Classical Kloosterman Sums: Representation Theory, Magic Squares, and Ramanujan Multigraphs

Authors

Patrick S. Fleming, South Dakota School of Mines and Technology
Stephan Ramon Garcia, Pomona CollegeFollow
Gizem Karaali, Pomona CollegeFollow

Document Type

Article - preprint

Department

Mathematics (Pomona)

Publication Date

2011

Keywords

Kloosterman sum, Representation, Magic square, Panmagic square, Ramanujan multigraph, Ramanujan graph, Multigraph, Eigenvalues, Weil bound

Abstract

We consider a certain finite group for which Kloosterman sums appear as character values. This leads us to consider a concrete family of commuting hermitian matrices which have Kloosterman sums as eigenvalues. These matrices satisfy a number of “magical” combinatorial properties and they encode various arithmetic properties of Kloosterman sums. These matrices can also be regarded as adjacency matrices for multigraphs which display Ramanujan-like behavior.

Comments

Pre-print from http://arxiv.org/abs/1004.3550

Final publication can be found at:

Fleming, P., Garcia, S.R., Karaali, G., Classical Kloosterman sums: representation theory, magic squares, and Ramanujan multigraphs, J. Number Theory 131 (2011), no. 4, 661–680. MR2753270 (2012a:11114)

Rights Information

© 2010 Elsevier B.V.

Terms of Use & License Information

Terms of Use for work posted in Scholarship@Claremont.

DOI

10.1016/j.jnt.2010.10.009

Recommended Citation

Patrick S. Fleming, Stephan Ramon Garcia, Gizem Karaali, Classical Kloosterman sums: Representation theory, magic squares, and Ramanujan multigraphs, Journal of Number Theory, Volume 131, Issue 4, April 2011, Pages 661-680, ISSN 0022-314X, 10.1016/j.jnt.2010.10.009. (http://www.sciencedirect.com/science/article/pii/S0022314X10002684)