In
mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
, the Selberg trace formula, introduced by , is an expression for the character of the
unitary representation In mathematics, a unitary representation of a group ''G'' is a linear representation π of ''G'' on a complex Hilbert space ''V'' such that π(''g'') is a unitary operator for every ''g'' ∈ ''G''. The general theory is well-developed in case ''G ...
of a
Lie group
In mathematics, a Lie group (pronounced ) is a group that is also a differentiable manifold. A manifold is a space that locally resembles Euclidean space, whereas groups define the abstract concept of a binary operation along with the additio ...
on the space of
square-integrable function
In mathematics, a square-integrable function, also called a quadratically integrable function or L^2 function or square-summable function, is a real- or complex-valued measurable function for which the integral of the square of the absolute value i ...
s, where is a cofinite
discrete group
In mathematics, a topological group ''G'' is called a discrete group if there is no limit point in it (i.e., for each element in ''G'', there is a neighborhood which only contains that element). Equivalently, the group ''G'' is discrete if and on ...
. The character is given by the trace of certain functions on .
The simplest case is when is
cocompact
Cocompact may refer to:
* Cocompact group action
* Cocompact Coxeter group
* Cocompact embedding
* Cocompact lattice
{{dab ...
, when the representation breaks up into discrete summands. Here the trace formula is an extension of the
Frobenius formula for the character of an
induced representation
In group theory, the induced representation is a representation of a group, , which is constructed using a known representation of a subgroup . Given a representation of '','' the induced representation is, in a sense, the "most general" represent ...
of finite groups. When is the cocompact subgroup of the real numbers , the Selberg trace formula is essentially the
Poisson summation formula
In mathematics, the Poisson summation formula is an equation that relates the Fourier series coefficients of the periodic summation of a function to values of the function's continuous Fourier transform. Consequently, the periodic summation of a ...
.
The case when is not compact is harder, because there is a
continuous spectrum, described using
Eisenstein series
Eisenstein series, named after German mathematician Gotthold Eisenstein, are particular modular forms with infinite series expansions that may be written down directly. Originally defined for the modular group, Eisenstein series can be generaliz ...
. Selberg worked out the non-compact case when is the group ; the extension to higher rank groups is the
Arthur–Selberg trace formula In mathematics, the Arthur–Selberg trace formula is a generalization of the Selberg trace formula from the group SL2 to arbitrary reductive groups over global fields, developed by James Arthur (mathematician), James Arthur in a long series of pape ...
.
When is the fundamental group of a
Riemann surface
In mathematics, particularly in complex analysis, a Riemann surface is a connected one-dimensional complex manifold. These surfaces were first studied by and are named after Bernhard Riemann. Riemann surfaces can be thought of as deformed vers ...
, the Selberg trace formula describes the spectrum of differential operators such as the
Laplacian
In mathematics, the Laplace operator or Laplacian is a differential operator given by the divergence of the gradient of a scalar function on Euclidean space. It is usually denoted by the symbols \nabla\cdot\nabla, \nabla^2 (where \nabla is the ...
in terms of geometric data involving the lengths of geodesics on the Riemann surface. In this case the Selberg trace formula is formally similar to the
explicit formulas relating the zeros of the
Riemann zeta function
The Riemann zeta function or Euler–Riemann zeta function, denoted by the Greek letter (zeta), is a mathematical function of a complex variable defined as \zeta(s) = \sum_^\infty \frac = \frac + \frac + \frac + \cdots for \operatorname(s) > ...
to prime numbers, with the zeta zeros corresponding to eigenvalues of the Laplacian, and the primes corresponding to geodesics. Motivated by the analogy, Selberg introduced the
Selberg zeta function
The Selberg zeta-function was introduced by . It is analogous to the famous Riemann zeta function
: \zeta(s) = \prod_ \frac
where \mathbb is the set of prime numbers. The Selberg zeta-function uses the lengths of simple closed geodesics inste ...
of a Riemann surface, whose analytic properties are encoded by the Selberg trace formula.
Early history
Cases of particular interest include those for which the space is a
compact Riemann surface
In mathematics, particularly in complex analysis, a Riemann surface is a connected one-dimensional complex manifold. These surfaces were first studied by and are named after Bernhard Riemann. Riemann surfaces can be thought of as deformed versio ...
