1549380232-Automorphic_Forms_and_Applications__Sarnak_

LECTURE 1. FINITE MODELS

spectrum finite upper half plane, p=353, a=3, g=1, h=5

40

20

-30 .L{) ·10 10 20 30 40

level spacing finite upper half plane p=353, a=3, g=1,h=5

200

150

100

50

0.2 0.4 0.6 0.8 1.2 1.4 1.6 1.8

Figure 9. Histograms of Spectra of Finite Upper Half Plane Graphs (without

multiplicity) ?353(3, 3). In our program to compute Soto-Andrade sums we

needed to know that a generator of the multiplicative group of IF353(V3) is

1 + 5V3. The top histogram is for the spectrum and the lower one is for the

unnormalized level spacing.

351

approach the semi-circle distribution as q goes to infinity. They use the Jacquet-
Langlands correspondence for GL(2) over function fields and the connection Winnie
Li has made between the finite upper half plane graphs and Morgenstern's function
field analogues (see [61]) of the Lubotzky, Phillips and Sarnak graphs. See Li's
article in [37] pp. 387-403. Note that the result of Chai and Li does not follow
from the work of McKay [58] because the degrees of the finite upper h alf plane
graphs approach infinity with q. The Poisson behavior of the level spacings for
finite upper half planes is still only conjectural.
However if you r eplace the finite field lF q with a finite ring like 'll/ q'll, for q =
pr, p =prime, r > 1, the spectral histograms change to those in Figure 10 which
do not resemble the semi-circle at all.
Here we view finite upper half planes as providing a "toy" symmetric space.
But an application has been found (see Tiu and Wallace [88] ).

Exercise 6. Define finite upper half plane graphs over the finite fields of charac-
teristic 2. The spectra of these graphs have been considered in Angel [3] and Evans
[27]. Using a result of Katz, the non-trivial graphs have also been proved to be
Ramanujan.