LECTURE 2. THREE SYlVIMETRIC SPACES 365

r c G ; r = GL(2, 1Fp)

conjugacy clas s e s in r

central { ~ ~ ) , a E lF;

hy p erbolic ( ~ ~ ) ,

a =f. b E lF~

p arabolic ( ~^1 a ) , a E lF* P

elliptic { ~ b! )

a,bElFp,b::/=0,

~=f.u^2 ,uE1Fp

S elberg trace formula

(q = p2)

2= J(7r)mult(7r, Ind~l)

7rEGK

= lr\Gl(P - l)f( Vl)

+ (q+l)(q-1)2 L F(c)

2(p-l)

cEF;

c#l

+ q(q

2

-^1 l {F(l ) -F(v'o)}

+/-1 2 L F(a+bry) a-bry '

a ,bEIFv

b#O

where ry^2 = ~, fJ E 1Fq

Unknown if there is an an alogue of Selberg zeta function

Table 3. Part 2. Trace formula for the the fin ite symmetric spaces.

T he Selberg trace formula provides a correspondence between t he length spec-
trum and the Laplacian spectrum. It thus gives analogues of many of t he explicit
formulas in analytic number theory. There are many applications; e.g. to prove t he
Weyl law for t he number of l>-nl ::; x as x ---) oo. The Weyl law says

{

/\ ' n I I , I /\n ::; x } rv area(r\47r H )x , as x --+ ()().