. The initial publication in 1956 of
Atle Selberg
Atle Selberg (14 June 1917 – 6 August 2007) was a Norwegian mathematician known for his work in analytic number theory and the theory of automorphic forms, and in particular for bringing them into relation with spectral theory. He was awarded t ...
dealt with this case, its
Laplacian
In mathematics, the Laplace operator or Laplacian is a differential operator given by the divergence of the gradient of a scalar function on Euclidean space. It is usually denoted by the symbols \nabla\cdot\nabla, \nabla^2 (where \nabla is the ...
differential operator and its powers. The traces of powers of a Laplacian can be used to define the
Selberg zeta function
The Selberg zeta-function was introduced by . It is analogous to the famous Riemann zeta function
: \zeta(s) = \prod_ \frac
where \mathbb is the set of prime numbers. The Selberg zeta-function uses the lengths of simple closed geodesics inste ...
. The interest of this case was the analogy between the formula obtained, and the
explicit formulae of
prime number
A prime number (or a prime) is a natural number greater than 1 that is not a product of two smaller natural numbers. A natural number greater than 1 that is not prime is called a composite number. For example, 5 is prime because the only ways ...
theory. Here the
closed geodesic In differential geometry and dynamical systems, a closed geodesic on a Riemannian manifold is a geodesic that returns to its starting point with the same tangent direction. It may be formalized as the projection of a closed orbit of the geodesic flo ...
s on play the role of prime numbers.
At the same time, interest in the traces of
Hecke operator In mathematics, in particular in the theory of modular forms, a Hecke operator, studied by , is a certain kind of "averaging" operator that plays a significant role in the structure of vector spaces of modular forms and more general automorphic repr ...
s was linked to the Eichler–Selberg trace formula, of Selberg and
Martin Eichler, for a Hecke operator acting on a vector space of
cusp form In number theory, a branch of mathematics, a cusp form is a particular kind of modular form with a zero constant coefficient in the Fourier series expansion.
Introduction
A cusp form is distinguished in the case of modular forms for the modular gro ...
s of a given weight, for a given
congruence subgroup
In mathematics, a congruence subgroup of a matrix group with integer entries is a subgroup defined by congruence conditions on the entries. A very simple example would be invertible matrix, invertible 2 × 2 integer matrices of determinan ...
of the
modular group
In mathematics, the modular group is the projective special linear group of matrices with integer coefficients and determinant 1. The matrices and are identified. The modular group acts on the upper-half of the complex plane by fractional l ...
. Here the trace of the identity operator is the dimension of the vector space, i.e. the dimension of the space of modular forms of a given type: a quantity traditionally calculated by means of the
Riemann–Roch theorem
The Riemann–Roch theorem is an important theorem in mathematics, specifically in complex analysis and algebraic geometry, for the computation of the dimension of the space of meromorphic functions with prescribed zeros and allowed poles. It rel ...
.
Applications
The trace formula has applications to
arithmetic geometry
In mathematics, arithmetic geometry is roughly the application of techniques from algebraic geometry to problems in number theory. Arithmetic geometry is centered around Diophantine geometry, the study of rational points of algebraic variety, alg ...
and
number theory
Number theory (or arithmetic or higher arithmetic in older usage) is a branch of pure mathematics devoted primarily to the study of the integers and arithmetic function, integer-valued functions. German mathematician Carl Friedrich Gauss (1777â ...
. For instance, using the trace theorem,
Eichler and Shimura calculated the
Hasse–Weil L-functions associated to
modular curve
In number theory and algebraic geometry, a modular curve ''Y''(Γ) is a Riemann surface, or the corresponding algebraic curve, constructed as a quotient of the complex upper half-plane H by the action of a congruence subgroup Γ of the modular grou ...
s;
Goro Shimura
was a Japanese mathematician and Michael Henry Strater Professor Emeritus of Mathematics at Princeton University who worked in number theory, automorphic forms, and arithmetic geometry. He was known for developing the theory of complex multipli ...
's methods by-passed the analysis involved in the trace formula. The development of
parabolic cohomology (from
Eichler cohomology Several people are named Eichler:
* August W. Eichler (1839–1887), German botanist
* Caroline Eichler (1808/9–1843), German inventor, first woman to be awarded a patent (for her leg prosthesis)
* Eunice Eichler (1932–2017), New Zealand Sal ...
) provided a purely algebraic setting based on
group cohomology
In mathematics (more specifically, in homological algebra), group cohomology is a set of mathematical tools used to study groups using cohomology theory, a technique from algebraic topology. Analogous to group representations, group cohomology loo ...
