Hilbert's twelfth problem
Encyclopedia
Kronecker's Jugendtraum or Hilbert's twelfth problem, of the 23 mathematical Hilbert problems, is the extension of Kronecker–Weber theorem
on abelian extension
s of the rational number
s, to any base number field. That is, it asks for analogues of the roots of unity, as complex numbers that are particular values of the exponential function
; the requirement is that such numbers should generate a whole family of further number fields that are analogues of the cyclotomic field
s and their subfields.
The classical theory of complex multiplication
, now often known as the Kronecker Jugendtraum, does this for the case of any imaginary quadratic field, by using modular functions and elliptic function
s chosen with a particular period lattice related to the field in question. Shimura extended this to CM fields. The general case is still open . Leopold Kronecker
described the complex multiplication issue as his or “dearest dream of his youth”.
is to describe the fields of algebraic numbers
. The work of Galois
made it clear that field extensions are controlled by certain groups
, the Galois group
s. The simplest situation, which is already at the boundary of what we can do, is when the group in question is abelian
. All quadratic extensions, obtained by adjoining the roots of a quadratic equation, are abelian, and their study was commenced by Gauss. Another type of abelian extension of the field Q of rational numbers is given by adjoining the nth roots of unity, resulting in the cyclotomic field
s. Already Gauss had shown that, in fact, every quadratic field
is contained in a larger cyclotomic field. The Kronecker–Weber theorem
shows that any finite abelian extension of Q is contained in a cyclotomic field. Kronecker's (and Hilbert's) question addresses the situation of a more general algebraic number field K: what are the algebraic numbers necessary to construct all abelian extensions of K? The complete answer to this question has been completely worked out only when K is an imaginary quadratic field or its generalization, a CM-field
.
Hilbert's original statement of his 12th problem is rather misleading: he seems to imply that the abelian extensions of imaginary quadratic fields are generated by special values of elliptic modular functions, which is not correct. (It is hard to tell exactly what Hilbert was saying, one problem being that he may have been using the term "elliptic function" to mean both the elliptic function ℘ and the elliptic modular function j.)
First it is also necessary to use roots of unity, though Hilbert may have implicitly meant to include these. More seriously, while values of elliptic modular functions generate the Hilbert class field, for more general abelian extensions one also needs to use values of elliptic functions. For example, the abelian extension
is not generated by singular moduli and roots of unity.
One particularly appealing way to state the Kronecker–Weber theorem is by saying that the maximal abelian extension of Q can be obtained by adjoining the special values exp(2πi/n) of the exponential function
. Similarly, the theory of complex multiplication
shows that the maximal abelian extension of Q(τ), where τ is an imaginary quadratic irrationality, can be obtained by adjoining the special values of ℘(τ,z) and j(τ) of modular functions j and elliptic functions ℘, and roots of unity, where τ is in the imaginary quadratic field and z represents a torsion point on the corresponding elliptic curve. One interpretation of Hilbert's twelfth problem asks to provide a suitable analogue of exponential, elliptic, or modular functions, whose special values would generate the maximal abelian extension Kab of a general number field K. In this form, it remains unsolved. A description of the field Kab was obtained in the class field theory
, developed by Hilbert
himself, Emil Artin
, and others in the first half of the 20th century. However the construction of Kab in class field theory involves first constructing larger non-abelian extensions using Kummer theory, and then cutting down to the abelian extensions, so does not really solve Hilbert's problem which asks for a more direct construction of the abelian extensions.
and Taniyama
. This gives rise to abelian extensions of CM-field
s in general. The question of which extensions can be found is that of the Tate module
s of such varieties, as Galois representations. Since this is the most accessible case of l-adic cohomology, these representations have been studied in depth.
Robert Langlands
argued in 1973 that the modern version of the should deal with Hasse-Weil zeta function
s of Shimura varieties
. While he envisaged a grandiose program
that would take the subject much further, more than thirty years later serious doubts remain concerning its import for the question that Hilbert asked.
