Class number problem for imaginary quadratic fields
Encyclopedia
In mathematics
, the Gauss class number problem (for imaginary quadratic fields), 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 discriminant
s. There are related questions for real quadratic fields and the behavior as
.
The difficulty is in effective computation of bounds: for a given discriminant, it is easy to compute the class number, and there are several ineffective lower bounds on class number (meaning that they involve a constant that is not computed), but effective bounds (and explicit proofs of completeness of lists) are harder.
of 1801 (Section V, Articles 303 and 304).
Gauss discusses imaginary quadratic fields in Article 303, stating the first two conjectures, and discusses real quadratic fields in Article 304, stating the third conjecture.
Gauss Conjecture (Class number tends to infinity):
Gauss Class Number Problem (Low class number lists): For given low class number (such as 1, 2, and 3), Gauss gives lists of imaginary quadratic fields with the given class number and believes them to be complete.
Infinitely many real quadratic fields with class number one: Gauss conjectures that there are infinitely many real quadratic fields with class number one.
The original Gauss class number problem for imaginary quadratic fields is significantly different and easier than the modern statement: he restricted to even discriminants, and allowed non-fundamental discriminants.
Low class number lists: Class number 1: solved, Baker (1966), Stark (1967), Heegner (1952).
Infinitely many real quadratic fields with class number one: Open.
The non-fundamental discriminants of class number 1 are:
Thus, the even discriminants of class number 1, fundamental and non-fundamental (Gauss's original question) are:
proved the Gauss Conjecture. Equivalently, for any given class number, there are only finitely many imaginary quadratic number fields with that class number.
Also in 1934, Heilbronn and Edward Linfoot
showed that there were at most 10 imaginary quadratic number fields with class number 1 (the 9 known ones, and at most one further).
The result was ineffective (see effective results in number theory
): it did not allow bounds on the size of the remaining field.
In later developments, the case n = 1 was first discussed by Kurt Heegner
, using modular form
s and modular equation
s to show that no further such field could exist. This work was not initially accepted; only with later work of Harold Stark
and Bryan Birch was the position clarified, and Heegner's work understood. See Stark–Heegner theorem
, Heegner number
. Practically simultaneously, Alan Baker proved an Baker's theorem
on linear forms in logarithms
of algebraic number
s which resolved the problem by a completely different method. The case n = 2 was tackled shortly afterwards, at least in principle, as an application of Baker's work.
The complete list of imaginary quadratic fields with class number one is
with k one of
The general case awaited the discovery of Dorian Goldfeld that the class number problem could be connected to the L-function
s of elliptic curve
s. This reduced the question, in principle, of effective determination, to one about establishing the existence of a multiple zero of such an L-function. This could be done on the basis of the later Gross-Zagier theorem. So at that point one could specify a finite calculation, the result of which would be a complete list for a given class number. In fact in practice such lists that are probably complete can be made by relatively simple methods; what is at issue is certainty. The cases up to n = 100 have now (2004) been done: see Watkins (2004).
. This extra factor is hard to control. It may well be the case that class number 1 for real quadratic fields occurs infinitely often.
The Cohen-Lenstra heuristics are a set of more precise conjectures about the structure of class groups of quadratic fields. For real fields they predict that about 75.446% will have class number 1, a result that agrees with computations.
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 Gauss class number problem (for imaginary quadratic fields), 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
Carl 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 can also be stated in terms of 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...
s. There are related questions for real quadratic fields and the behavior as
.The difficulty is in effective computation of bounds: for a given discriminant, it is easy to compute the class number, and there are several ineffective lower bounds on class number (meaning that they involve a constant that is not computed), but effective bounds (and explicit proofs of completeness of lists) are harder.
Gauss's original conjectures
The problems are posed in Gauss's Disquisitiones ArithmeticaeDisquisitiones Arithmeticae
The Disquisitiones Arithmeticae is a textbook of number theory written in Latin by Carl Friedrich Gauss in 1798 when Gauss was 21 and first published in 1801 when he was 24...
of 1801 (Section V, Articles 303 and 304).
Gauss discusses imaginary quadratic fields in Article 303, stating the first two conjectures, and discusses real quadratic fields in Article 304, stating the third conjecture.
Gauss Conjecture (Class number tends to infinity):
Gauss Class Number Problem (Low class number lists): For given low class number (such as 1, 2, and 3), Gauss gives lists of imaginary quadratic fields with the given class number and believes them to be complete.
Infinitely many real quadratic fields with class number one: Gauss conjectures that there are infinitely many real quadratic fields with class number one.
