Spherical functions for finite upper half planes with characteristic 2
This paper along with that of \textit{N. Katz} [Finite Fields Appl. 1, 395-398 (1995)]\ extends results of \textit{J. Angel}, \textit{S. Poulos}, the reviewer, \textit{C. Trimble} and \textit{E. Velasquez} [Contemp. Math. 173, 15-70 (1994; Zbl 0813.11034)]\ and \textit{N. Katz} [J. Reine Angew. Math. 438, 143-161 (1993; Zbl 0798.11053)]\ to the case of fields of characteristic 2. This is part of an attempt to study finite analogues of the real symmetric spaces such as the Poincaré upper half plane. The reviewer [Experimental Math. (in press)]\ has written a survey paper on this subject (for fields of odd characteristic). The non-Euclidean finite upper half plane graphs were first considered by Jeff Angel for fields of characteristic 2. See \textit{J. Angle} [Ph.D. Thesis, U.C.S.D. (1993)]\ and \textit{J. Angel} [Finite Fields Appl. 2, 62-86 (1996)]. However, there remained the question of whether these graphs were Ramanujan in the sense of Lubotzky, Phillips and Sarnak. See \textit{A. Lubotzky} [Discrete groups, expanding graphs and invariant measures (Birkhäuser, 1994; Zbl 0826.22012)]. Ramanujan graphs are of interest because they are good expanders and because the simple random walk on these graphs converges particularly rapidly to uniform. The present paper along with the above-mentioned paper of Katz shows that the graphs in question are indeed Ramanujan. In the case of finite fields of characteristic 2, the formulas for the eigenvalues of the adjacency operators of the graphs are slightly different from those of \textit{J. Soto-Andrade} [Proc. Symp. Pure Math. 47, 305-316 (1987; Zbl 0652.20047)]\ in that they involve additive characters as well as multiplicative. Note that when fields with \(q\) elements are replaced by rings \(\mathbb{Z}/ q\mathbb{Z}\), the corresponding graphs are not usually Ramanujan (at least for odd \(q\)); see \textit{J. Angel}, \textit{C. Trimble}, \textit{B. Shook}, and \textit{A. Terras} [Linear Algebra Appl. 226-228, 423-457 (1995; Zbl 0833.11059)].
non-Euclidean finite upper half plane graphs
scientific article; zbMATH DE number 796756