Stark–Heegner theorem
Encyclopedia
In number theory
, a branch of mathematics
, the Stark–Heegner theorem states precisely which quadratic imaginary number fields
admit unique factorisation
in their ring of integers
. It solves a special case of Gauss's class number problem
of determining the number of imaginary quadratic fields that have a given fixed class number
.
Let Q denote the set of rational number
s, and let d be a square-free integer
(i.e., a product of distinct primes
) other than 1. Then Q(√d)
is a finite extension
of Q, called a quadratic extension. The class number
of Q(√d) is the number of equivalence classes of ideals of the ring of integers of Q(√d), where two ideals I and J are equivalent if and only if
there exist principal ideal
s (a) and (b) such that (a)I = (b)J. Thus, the ring of integers of Q(√d) is a principal ideal domain
(and hence a unique factorization domain
) if and only if the class number of Q(√d) is equal to 1. The Stark–Heegner theorem can then be stated as follows:
These are known as the Heegner number
s.
This list is also written, replacing −1 with −4 and −2 with −8 (which does not change the field), as:
where D is interpreted as the discriminant
(either of the number field or of an elliptic curve
with complex multiplication
).
. It was essentially proven by Kurt Heegner
in 1952, but Heegner's proof had some minor gaps and the theorem was not accepted until Harold Stark
gave a complete proof in 1967, which Stark showed was actually equivalent to Heegner's. Heegner "died before anyone really understood what he had done". Stark formally filled in the gap in Heegner's proof in 1969. Alan Baker gave a completely different proof at about the same time (or more precisely reduced the result to a finite amount of computation). Stark's 1969 paper also cited the widely-known 1895 text by Heinrich Martin Weber and noted that had Weber "only made the observation that the reducibility of [a certain equation] would lead to a Diophantine equation
, the class-number one problem would have been solved 60 years ago".
In 1985, Monsur Kenku gave a novel proof using the Klein quartic
. Noam Elkies
gives an exposition of this result.
Number theory
Number theory is a branch of pure mathematics devoted primarily to the study of the integers. Number theorists study prime numbers as well...
, a branch of mathematics
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 Stark–Heegner theorem states precisely which quadratic imaginary number fields
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...
admit unique factorisation
Unique factorization domain
In mathematics, a unique factorization domain is, roughly speaking, a commutative ring in which every element, with special exceptions, can be uniquely written as a product of prime elements , analogous to the fundamental theorem of arithmetic for the integers...
in their ring of integers
Ring of integers
In mathematics, the ring of integers is the set of integers making an algebraic structure Z with the operations of integer addition, negation, and multiplication...
. It solves a special case of Gauss's class number problem
Class number problem for imaginary quadratic fields
In mathematics, the Gauss class number problem , as usually understood, is to provide for each n ≥ 1 a complete list of imaginary quadratic fields with class number n. It is named after the great mathematician Carl Friedrich Gauss. It can also be stated in terms of discriminants...
of determining the number of imaginary quadratic fields that have a given fixed class number
Ideal class group
In mathematics, the extent to which unique factorization fails in the ring of integers of an algebraic number field can be described by a certain group known as an ideal class group...
.
Let Q denote the set of 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, and let d be a square-free integer
Integer
The integers are formed by the natural numbers together with the negatives of the non-zero natural numbers .They are known as Positive and Negative Integers respectively...
(i.e., a product of distinct primes
Prime number
A prime number is a natural number greater than 1 that has no positive divisors other than 1 and itself. A natural number greater than 1 that is not a prime number is called a composite number. For example 5 is prime, as only 1 and 5 divide it, whereas 6 is composite, since it has the divisors 2...
