### The Selberg trace formula of compact Riemann surfaces

```THE SELBERG TRACE FORMULA OF COMPACT
RIEMANN SURFACES
IGOR PROKHORENKOV
1. Introduction to the Selberg Trace Formula
This is a talk about the paper H. P. McKean: Selberg’s Trace Formula as applied to a compact Riemann surface (1972). For simplicity,
assume that M is a compact Riemannian manifold. Consider classical
mechanics on M , where a free particle on M moves along geodesics. If
M has infinite fundamental group, then in each free homotopy class of
a curve on M , there is a unique closed geodesic. If M is a Riemann
surface of genus ≥ 1, we can look at the length of the (unique) closed
shortest geodesic in each equivalence class from π1 (M ).
Next, consider quantum mechanics. Eigenvalues of the Laplacian on
the manifold
0 = γ1 < γ2 ≤ γ3 ≤ ... ↑ +∞
γj is the energy of the j th “pure” state. We expect that there is a
relation between the classical data (lengths of closed geodesics) and
quantum data (eigenvalues). The Selberg trace formula provides this
link. So there should be a “Selberg trace formula” on any manifold.
There are many examples of this. When the manifold has a lot of
symmetry (eg hyperbolic space mod a subgroup of P SL (2, R)), there
is an example.
Starting with the 19th century: the Poisson summation formula.
Let M = CL be the 2-torus, where L is an integral lattice. We let
L be the integral span of 1 and a + ib. Consider the Laplacian on this
surface. Then the Laplacian will be
2
∂
∂2
+
∆=−
∂x21 ∂x22
acting on L-periodic functions. The eigenfunctions are
f (x) = exp (2πiω 0 · x) ,
where ω 0 is an element of the dual lattice: that is ω 0 ·ω ∈ Z for all ω ∈ L.
The corresponding eigenvalues are 4π 2 |ω 0 |2 . The Poisson summation
1
2
IGOR PROKHORENKOV
formula relates the theta functions of L and its dual lattice L0 , as
follows:
!
area (M ) X
X
|ω|2
2
0 2
exp −4π |ω | t =
exp −
,
4πt ω∈L
4t
ω 0 ∈L0
or
X
λj
|
!
X
area (M )
|ω|2
exp (−λj t) =
exp −
4πt
4t
|ω| length of closed geodesics
{z
} |
{z
}
quantum side
classical side
The theta function in physics language is the partition function.
Also, this is the trace of the heat kernel:
X
exp (−λj t) ,
tr (exp (−t∆)) =
λj
where exp (−t∆) is the heat operator. The heat equation is
∂
K = −∆x K; K (0, x, y) = δ (x − y) ,
∂t
Z
tr (exp (−t∆)) =
K (t, x, x) dx
M
In the case of the torus,
K T (t, x, x) =
X
2
K R (t, γ (x) , x) .
γ∈L
On the circle,
1
K S (t, x, x) =
X
K R (t, x + n, x) .
n∈Z
To prove the Poisson summation formula, one expands the left side
in terms of the Fourier series 2and
uses the known heat kernel for R :
|x−y|
1
R
√
K (t, x, y) = 4πt exp − 4t .
The knowledge of the spectrum of the Laplacian determines |a| and
|b|, as follows. For our torus with the lattice above, area(M ) = |b|.
After this, subtract terms from both sides the parts corresponding
to
q
geodesics with length 1, 2, 3, ... The next geodesic will be |a|2 + |b|2 ,
so we can find |a|. So we can determine the torus up to reflection.
Next, we generalize to a hyperbolic Riemann surface M with genus
g ≥ 2. Then the spectrum σ (M ) of the Laplacian is
0 = γ1 < γ2 ≤ γ3 ≤ ... → +∞.
THE SELBERG TRACE FORMULA OF COMPACT RIEMANN SURFACES
3
The Selberg trace formula relates the trace
∞
X
tr (exp (−t∆)) =
exp (−tλj )
j=1
of the heat kernel to the kind of dual theta function. The role of the
lattice L is taken over by the conjugacy classes Q of G = π1 (M ),
identified with a subgroup of SL (2, R). Then
M = SL (2, R) G.
The numbers |w| are replaced by
l (Q) = 2 cosh
−1
1
tr (Q) .
2
Here, Q is a free deformation class of closed paths on M , and l (Q)
is the length of the shortest path in this class. There is a famous
(noncompact) cases that we will not cover
M = SL (2, R) SL (2, Z)
or
M = SL (2, R) Γ
where Γ is an algebraic subgroup of SL (2, Z). Audrey Terras and Serge
Lang have good books on the subject. Also there is a survey paper by
Werner M¨
uller.
2. Riemann surface formula
Let M be a compact Riemann surface of genus g ≥ 2. By the
Riemann uniformization theorem, the universal cover is the upper half
plane H.
