholar
[3] Ballantine, C., A Hypergraph With Commuting Partial Laplacians. Canad. Math. Bull., to appear.Google Scholar
[4] Bollobas, B., Extremal Graph Theory. London Math. Soc. Monographs 11, Academic Press, 1978.Google Scholar
[5] Borel, A., Automorphic L-functions. In: Automorphic forms, representations, and L-functions, Proc. Sympos. Pure Math. 33, Part II (eds. Borel, A. and Casselman, W.), AMS, Providence, RI, 1979, 2761.Google Scholar
[6] Borel, A., Some finiteness properties of adele groups over number fields. Inst. Hautes Études Sci. Publ. Math. 16(1963), 530.Google Scholar
[7] Brown, K. S., Buildings. Springer-Verlag, New York, 1989.Google Scholar
[8] Cartier, P., Representations of p-adic groups: A survey. In: Automorphic forms, representations, and Lfunctions, Proc. Sympos. Pure Math. 33, Part I (eds. Borel, A. and Casselman, W.), AMS, Providence, RI, 1979, 111155.Google Scholar
[9] Cassels, J. W. S. and Föhlich, A., Algebraic Number Theory. Academic Press, 1967.Google Scholar
[10] Flath, D., Decomposition of representations into tensor products. In: Automorphic forms, representations, and L-functions, Proc. Sympos. Pure Math. 33, Part I (eds. Borel, A. and Casselman, W.), AMS, Providence, RI, 1979, 179184.Google Scholar
[11] Jacquet, H. and Shalika, J., On Euler products and the classification of automorphic forms, I and II. Amer. J. Math. 103(1981), no. 3 and no. 4, 499–558 and 777815.Google Scholar
[12] Deligne, P., La Conjecture de Weil I. Inst. Hautes Études Sci. Publ. Math. 43(1974), 273308.Google Scholar
[13] Knapp, A., Representation theory of semisimple groups. Princeton Univ. Press, Princeton, NJ, 1986.Google Scholar
[14] Langlands, R. P., On the notion of an automorphic representation. In: Automorphic forms, representations, and L-functions, Proc. Sympos. Pure Math. 33, Part I (eds. Borel, A. and Casselman, W.), AMS, Providence, RI, 1979, 203207.Google Scholar
[15] Labesse, J.-P. and Langlands, R. P., L-Indistinguishability for SL(2). Canad. J. Math. (4) 31(1979), 726785.Google Scholar
[16] Langlands, R. P. and Ramakrishnan, D., The description of the theorem. In: The zeta functions of Picard modular surfaces (eds. Langlands, R. P. and Ramakrishnan, D.), Univ. Montreal, Montreal, 1992, 255301.Google Scholar
[17] Lubotzky, A., Discrete groups, expanding graphs and invariant measures. Progr.Math. 125, Birkhauser Verlag, 1994.Google Scholar
[18] Lubotzky, A., Phillips, R. and Sarnak, P., Ramanujan graphs. Combinatorica 8(1988), 261277.Google Scholar
[19] Rogawski, J. D., Automorphic representations of the unitary group in three variables. Ann. ofMath. Stud. 123, Princeton Univ. Press, Princeton, NJ, 1990.Google Scholar
[20] Rogawski, J. D., The multiplicity formula for A-packets. In: The zeta functions of Picard modular surfaces, Univ. Montreal,Montreal, PQ 1992, 395–419.Google Scholar
[21] Ronan, M., Lectures on Buildings. Academic Press, 1989.Google Scholar
[22] Selberg, A., On discontinuous groups in higher-dimensional symmetric spaces. In: Contributions to Function Theory, International Colloquia on Function Theory (Bombay, 1960), Tata Institute of Fundamental Research, Bombay, 1960, 147164.Google Scholar
[23] Steinberg, R., Endomorphisms of linear algebraic groups. Mem. Amer. Math. Soc. 80, AMS, 1960.Google Scholar
[24] Tate, J., The Harish transform on GL(n). Mimeographed notes.Google Scholar
[25] Tate, J., Number theoretic background. In: Automorphic forms, representations, and L-functions, Proc. Sympos. Pure Math. 33, Part II (eds. Borel, A. and Casselman, W.), AMS, Providence, RI, 1979, 326.Google Scholar
[26] Tits, J., Reductive groups over local fields. In: Automorphic forms, representations, and L-functions, Proc. Sympos. Pure Math. 33, Part I (eds. Borel, A. and Casselman, W.), AMS, Providence, RI, 1979, 2969.Google Scholar