, taking account of the
cusps characteristic of non-compact Riemann surfaces and modular curves.
The trace formula also has purely
differential-geometric applications. For instance, by a result of Buser, the
length spectrum
Length is a measure of distance. In the International System of Quantities, length is a quantity with dimension distance. In most systems of measurement a base unit for length is chosen, from which all other units are derived. In the Intern ...
of a
Riemann surface
In mathematics, particularly in complex analysis, a Riemann surface is a connected one-dimensional complex manifold. These surfaces were first studied by and are named after Bernhard Riemann. Riemann surfaces can be thought of as deformed vers ...
is an isospectral invariant, essentially by the trace formula.
Later work
The general theory of
Eisenstein series
Eisenstein series, named after German mathematician Gotthold Eisenstein, are particular modular forms with infinite series expansions that may be written down directly. Originally defined for the modular group, Eisenstein series can be generaliz ...
was largely motivated by the requirement to separate out the
continuous spectrum, which is characteristic of the non-compact case.
The trace formula is often given for algebraic groups over the adeles rather than for Lie groups, because this makes the corresponding discrete subgroup into an algebraic group over a field which is technically easier to work with. The case of SL
2(C) is discussed in and . Gel'fand et al also treat SL
2() where is a locally compact topological field with
ultrametric norm, so a finite extension of the
p-adic numbers
In mathematics, the -adic number system for any prime number extends the ordinary arithmetic of the rational numbers in a different way from the extension of the rational number system to the real and complex number systems. The extension ...
Q
''p'' or of the
formal Laurent series
In mathematics, a formal series is an infinite sum that is considered independently from any notion of convergence, and can be manipulated with the usual algebraic operations on series (addition, subtraction, multiplication, division, partial sums ...
F
''q''((''T'')); they also handle the adelic case in characteristic 0, combining all completions R and Q
''p'' of the
rational numbers
In mathematics, a rational number is a number that can be expressed as the quotient or fraction of two integers, a numerator and a non-zero denominator . For example, is a rational number, as is every integer (e.g. ). The set of all rationa ...
Q.
Contemporary successors of the theory are the
Arthur–Selberg trace formula In mathematics, the Arthur–Selberg trace formula is a generalization of the Selberg trace formula from the group SL2 to arbitrary reductive groups over global fields, developed by James Arthur (mathematician), James Arthur in a long series of pape ...
applying to the case of general semisimple ''G'', and the many studies of the trace formula in the
Langlands philosophy
In representation theory and algebraic number theory, the Langlands program is a web of far-reaching and influential conjectures about connections between number theory and geometry. Proposed by , it seeks to relate Galois groups in algebraic num ...
(dealing with technical issues such as
endoscopy
An endoscopy is a procedure used in medicine to look inside the body. The endoscopy procedure uses an endoscope to examine the interior of a hollow organ or cavity of the body. Unlike many other medical imaging techniques, endoscopes are insert ...
). The Selberg trace formula can be derived from the Arthur–Selberg trace formula with some effort.
Selberg trace formula for compact hyperbolic surfaces
A compact hyperbolic surface can be written as the space of orbits
where is a subgroup of , and is the
upper half plane, and acts on by
linear fractional transformations.
The Selberg trace formula for this case is easier than the general case because the surface is compact so there is no continuous spectrum, and the group has no parabolic or elliptic elements (other than the identity).
Then the spectrum for the
Laplace–Beltrami operator
In differential geometry, the Laplace–Beltrami operator is a generalization of the Laplace operator to functions defined on submanifolds in Euclidean space and, even more generally, on Riemannian and pseudo-Riemannian manifolds. It is named af ...
on is discrete and real, since the Laplace operator is self adjoint with compact
resolvent; that is
where the eigenvalues correspond to -invariant eigenfunctions in of the Laplacian; in other words
Using the variable substitution
the eigenvalues are labeled
Then the Selberg trace formula is given by
The right hand side is a sum over conjugacy classes of the group , with the first term corresponding to the identity element and the remaining terms forming a sum over the other conjugacy classes (which are all hyperbolic in this case). The function has to satisfy the following:
* be analytic on ;
* ;
* there exist positive constants and such that:
The function is the Fourier transform of , that is,
Notes
References
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External links
Selberg trace formula resource page
{{Authority control
Automorphic forms