H = {(x1 + ix2 ) : x2 > 0}
This is also called the Poincar´e hyperbolic plane, with metric
ds2 =
dx21 + dx22
x22
(You can also realize this as the Poincare disk.) The fundamental
group π1 (M ) acts by deck
on H that are isometries
transformations
a b
z 7→ az+b
, such that det
= 1 with a, b, c, d ∈ R. So SL (2, R)
cz+d
c d
is the group of isometries of H. Thus, we can identify π1 (M ) with a
subgroup G of SL (2, R). The G has a fundamental region in H that is
a hyperbolic polygon with 4g sides.
4
IGOR PROKHORENKOV
What must be true about G in order that HG is a compact Riemann
that SL (2, R) = KAN , where K = SO (2) =
surface? Note
cos (θ) − sin θ
is the stabilizer of i. The group A is the group
sin θ
cos θ
a 0
: a > 0 . The group N is the group of
of magnifications
0 a−1
1 b
horizontal translations, N =
:b∈R .
0 1
Proposition 1. For any g ∈ SL (2, R), g is conjugate to
• rotation iff tr (g) < 2 (elliptic)
• magnification iff tr (g) > 2 (hyperbolic)
• translation iff tr (g) = 2 (parabolic)
The hyperbolic distance d (x, y) satisfies
d (x, gx) = d (kx, kgx)
= d kx, kgk −1 kx ,
for all k, g in SL (2, R). So
inf d (x, gx) = inf d kx, kgk −1 kx
x
x
= inf d x, kgk −1 x
x
Think of elements of G as homotopy classes of closed paths on M
with fixed base point.
Free homotopy classes of M are identified with the conjugacy class
Q = {kgk −1 : k ∈ G}.
Every nontrivial element of G is conjugate to a hyperbolic element.
(Proof: if an element g of G is not the identity, then there is a geodesic
of minimum length connecting some x to gx, but if g is parabolic or
elliptic, this length can go to zero.) Thus it is conjugate to a magnification, and thus ` (g n ) = |n| ` (g) is true, where ` (g) = inf x d (x, gx) is
the length of the shortest path.
For every g ∈ G that is nontrivial, it can be expressed in a unique
way as the positive power of a primitive element p ∈ G (primitive: it
is not the power of any other element of G).
Proposition 2. As p runs through the inconjugate primitive elements
in G and n through the positive integers, the conjugacy class
Q = kpn k −1 : k ∈ GGp
runs through the conjugacy classes of G. Here, Gp is the centralizer of
p. Moreover, for fixed p, n, elements kpk −1 run once through Q as k
runs through GGp .
THE SELBERG TRACE FORMULA OF COMPACT RIEMANN SURFACES
Note that d (x, y) = cosh−1 1 +
m 0
p∼
.
0 m−1
kx−yk
2x2 y2
5
, ` (pn ) = n |log m2 | where
Theorem 3. (Selberg trace formula) Start with a function K : R → R
that decays sufficiently rapidly as x → ∞. Then KH (x, y) = K (cosh d (x, y))
is a function on H × H. It induces a symmetric kernel on M × M via
X
KM (x, y) =
KH (x, gy) .
g∈G
Then
KM (x, hy) = KM (x, y)
for all h ∈ G. Then, with dx =
Z
KM (x, x) dx
trKM : =
dx1 dx2
x22
= hyperbolic volume element
M
= area (M ) K (1) +
∞
X
` (p)
X
p
cosh ` (pn ) − 1
n=1 inconjugate
Z
∞
cosh `(pn )
K (b) db
p
.
b − cosh ` (pn )
primitive p
Proof. Let F be a fundamental domain of M . We have
Z
KM (x, x) dx
trKM =
F
XZ
K (cosh d (x, gx)) dx
=
g∈G
M
= area (M ) K (1) +
= area (M ) K (1) +
= area (M ) K (1) +
∞
X
X
X Z
n=1 inconjugate k∈GGp
primitive p
∞
X X
X
n=1 inconjugate k∈GGp
primitive p
∞
X X Z
K cosh d x, kpn k −1 x
F
Z
K (cosh d (x, pn x)) dx
F
K (cosh d (x, pn x)) dx
n=1 inconjugate
primitive p
Fp
where Fp is a fundamental domain for Gp . Then p is conjugate to
some magnification x 7→ m2 x. Then Fp = {x1 ∈ R : 1 ≤ x2 ≤ m2 } .
The formula follows from a direct calculation.
dx
6
IGOR PROKHORENKOV
In the particular case where KH (t) is the fundamental solution of
the heat equation on H. Then KH (t) = exp (−t∆). Then
∞
X
tr (exp (−t∆)) =
exp (−tγn )
n=0
= area (M )
e−t/4
Z
∞
2
be−b /4t
db
sinh 21 b
(4πt)3/2 0
∞
1X X
e−t/4 −|`(pn )|2 /4t
` (p)
e
.
+
2 n=1 inconjugate sinh 21 ` (pn ) (4πt)1/2
primitive p
```