1549380232-Automorphic_Forms_and_Applications__Sarnak_

LECTURE 1. FINITE MODELS

Spectrum for E48011~,1)

1000

500

0

·1 .0.8 -06 -04 - 02 0 .2 04 0.6 0.8

Level Spacing for E48011 ~.1)

4000

0'--~'------'~--'~----'-~-~-l----'-~-'

0 02 0.4 0.6 0 .8 1 12 1.4

Figure 8. The top histogram is that of the spectrum (without multiplicity)

of the adjacency matrix of the finite Euclidean 3-space graph X(IF~, S) , where

p = 48611 and S consists of solutions to the congruence xi + x~ + x~ =

1 (modp). The bottom histogram is the normalized level spacing histogram for

the same graph.

349

When p = l(mod4), on the other hand, sometimes the graphs are not Ramanu-
jan; e.g., when p=l 7 and 53. The list of primes for which these graphs are not
Ramanujan is a bit mysterious at this point. See Bannai et al [7].

4.2. Finite Upper Half Planes

We can replace the finite Euclidean plane lF~ with the finite non-Euclidean "up-

per" half plane Hq C lF q ( v'o) where 8 is a non-square in the finite field lF q (for q
odd) and

Hq={z=x+y05"!x,yElFq, y#O}.

Our finite analogue of the complex numbers is lF q ( v'o) = lF q2. We have all the usual
algebraic rules for computing in the field of complex numbers. For example, define,
for z = x+yv'o, the imaginary part of z to bey= Im(z). And define the conjugate
z = x -yv'o, the norm N z = zz.
We define a non-Euclidean "distance" by

(4.5)

N(z -w)

d(z, w) = Im(z)Im(w) ·