) other than 1. Then Q(√d)
Algebraic number field
In mathematics, an algebraic number field F is a finite field extension of the field of rational numbers Q...
is a finite extension
Field extension
In abstract algebra, field extensions are the main object of study in field theory. The general idea is to start with a base field and construct in some manner a larger field which contains the base field and satisfies additional properties...
of Q, called a quadratic extension. The class number
Ideal class group
In mathematics, the extent to which unique factorization fails in the ring of integers of an algebraic number field can be described by a certain group known as an ideal class group...
of Q(√d) is the number of equivalence classes of ideals of the ring of integers of Q(√d), where two ideals I and J are equivalent if and only if
If and only if
In logic and related fields such as mathematics and philosophy, if and only if is a biconditional logical connective between statements....
there exist principal ideal
Principal ideal
In ring theory, a branch of abstract algebra, a principal ideal is an ideal I in a ring R that is generated by a single element a of R.More specifically:...
s (a) and (b) such that (a)I = (b)J. Thus, the ring of integers of Q(√d) is a principal ideal domain
Principal ideal domain
In abstract algebra, a principal ideal domain, or PID, is an integral domain in which every ideal is principal, i.e., can be generated by a single element. More generally, a principal ideal ring is a nonzero commutative ring whose ideals are principal, although some authors refer to PIDs as...
(and hence a unique factorization domain
Unique factorization domain
In mathematics, a unique factorization domain is, roughly speaking, a commutative ring in which every element, with special exceptions, can be uniquely written as a product of prime elements , analogous to the fundamental theorem of arithmetic for the integers...
) if and only if the class number of Q(√d) is equal to 1. The Stark–Heegner theorem can then be stated as follows:
- If d < 0, then the class number of Q(√d) is equal to 1 if and only if
These are known as the Heegner number
Heegner number
In number theory, a Heegner number is a square-free positive integer d such that the imaginary quadratic field Q has class number 1...
s.
This list is also written, replacing −1 with −4 and −2 with −8 (which does not change the field), as:
where D is interpreted as the discriminant
Discriminant of an algebraic number field
In mathematics, the discriminant of an algebraic number field is a numerical invariant that, loosely speaking, measures the size of the algebraic number field...
(either of the number field or of an elliptic curve
Elliptic curve
In mathematics, an elliptic curve is a smooth, projective algebraic curve of genus one, on which there is a specified point O. An elliptic curve is in fact an abelian variety — that is, it has a multiplication defined algebraically with respect to which it is a group — and O serves as the identity...
with 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...
).
History
This result was first conjectured by GaussCarl Friedrich Gauss
Johann Carl Friedrich Gauss was a German mathematician and scientist who contributed significantly to many fields, including number theory, statistics, analysis, differential geometry, geodesy, geophysics, electrostatics, astronomy and optics.Sometimes referred to as the Princeps mathematicorum...
. It was essentially proven by Kurt Heegner
Kurt Heegner
Kurt Heegner was a German private scholar from Berlin, who specialized inradio engineering and mathematics...
in 1952, but Heegner's proof had some minor gaps and the theorem was not accepted until 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...
gave a complete proof in 1967, which Stark showed was actually equivalent to Heegner's. Heegner "died before anyone really understood what he had done". Stark formally filled in the gap in Heegner's proof in 1969. Alan Baker gave a completely different proof at about the same time (or more precisely reduced the result to a finite amount of computation). Stark's 1969 paper also cited the widely-known 1895 text by Heinrich Martin Weber and noted that had Weber "only made the observation that the reducibility of [a certain equation] would lead to a Diophantine equation
Diophantine equation
In mathematics, a Diophantine equation is an indeterminate polynomial equation that allows the variables to be integers only. Diophantine problems have fewer equations than unknown variables and involve finding integers that work correctly for all equations...
, the class-number one problem would have been solved 60 years ago".
In 1985, Monsur Kenku gave a novel proof using the Klein quartic
Klein quartic
In hyperbolic geometry, the Klein quartic, named after Felix Klein, is a compact Riemann surface of genus 3 with the highest possible order automorphism group for this genus, namely order 168 orientation-preserving automorphisms, and 336 automorphisms if orientation may be reversed...
. Noam Elkies
Noam Elkies
Noam David Elkies is an American mathematician and chess master.At age 14, Elkies received a gold medal with a perfect score at the International Mathematical Olympiad, the youngest ever to do so...
gives an exposition of this result.