A separate development was Stark's conjecture
(Harold Stark
), which in contrast dealt directly with the question of finding interesting, particular units in number fields. This has seen a large conjectural development for L-function
s, and is also capable of producing concrete, numerical results.
Kronecker–Weber theorem
In algebraic number theory, the Kronecker–Weber theorem states that every finite abelian extension of the field of rational numbers Q, or in other words, every algebraic number field whose Galois group over Q is abelian, is a subfield of a cyclotomic field, i.e. a field obtained by adjoining a root...
on abelian extension
Abelian extension
In abstract algebra, an abelian extension is a Galois extension whose Galois group is abelian. When the Galois group is a cyclic group, we have a cyclic extension. More generally, a Galois extension is called solvable if its Galois group is solvable....
s of the rational number
Rational number
In mathematics, a rational number is any number that can be expressed as the quotient or fraction a/b of two integers, with the denominator b not equal to zero. Since b may be equal to 1, every integer is a rational number...
s, to any base number field. That is, it asks for analogues of the roots of unity, as complex numbers that are particular values of the exponential function
Exponential function
In mathematics, the exponential function is the function ex, where e is the number such that the function ex is its own derivative. The exponential function is used to model a relationship in which a constant change in the independent variable gives the same proportional change In mathematics,...
; the requirement is that such numbers should generate a whole family of further number fields that are analogues of the cyclotomic field
Cyclotomic field
In number theory, a cyclotomic field is a number field obtained by adjoining a complex primitive root of unity to Q, the field of rational numbers...
s and their subfields.
The classical theory of complex multiplication
Complex multiplication
In mathematics, complex multiplication is the theory of elliptic curves E that have an endomorphism ring larger than the integers; and also the theory in higher dimensions of abelian varieties A having enough endomorphisms in a certain precise sense In mathematics, complex multiplication is the...
, now often known as the Kronecker Jugendtraum, does this for the case of any imaginary quadratic field, by using modular functions and elliptic function
Elliptic function
In complex analysis, an elliptic function is a function defined on the complex plane that is periodic in two directions and at the same time is meromorphic...
s chosen with a particular period lattice related to the field in question. Shimura extended this to CM fields. The general case is still open . Leopold Kronecker
Leopold Kronecker
Leopold Kronecker was a German mathematician who worked on number theory and algebra.He criticized Cantor's work on set theory, and was quoted by as having said, "God made integers; all else is the work of man"...
described the complex multiplication issue as his or “dearest dream of his youth”.
Description of the problem
The fundamental problem of algebraic number theoryAlgebraic number theory
Algebraic number theory is a major branch of number theory which studies algebraic structures related to algebraic integers. This is generally accomplished by considering a ring of algebraic integers O in an algebraic number field K/Q, and studying their algebraic properties such as factorization,...
is to describe the fields of algebraic numbers
Algebraic number field
In mathematics, an algebraic number field F is a finite field extension of the field of rational numbers Q...
. The work of Galois
Évariste Galois
Évariste Galois was a French mathematician born in Bourg-la-Reine. While still in his teens, he was able to determine a necessary and sufficient condition for a polynomial to be solvable by radicals, thereby solving a long-standing problem...
made it clear that field extensions are controlled by certain groups
Group (mathematics)
In mathematics, a group is an algebraic structure consisting of a set together with an operation that combines any two of its elements to form a third element. To qualify as a group, the set and the operation must satisfy a few conditions called group axioms, namely closure, associativity, identity...
, the Galois group
Galois group
In mathematics, more specifically in the area of modern algebra known as Galois theory, the Galois group of a certain type of field extension is a specific group associated with the field extension...
s. The simplest situation, which is already at the boundary of what we can do, is when the group in question is abelian
Abelian group
In abstract algebra, an abelian group, also called a commutative group, is a group in which the result of applying the group operation to two group elements does not depend on their order . Abelian groups generalize the arithmetic of addition of integers...
