Automorphic form
Encyclopedia
In mathematics
, the general notion of automorphic form is the extension to analytic function
s, perhaps of several complex variables
, of the theory of modular form
s. It is in terms of a Lie group
, to generalise the groups SL2(R) or PSL2 (R) of modular forms, and a discrete group
, to generalise the modular group
, or one of its congruence subgroup
s.
for 
, which is a type of 1-cocycle in the language of group cohomology
. The values of
may be complex numbers, or in fact complex square matrices, corresponding to the possibility of vector-valued automorphic forms. The cocycle condition imposed on the factor of automorphy is something that can be routinely checked, when 
is derived from a Jacobian matrix, by means of the chain rule
.
In the general setting, then, an automorphic form is a function
on 
(with values in some fixed finite-dimensional vector space 
, in the vector-valued case), subject to three kinds of conditions:
It is the first of these that makes
automorphic, that is, satisfy an interesting functional equation
relating
with 
for 
. In the vector-valued case the specification can involve a finite-dimensional group representation
ρ acting on the components to 'twist' them. The Casimir operator condition says that some Laplacians have
as eigenfunction; this ensures that 
has excellent analytic properties, but whether it is actually a complex-analytic function depends on the particular case. The third condition is to handle the case where 
is not compact
but has cusp
s.
a Fuchsian group
had already received attention before 1900 (see below). The Hilbert modular forms (Hilbert-Blumenthal, as one should say) were proposed not long after that, though a full theory was long in coming. The Siegel modular form
s, for which
is a symplectic group
, arose naturally from considering moduli space
s and theta functions. The post-war interest in several complex variables made it natural to pursue the idea of automorphic form in the cases where the forms are indeed complex-analytic. Much work was done, in particular by Ilya Piatetski-Shapiro, in the years around 1960, in creating such a theory. The theory of the Selberg trace formula
, as applied by others, showed the considerable depth of the theory. Robert Langlands
showed how (in generality, many particular cases being known) the Riemann-Roch theorem could be applied to the calculation of dimensions of automorphic forms; this is a kind of post hoc check on the validity of the notion. He also produced the general theory of Eisenstein series
, which corresponds to what in spectral theory
terms would be the 'continuous spectrum' for this problem, leaving the cusp form
or discrete part to investigate. From the point of view of number theory, the cusp forms had been recognised, since Srinivasa Ramanujan
, as the heart of the matter.
an algebraic group
, treated as an adelic algebraic group
. It does not completely include the automorphic form idea introduced above, in that the adele
approach is a way of dealing with the whole family of congruence subgroup
s at once. Inside an
space for a quotient of the adelic form of 
, an automorphic representation is a representation that is an infinite tensor product
of representations of p-adic groups, with specific enveloping algebra
representations for the infinite prime(s). One way to express the shift in emphasis is that the Hecke operator
s are here in effect put on the same level as the Casimir operators; which is natural from the point of view of functional analysis
, though not so obviously for the number theory. It is this concept that is basic to the formulation of the Langlands philosophy.
's first area of interest in mathematics, dating to the 1880s, was automorphic forms. He named them Fuchsian functions, after the mathematician Lazarus Fuchs
, because Fuchs was known for being a good teacher and had researched on differential equations and the theory of functions. Poincaré actually developed the concept of these functions as part of his doctoral thesis. Under Poincaré's definition, an automorphic function is one which is analytic in its domain and is invariant under a denumerable infinite group of linear fractional transformations. Automorphic functions then generalize both trigonometric and elliptic functions.
Poincaré explains how he discovered Fuchsian functions:
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
, the general notion of automorphic form is the extension to analytic function
Analytic function
In mathematics, an analytic function is a function that is locally given by a convergent power series. There exist both real analytic functions and complex analytic functions, categories that are similar in some ways, but different in others...
s, perhaps of several complex variables
Several complex variables
The theory of functions of several complex variables is the branch of mathematics dealing with functionson the space Cn of n-tuples of complex numbers...
, of the theory of modular form
Modular form
In mathematics, a modular form is a analytic function on the upper half-plane satisfying a certain kind of functional equation and growth condition. The theory of modular forms therefore belongs to complex analysis but the main importance of the theory has traditionally been in its connections...
s. It is in terms of a Lie group
Lie group
In mathematics, a Lie group is a group which is also a differentiable manifold, with the property that the group operations are compatible with the smooth structure...
, to generalise the groups SL2(R) or PSL2 (R) of modular forms, and a discrete groupDiscrete group
In mathematics, a discrete group is a group G equipped with the discrete topology. With this topology G becomes a topological group. A discrete subgroup of a topological group G is a subgroup H whose relative topology is the discrete one...
, to generalise the modular groupModular group
In mathematics, the modular group Γ is a fundamental object of study in number theory, geometry, algebra, and many other areas of advanced mathematics...