The original Gauss class number problem for imaginary quadratic fields is significantly different and easier than the modern statement: he restricted to even discriminants, and allowed non-fundamental discriminants.
Status
Gauss Conjecture: Solved, Heilbronn, 1934.Low class number lists: Class number 1: solved, Baker (1966), Stark (1967), Heegner (1952).
- Class number 2: solved, Baker (1971), Stark (1971)
- Class number 3: solved, 1985
- Class numbers h up to 100: solved, Watkins 2004
Infinitely many real quadratic fields with class number one: Open.
Lists of discriminants of class number 1
For imaginary quadratic number fields, the (fundamental) discriminants of class number 1 are:The non-fundamental discriminants of class number 1 are:
Thus, the even discriminants of class number 1, fundamental and non-fundamental (Gauss's original question) are:
Modern developments
In 1934, Hans HeilbronnHans Heilbronn
Hans Arnold Heilbronn was a mathematician.He was born into a German-Jewish family. He was a student at the universities of Berlin, Freiburg and Göttingen, where he met Edmund Landau, who supervised his doctorate...
proved the Gauss Conjecture. Equivalently, for any given class number, there are only finitely many imaginary quadratic number fields with that class number.
Also in 1934, Heilbronn and Edward Linfoot
Edward Linfoot
Edward Hubert Linfoot was a British mathematician, primarily known for his work on optics, but also noted for his work in pure mathematics.- Early life and career :...
showed that there were at most 10 imaginary quadratic number fields with class number 1 (the 9 known ones, and at most one further).
The result was ineffective (see effective results in number theory
Effective results in number theory
For historical reasons and in order to have application to the solution of Diophantine equations, results in number theory have been scrutinised more than in other branches of mathematics to see if their content is effectively computable...
): it did not allow bounds on the size of the remaining field.
In later developments, the case n = 1 was first discussed by Kurt Heegner
Kurt Heegner
Kurt Heegner was a German private scholar from Berlin, who specialized inradio engineering and mathematics...
, using 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 and modular equation
Modular equation
In mathematics, a modular equation is an algebraic equation satisfied by moduli, in the sense of moduli problem. That is, given a number of functions on a moduli space, a modular equation is an equation holding between them, or in other words an identity for moduli.The most frequent use of the term...
s to show that no further such field could exist. This work was not initially accepted; only with later work of 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...
and Bryan Birch was the position clarified, and Heegner's work understood. See Stark–Heegner theorem
Stark–Heegner theorem
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...
, 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...
. Practically simultaneously, Alan Baker proved an Baker's theorem
Baker's theorem
In transcendence theory, a mathematical discipline, Baker's theorem gives a lower bound for linear combinations of logarithms of algebraic numbers...
on linear forms in logarithms
Linear forms in logarithms
In number theory the method of linear forms in logarithms is the application of estimatesfor the magnitude of a finite sum\sum \beta_i\log\alpha_i=\Lambda,...
of algebraic number
Algebraic number
In mathematics, an algebraic number is a number that is a root of a non-zero polynomial in one variable with rational coefficients. Numbers such as π that are not algebraic are said to be transcendental; almost all real numbers are transcendental...
s which resolved the problem by a completely different method. The case n = 2 was tackled shortly afterwards, at least in principle, as an application of Baker's work.
The complete list of imaginary quadratic fields with class number one is
with k one ofThe general case awaited the discovery of Dorian Goldfeld that the class number problem could be connected to the 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 of 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...
s. This reduced the question, in principle, of effective determination, to one about establishing the existence of a multiple zero of such an L-function. This could be done on the basis of the later Gross-Zagier theorem. So at that point one could specify a finite calculation, the result of which would be a complete list for a given class number. In fact in practice such lists that are probably complete can be made by relatively simple methods; what is at issue is certainty. The cases up to n = 100 have now (2004) been done: see Watkins (2004).
Real quadratic fields
The contrasting case of real quadratic fields is very different, and much less is known. That is because what enters the analytic formula for the class number is not h, the class number, on its own — but h log ε, where ε is a fundamental unitFundamental unit (number theory)
In algebraic number theory, a fundamental unit is a generator for the torsion-free unit group of the ring of integers of a number field, when that group is infinite cyclic...
. This extra factor is hard to control. It may well be the case that class number 1 for real quadratic fields occurs infinitely often.
The Cohen-Lenstra heuristics are a set of more precise conjectures about the structure of class groups of quadratic fields. For real fields they predict that about 75.446% will have class number 1, a result that agrees with computations.