. All quadratic extensions, obtained by adjoining the roots of a quadratic equation, are abelian, and their study was commenced by Gauss. Another type of abelian extension of the field Q of rational numbers is given by adjoining the nth roots of unity, resulting in the cyclotomic field
Cyclotomic field
In number theory, a cyclotomic field is a number field obtained by adjoining a complex primitive root of unity to Q, the field of rational numbers...
s. Already Gauss had shown that, in fact, every quadratic field
Quadratic field
In algebraic number theory, a quadratic field is an algebraic number field K of degree two over Q. It is easy to show that the map d ↦ Q is a bijection from the set of all square-free integers d ≠ 0, 1 to the set of all quadratic fields...
is contained in a larger cyclotomic field. The Kronecker–Weber theorem
Kronecker–Weber theorem
In algebraic number theory, the Kronecker–Weber theorem states that every finite abelian extension of the field of rational numbers Q, or in other words, every algebraic number field whose Galois group over Q is abelian, is a subfield of a cyclotomic field, i.e. a field obtained by adjoining a root...
shows that any finite abelian extension of Q is contained in a cyclotomic field. Kronecker's (and Hilbert's) question addresses the situation of a more general algebraic number field K: what are the algebraic numbers necessary to construct all abelian extensions of K? The complete answer to this question has been completely worked out only when K is an imaginary quadratic field or its generalization, a CM-field
CM-field
In mathematics, a CM-field is a particular type of number field K, so named for a close connection to the theory of complex multiplication. Another name used is J-field. Specifically, K is a quadratic extension of a totally real field which is totally imaginary, i.e...
.
Hilbert's original statement of his 12th problem is rather misleading: he seems to imply that the abelian extensions of imaginary quadratic fields are generated by special values of elliptic modular functions, which is not correct. (It is hard to tell exactly what Hilbert was saying, one problem being that he may have been using the term "elliptic function" to mean both the elliptic function ℘ and the elliptic modular function j.)
First it is also necessary to use roots of unity, though Hilbert may have implicitly meant to include these. More seriously, while values of elliptic modular functions generate the Hilbert class field, for more general abelian extensions one also needs to use values of elliptic functions. For example, the abelian extension
is not generated by singular moduli and roots of unity.One particularly appealing way to state the Kronecker–Weber theorem is by saying that the maximal abelian extension of Q can be obtained by adjoining the special values exp(2πi/n) of the exponential function
Exponential function
In mathematics, the exponential function is the function ex, where e is the number such that the function ex is its own derivative. The exponential function is used to model a relationship in which a constant change in the independent variable gives the same proportional change In mathematics,...
. Similarly, the theory of complex multiplication
Complex multiplication
In mathematics, complex multiplication is the theory of elliptic curves E that have an endomorphism ring larger than the integers; and also the theory in higher dimensions of abelian varieties A having enough endomorphisms in a certain precise sense In mathematics, complex multiplication is the...
shows that the maximal abelian extension of Q(τ), where τ is an imaginary quadratic irrationality, can be obtained by adjoining the special values of ℘(τ,z) and j(τ) of modular functions j and elliptic functions ℘, and roots of unity, where τ is in the imaginary quadratic field and z represents a torsion point on the corresponding elliptic curve. One interpretation of Hilbert's twelfth problem asks to provide a suitable analogue of exponential, elliptic, or modular functions, whose special values would generate the maximal abelian extension Kab of a general number field K. In this form, it remains unsolved. A description of the field Kab was obtained in the class field theory
Class field theory
In mathematics, class field theory is a major branch of algebraic number theory that studies abelian extensions of number fields.Most of the central results in this area were proved in the period between 1900 and 1950...
