LECTURE 2. THREE SYMMETRIC SPACES 357
and regular. Here "GUE" means that the spacing between pairs of zeros/poles is
that of the eigenvalues of a random Hermitian matrix. Columns 1-3 are essentially
taken from Katz and Sarnak. Column 4 is ours. One should also make a column
for Selberg zeta and L-functions of Riemannian manifolds. We have omitted the
last row of the Katz and Sarnak table. It concerns the monodromy or symmetry
group of the family of zeta functions. See Katz and Sarnak [44] for an explanation
of that row.
type of (or number function (regular)
L-function field field graph theoretic
functional equation yes yes yes, many
spectral interpretation
of O's/poles? yes yes
RH expect it yes iff graph Ramanujan
level spacing of high yes for
zeros/poles GUE expect it almost all curves?
Table 1. The Zoo of Zetas - A New Column for Table 2 in Katz and
Sarnak [45].
Hashimoto [34], p. 255, shows that the congruence zeta function of the modular
curve X 0 (f) over a finite field lFP is essentially the Ihara zeta function of a certain
graph X attached to the curve. And he finds that the number of lF P rational points
of the Jacobian variety of X 0 (f) is the class number of the function field of the
modular curve (p -=/= f) and is related to the complexity of X. Thus in some cases
our 4th column is the same as the function field zeta column in [45].
- Comparisons of the Three Types of Symmetric Spaces
We arrange our comparisons of symmetric spaces G / K in a series of figures and
tables. The goal is to compare the Selberg trace formula in the 3 spaces. Figure
1 shows the three spaces with the Poincare upper half plane on the upper left,
the 3-regular tree on bottom, and the finite upper half space over the field with 3
elements on the upper right. Of course, we cannot draw all of the infinite spaces H
and T. In Table 2, the first column belongs to the Poincare upper half plane, the
second column to the (q + 1)-regular tree, the third column to the finite upper half
plane over 1Fq. Note that we split the table into two parts, with part 1 containing
the first 2 columns and part 2 containing the 3rd column. Here lF q denotes a finite
field with an odd number q of elements. Table 2 can be viewed as a dictionary for
the 3 languages.
Symmetric spaces G/K can be described in terms of Gelfand pairs (G,K) of
group G and subgroup K. This means that the convolution algebra of functions
on G which are K bi-invariant is a commutative algebra. Here convolution of
functions f, g : G --+ C is given by formula ( 4.6) for discrete G Replace the sum
in ( 4.6) by an integral with respect to Haar measure on continuous G
For the Poincare upper half plane the group G is the special linear group
SL(2, JR.) which consists of 2 x 2 real matrices of determinant 1 acting on H by
fractional linear transformation. In the case of the finite upper half plane the