, or one of its congruence subgroup
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 2x2 integer matrices of determinant 1, such that the off-diagonal entries are even.An importance class of congruence...
s.
Formulation
The formulation requires the general notion of factor of automorphyFactor of automorphy
In mathematics, the notion of factor of automorphy arises for a group acting on a complex-analytic manifold. Suppose a group G acts on a complex-analytic manifold X. Then, G also acts on the space of holomorphic functions from X to the complex numbers. A function f is termed an automorphic form if...
for , which is a type of 1-cocycle in the language of group cohomologyGroup cohomology
In abstract algebra, homological algebra, algebraic topology and algebraic number theory, as well as in applications to group theory proper, group cohomology is a way to study groups using a sequence of functors H n. The study of fixed points of groups acting on modules and quotient modules...
. The values of
may be complex numbers, or in fact complex square matrices, corresponding to the possibility of vector-valued automorphic forms. The cocycle condition imposed on the factor of automorphy is something that can be routinely checked, when is derived from a Jacobian matrix, by means of the chain ruleChain rule
In calculus, the chain rule is a formula for computing the derivative of the composition of two or more functions. That is, if f is a function and g is a function, then the chain rule expresses the derivative of the composite function in terms of the derivatives of f and g.In integration, the...
.
In the general setting, then, an automorphic form is a function
on (with values in some fixed finite-dimensional vector space , in the vector-valued case), subject to three kinds of conditions:- to transform under translation by elements
according to the given automorphy factor 
; - to be an eigenfunction of certain Casimir operators on
; and - to satisfy some conditions on growth at infinity.
It is the first of these that makes
automorphic, that is, satisfy an interesting functional equationFunctional equation
In mathematics, a functional equation is any equation that specifies a function in implicit form.Often, the equation relates the value of a function at some point with its values at other points. For instance, properties of functions can be determined by considering the types of functional...
relating
with for . In the vector-valued case the specification can involve a finite-dimensional group representationGroup representation
In the mathematical field of representation theory, group representations describe abstract groups in terms of linear transformations of vector spaces; in particular, they can be used to represent group elements as matrices so that the group operation can be represented by matrix multiplication...
ρ acting on the components to 'twist' them. The Casimir operator condition says that some Laplacians have
as eigenfunction; this ensures that has excellent analytic properties, but whether it is actually a complex-analytic function depends on the particular case. The third condition is to handle the case where is not compactCompact space
In mathematics, specifically general topology and metric topology, a compact space is an abstract mathematical space whose topology has the compactness property, which has many important implications not valid in general spaces...
but has cusp
Cusp form
In number theory, a branch of mathematics, a cusp form is a particular kind of modular form, distinguished in the case of modular forms for the modular group by the vanishing in the Fourier series expansion \Sigma a_n q^n...
s.
History
Before this very general setting was proposed (around 1960), there had already been substantial developments of automorphic forms other than modular forms. The case of a Fuchsian groupFuchsian group
In mathematics, a Fuchsian group is a discrete subgroup of PSL. The group PSL can be regarded as a group of isometries of the hyperbolic plane, or conformal transformations of the unit disc, or conformal transformations of the upper half plane, so a Fuchsian group can be regarded as a group acting...
had already received attention before 1900 (see below). The Hilbert modular forms (Hilbert-Blumenthal, as one should say) were proposed not long after that, though a full theory was long in coming. The Siegel modular form
Siegel modular form
In mathematics, Siegel modular forms are a major type of automorphic form. These stand in relation to the conventional elliptic modular forms as abelian varieties do in relation to elliptic curves; the complex manifolds constructed as in the theory are basic models for what a moduli space for...
s, for which
is a symplectic groupSymplectic group
In mathematics, the name symplectic group can refer to two different, but closely related, types of mathematical groups, denoted Sp and Sp. The latter is sometimes called the compact symplectic group to distinguish it from the former. Many authors prefer slightly different notations, usually...
, arose naturally from considering moduli space
Moduli space
In algebraic geometry, a moduli space is a geometric space whose points represent algebro-geometric objects of some fixed kind, or isomorphism classes of such objects...
s and theta functions. The post-war interest in several complex variables made it natural to pursue the idea of automorphic form in the cases where the forms are indeed complex-analytic. Much work was done, in particular by Ilya Piatetski-Shapiro, in the years around 1960, in creating such a theory. The theory of the Selberg trace formula
Selberg trace formula
In mathematics, the Selberg trace formula, introduced by , is an expression for the character of the unitary representation of G on the space L2 of square-integrable functions, where G is a Lie group and Γ a cofinite discrete group...