, developed by Hilbert
David Hilbert
David Hilbert was a German mathematician. He is recognized as one of the most influential and universal mathematicians of the 19th and early 20th centuries. Hilbert discovered and developed a broad range of fundamental ideas in many areas, including invariant theory and the axiomatization of...
himself, Emil Artin
Emil Artin
Emil Artin was an Austrian-American mathematician of Armenian descent.-Parents:Emil Artin was born in Vienna to parents Emma Maria, née Laura , a soubrette on the operetta stages of Austria and Germany, and Emil Hadochadus Maria Artin, Austrian-born of Armenian descent...
, and others in the first half of the 20th century. However the construction of Kab in class field theory involves first constructing larger non-abelian extensions using Kummer theory, and then cutting down to the abelian extensions, so does not really solve Hilbert's problem which asks for a more direct construction of the abelian extensions.
Modern development
Developments since around 1960 have certainly contributed. Before that in his dissertation used Hilbert modular forms to study abelian extensions of real quadratic fields. Complex multiplication of abelian varieties was an area opened up by the work of ShimuraGoro Shimura
is a Japanese mathematician, and currently a professor emeritus of mathematics at Princeton University.Shimura was a colleague and a friend of Yutaka Taniyama...
and Taniyama
Yutaka Taniyama
Yutaka Taniyama was a Japanese mathematician known for the Taniyama-Shimura conjecture.-Contribution:...
. This gives rise to abelian extensions of CM-field
CM-field
In mathematics, a CM-field is a particular type of number field K, so named for a close connection to the theory of complex multiplication. Another name used is J-field. Specifically, K is a quadratic extension of a totally real field which is totally imaginary, i.e...
s in general. The question of which extensions can be found is that of the Tate module
Tate module
In mathematics, a Tate module of an abelian group, named for John Tate, is a module constructed from an abelian group A. Often, this construction is made in the following situation: G is a commutative group scheme over a field K, Ks is the separable closure of K, and A = G...
s of such varieties, as Galois representations. Since this is the most accessible case of l-adic cohomology, these representations have been studied in depth.
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...
argued in 1973 that the modern version of the should deal with Hasse-Weil zeta function
Hasse-Weil zeta function
In mathematics, the Hasse–Weil zeta function attached to an algebraic variety V defined over an algebraic number field K is one of the two most important types of L-function. Such L-functions are called 'global', in that they are defined as Euler products in terms of local zeta functions...
s of Shimura varieties
Shimura variety
In number theory, a Shimura variety is a higher-dimensional analogue of a modular curve that arises as a quotient of a Hermitian symmetric space by a congruence subgroup of a reductive algebraic group defined over Q. The term "Shimura variety" applies to the higher-dimensional case, in the case of...
. While he envisaged a grandiose program
Langlands program
The Langlands program is a web of far-reaching and influential conjectures that relate Galois groups in algebraic number theory to automorphic forms and representation theory of algebraic groups over local fields and adeles. It was proposed by ....
that would take the subject much further, more than thirty years later serious doubts remain concerning its import for the question that Hilbert asked.
A separate development was Stark's conjecture
Stark's conjecture
In number theory, the Stark conjectures, the first of which were introduced in a series of papers in the 1970s by American mathematician Harold Stark and later greatly expanded by John Tate, give conjectural information about the coefficient of the leading term in the Taylor expansion of an Artin...
(Harold Stark
Harold Stark
Harold Mead Stark is an American mathematician, specializing in number theory. He is best known for his solution of the Gauss class number 1 problem, in effect correcting and completing the earlier work of Kurt Heegner; and for Stark's conjecture. He has recently collaborated with Audrey Terras on...
), which in contrast dealt directly with the question of finding interesting, particular units in number fields. This has seen a large conjectural development for L-function
L-function
The theory of L-functions has become a very substantial, and still largely conjectural, part of contemporary analytic number theory. In it, broad generalisations of the Riemann zeta function and the L-series for a Dirichlet character are constructed, and their general properties, in most cases...
s, and is also capable of producing concrete, numerical results.