Spectral Measure and Selberg's Trace Formula
By George Simpson, Supervised by Dr Ali Taheri
Mathematics Department at the University of Sussex
Results: Eisenstein Series
Introduction
In 1950, a Norwegian mathematician called Atle Selberg was awarded the
elds medal for his work in analytic number theory, harmonic analysis on
hyperbolic spaces, and spectral theory. He developed many stunning results,
and his idea involved bringing techniques from harmonic analysis to study
automorphic forms. This project will examine those results, and the
connections between these subjects.
Laplace Beltrami Operators on
Let
be a Fuchsian group. Dene the non-Euclidean Laplacian
I
y2
:=
n
I
I
H
@2
@2
+
@x
@y 2
2
H as
Let
H be the hyperbolic quotient space where a Fuchsian group and
H is the upper-half plane. Let ( H) be the space of bounded
automorphic forms. From the structure of the resolvent of , we can
deduce basic spectral information:
For
H compact, has discrete spectrum in [0; )
For
H non-compact, has absolutely continuous spectrum [1=4; )
and discrete spectrum contained in [0; )
If
H has innite-area, then there are no embedded eigenvalues, i.e. the
discrete spectrum is nite and contained in (0; 1=4):
C n
n
n
1
1
1
n
Spectral Measure
The von-Neumann spectral theorem states that if T is any self-adjoint
operator (bounded or unbounded), then we have a Stieltjes integral
representation
R +1
f(T)= 1 f ()dE
for some family of projection-valued measures dE and any Borel function
for R: If we can compute Green's function dened by
R
(T-I ) 1 (x ) =
G (; x ; y ) (y )dy
then given a convergent integral of the form
f() =
R +1
1 (
)
we have the spectral formula
dE
2 i d = (T
( + i 0)I ) 1
Spectral Resolution of in C (
1
g ()d (T
D:=
i 0)I )
(
1
n H)
f 2 C01( n H) : f
and f 2 L2( n H; dA)
By introducing invariant integral operators, we can dene an operator that
maps (
H) into itself and where it has pure point spectrum. Note that
the automorphic Laplacian has pure point spectrum in (
H) (spanned
by cusp forms).
The eigenspaces have nite dimension. For any complete orthonormal
system of cusp forms uj ; every f
( H) has the expansion
P
f(z)= j f ; uj uj (z )
C n
C n
f g
2C n
h i
converging in the norm topology in L2:
Spectral Decomposition of the Space of Eisenstein Series
Dene the weighted Eisenstein series
E n
Ea(z
j
)=
P
2
(Ima 1 z )
an
The space (
H) of incomplete Eisenstein series splits orthogonally into
invariant subspaces,
E(
n H) = R( n H) a Ea( n H)
Hence we have the expansion
P
h
i
P
R1
h
i Ea(z ; 1=2 + ir )dr
j f ; uj uj (z ) +
1 f ; Ea( ; 1=2 + ir )
which converges in the norm topology.
1
a 4
George Simpson JRA Candidate
The Selberg Trace Formula
I
I
I
To obtain the trace formula, we calculate the trace for parabolic, elliptic,
and hyperbolic conjugacy classes to get the following: Suppose
h is even
h is holomorphic in the strip t 12 + h(t ) ( t + 1) 2 : Let g be the Fourier transform of h: Then:
j= j jj
R1
P
j
F
j
' 1
1 R1
h
(rj ) + 4 1 h(r ) ' (2 + ir )dr = 4 1 h(r )r tanh( r )dr +
j
P P1 l =2
l =2) 1g (l log p ) log p +
(
p
p
PP Pl =1
cosh (1 2l =m)r
l 1 R 1
h(r ) cosh r dr
RR 1 0<l <m(2m sin m )
1
h
1 h(r ) (1 + ir )dr :
2
0
Dene the non-Euclidean Laplacian on the domain
Figure 1: This is a geometric interpretation of a holomorphic Eisenstein series G6 of weight 2k
for the modular group. Specically, it is the real part of G6 as a function of q on the unit disk.
The Selberg trace formula can be applied for the nite area case.For
example, it can be applied to the prime geodesic theorem due to its relation
to hyperbolic isometries:
l
P
# e
x Lix + Li(x sj ) where sj (1 sj )
f
g
are the eigenvalues in (0; 14 ): We also have the Weyl-Selberg asymptotic
formula
Area( nH)
1 R r 1=4 ' 1
# j
r 4 r 1=4 ' (2 + it )dt
r
4
fj j g
p
p
0
References
[1] Iwaniec, H. Spectral Methods of Automorphic Forms Graduate Studies in
Mathematics, 2nd Edition, 2002
[2] Terras, A. Harmonic Analysis on Symmetric Spaces-Euclidean Space, the
Sphere, and the Poincare Upper Half-Plane, Springer, 2nd Edition, 2013
[3] Sogge, C. Hangzhou Lectures on Eigenfunctions of the Laplacian, Princeton
University Press, 1st Edition, 2014
[4] Borthwick, D. Spectral Theory on Hyperbolic Surfaces, July 2010, retreived
(from https://math.dartmouth.edu/ specgeom/Borthwicks lides :pdf )
Acknowledgments
I would like to thank Ali Taheri for his patience and guidence during this
project.
[email protected]