1549380232-Automorphic_Forms_and_Applications__Sarnak_

364 AUDREY TERRAS, ARITHiVIETICAL QUANTUM CHAOS

r discrete subgroup of G r c G, funda mental group of X

with f \ H compact, X = f \ T finite

no fixed points (q + 1)-regular connected graph

conjugacy classes in r conjugacy classes in r

{'y} = {Xf'X-^1 lx E f}

center {±I} identity

hyperbolic/'""'± ( ~ t~l ) ' hyperbolic /' fixes geodesic C

t > 1, t -=I-r^1 shifts along C by v(!')

centralizer L 1 = { x E fix/' = f'X} f 1 ={xEflXf'=f'X}

=cyclic=< l'o > = < l'o >

l'o primitive hyperbolic l'o primitive hyperbolic

Spectrum 6. on F-(r\ G /I<) A<pn = An<t?ni An = qsn + ql Sn

6.<pn = A,,,<p,,,, n = 0, 1, 2, ... , I An I -=/:-q + 1 ==> 0 <R e (sn) < 1

An = sn(l - s,,,) :::; 0 X Ramanujan

{:::::::? for I An I -=/:-q + 1, Re (sn) = ~

Selberg trace formula Selberg trace formula

f(x) = f(d(x, 0))

f(sn) = area(f\H)f(i)

IXI ~

2= L f(si) = f(O)IXI

.A,.=s,.(1-sn) i=l

+ L N l~-gN/o N -- 1 F(N /' ) + 2= v(p) L Hf(ev(p))

{I } hyperbolic 12 - I^2 {p} hyperbolic e2: 1

F(~n) = c,,,Hf(n),

_ { 2n, n > 0

Cn - 1, n:::; 0

Hf(n) = f(lnl)

+(q - 1) 2= qj-l J(lnl + 2J)

j>l

Selberg zeta function Ihara zeta function

Z(s) = TI TI (1 - N,, 0 s-j) (l((s) = TI (1-uv(p))-1, U = q-s

ho} ]2'.0 {Po}

non-trivial zeros correspond (x(s)-1 =

to spectrum 6. on L^2 (f\ G /I<) (1 - u^2 y-^1 det(J - Au+ qu^2 I)

except for finite # of zeros r =rank r , r - 1 = IXl(g-l)

Riemann Hypothesis (R.H.) (x(s) satisfies R.H.

s E (0, 1), Re(s) = ~ {:::::::? X is R amanujan

Table 3. Part 1. Trace formulas for the the continuous and discrete symmetric

spaces.

corresponding to An is called a cuspidal Maass wave form. They a re much like
classical holomorphic automorphic forms. The Selberg trace formula given in Table
3 column 1 involves the spherical transform of a function f in £^2 (I<\ G /I<) as well
as the horocycle transform off. Since there are only two types of conjugacy classes
in r when the fundamental domain is compact and r acts without fixed points,
there are only two sorts of terms on the right-hand side of the formula. For a r like
SL(2, 'll) there will also be elliptic and parabolic conjugacy classes.