, as applied by others, showed the considerable depth of the theory. Robert Langlands
Robert Langlands
Robert Phelan Langlands is a mathematician, best known as the founder of the Langlands program. He is an emeritus professor at the Institute for Advanced Study...
showed how (in generality, many particular cases being known) the Riemann-Roch theorem could be applied to the calculation of dimensions of automorphic forms; this is a kind of post hoc check on the validity of the notion. He also produced the general theory of Eisenstein series
Real analytic Eisenstein series
In mathematics, the simplest real analytic Eisenstein series is a special function of two variables. It is used in the representation theory of SL and in analytic number theory...
, which corresponds to what in spectral theory
Spectral theory
In mathematics, spectral theory is an inclusive term for theories extending the eigenvector and eigenvalue theory of a single square matrix to a much broader theory of the structure of operators in a variety of mathematical spaces. It is a result of studies of linear algebra and the solutions of...
terms would be the 'continuous spectrum' for this problem, leaving the cusp form
Cusp form
In number theory, a branch of mathematics, a cusp form is a particular kind of modular form, distinguished in the case of modular forms for the modular group by the vanishing in the Fourier series expansion \Sigma a_n q^n...
or discrete part to investigate. From the point of view of number theory, the cusp forms had been recognised, since Srinivasa Ramanujan
Srinivasa Ramanujan
Srīnivāsa Aiyangār Rāmānujan FRS, better known as Srinivasa Iyengar Ramanujan was a Indian mathematician and autodidact who, with almost no formal training in pure mathematics, made extraordinary contributions to mathematical analysis, number theory, infinite series and continued fractions...
, as the heart of the matter.
Automorphic representations
The subsequent notion of automorphic representation has proved of great technical value for dealing with an algebraic groupAlgebraic group
In algebraic geometry, an algebraic group is a group that is an algebraic variety, such that the multiplication and inverse are given by regular functions on the variety...
, treated as an adelic algebraic group
Adelic algebraic group
In abstract algebra, an adelic algebraic group is a topological group defined by an algebraic group G over a number field K, and the adele ring A = A of K. It consists of the points of G having values in A; the definition of the appropriate topology is straightforward only in case G is a linear...
. It does not completely include the automorphic form idea introduced above, in that the adele
Adele ring
In algebraic number theory and topological algebra, the adele ring is a topological ring which is built on the field of rational numbers . It involves all the completions of the field....
approach is a way of dealing with the whole family of congruence subgroup
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 2x2 integer matrices of determinant 1, such that the off-diagonal entries are even.An importance class of congruence...
s at once. Inside an
space for a quotient of the adelic form of , an automorphic representation is a representation that is an infinite tensor productTensor product
In mathematics, the tensor product, denoted by ⊗, may be applied in different contexts to vectors, matrices, tensors, vector spaces, algebras, topological vector spaces, and modules, among many other structures or objects. In each case the significance of the symbol is the same: the most general...
of representations of p-adic groups, with specific enveloping algebra
Enveloping algebra
Enveloping algebra in mathematics may refer to:* The universal enveloping algebra of a Lie algebra* The enveloping algebra of a general non-associative algebra...
representations for the infinite prime(s). One way to express the shift in emphasis is that the Hecke operator
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 representations....
s are here in effect put on the same level as the Casimir operators; which is natural from the point of view of functional analysis
Functional analysis
Functional analysis is a branch of mathematical analysis, the core of which is formed by the study of vector spaces endowed with some kind of limit-related structure and the linear operators acting upon these spaces and respecting these structures in a suitable sense...
, though not so obviously for the number theory. It is this concept that is basic to the formulation of the Langlands philosophy.
Poincaré on psychology of discovery and his work on automorphic functions
PoincaréHenri Poincaré
Jules Henri Poincaré was a French mathematician, theoretical physicist, engineer, and a philosopher of science...
's first area of interest in mathematics, dating to the 1880s, was automorphic forms. He named them Fuchsian functions, after the mathematician Lazarus Fuchs
Lazarus Fuchs
Lazarus Immanuel Fuchs was a German mathematician who contributed important research in the field of linear differential equations...
, because Fuchs was known for being a good teacher and had researched on differential equations and the theory of functions. Poincaré actually developed the concept of these functions as part of his doctoral thesis. Under Poincaré's definition, an automorphic function is one which is analytic in its domain and is invariant under a denumerable infinite group of linear fractional transformations. Automorphic functions then generalize both trigonometric and elliptic functions.
Poincaré explains how he discovered Fuchsian functions:
- For fifteen days I strove to prove that there could not be any functions like those I have since called Fuchsian functions. I was then very ignorant; every day I seated myself at my work table, stayed an hour or two, tried a great number of combinations and reached no results. One evening, contrary to my custom, I drank black coffee and could not sleep. Ideas rose in crowds; I felt them collide until pairs interlocked, so to speak, making a stable combination. By the next morning I had established the existence of a class of Fuchsian functions, those which come from the hypergeometric series; I had only to write out the results, which took but a few hours.