1549380232-Automorphic_Forms_and_Applications__Sarnak_

LECTURE 1. FINITE MODELS 343

One says that r is arithmetic if there is a faithful rational representation (see

Borel [14], p. 6) p : G ---+ GL(n), the general linear group of n x n non-singular

matrices, such that p is defined over Q and such that p(r) is commensurable with

p(r)nGL(n, Z). Arithmetic and non-arithmetic subgroups of SL(2, q are discussed

by Elstrodt, Grunewald, and Mennicke in [26], Chapter 10.

nl sl nlsl

Figure 6. From Schmit [68]. The top shows the fundamental domain of an

arithmetic group which tessellates the Poincare disk. The lower right shows

the level spacing histogram for the Dirichlet problem on the triangle OLM using

the 1500 eigenvalues computed by Schmit using the method of collocation. The

lower left is the analogous histogram for a non-arithmetic triangle. The solid

line is the GOE distribution and the dashed one is Poisson (e-x ).

The top of Figure 6 shows the fundamental domain of an arithmetic group in
the unit disk. Note that the upper half plane can be identified with the unit disc
via the map sending z in H to ~:;:~. C. Schmit [68] found 1500 eigenvalues for
the Dirichlet problem of the Poincare Laplacian on the triangle OLM with angles
7r /8, 7r /2, 7r /3. The histogram of level spacings for this problem is the lower right
part of Figure 6. Schmit also considered the Dirichlet problem for a non-arithmetic
triangle with angles n /8, 7r /2, 67n /200. The level spacing histogram for this non-
arithmetic triangle is given in the lower left of Figure 6. Schmit concludes: "The
spectrum of the tessellating [arithmetic] triangle exhibits neither level repulsion nor
spectral rigidity and there are strong evidences that asymptotically the spectrum is
of Poisson type, although the billiard is known to be a strongly chaotic system. The