LECTURE 2. THREE SYMMETRIC SPACES 367terms are not so hard to deal with as in the continuous case since there can be
no divergent integrals. The elliptic terms are a bit annoying because they behave
differently depending on whether q is an even or odd power of p.
We could not fill in the last rows of the third column of table 3 because we
have not found an an alogue of the Selberg zeta function for finite upper h a lf plane
graphs. We have worked out a number of examples of the formula in [ 82 ].
One might question our definition of elliptic and hyperbolic here. In t he
classical case of SL(2, IR), hyperbolic means eigenvalues real and distinct while
elliptic means eigenvalues complex.
Next we consider f-Tessellations of the 3 Symmetric spaces. The first
(in Figure 6) is that of the Poincare upper ha lf plane given by using Mathematica
to give us a density plot of y^6 t imes the absolute value of the cusp form of weight12 known as .6.(x + iy), the discriminant. The next (in Figure 7) is the tessellation
of the 3 -regular tree obtained by defining a function which has values 1,2,3,4 on the
4 vertices of K 4. The last (in Figure 8) is a density plot for a n S L(2, lf 7 ) invariant
function on H 49.
Figure 6. Tessellation of the Poincare upper half plane corresponding to the
ab olute value of the modular form delta times y^6Exercise 10. a) Make columns for Tables 2 and 3 for the Euclidean spaces !Rn and
JF;.
b) D raw the Euclidean analogues of Figures 1-5.
Exercise 11. Compute the fundamental domain for GL(2, lf 5 ) acting on H 25.