Class numbers

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Gauss's class number problem

For a given , determine a complete list of fundamental binary quadratic form discriminants such that the class number is given by . Heegner (1952) gave a solution for , but it was not completely accepted due to a number of apparent gaps. However, subsequent examination of Heegner's proof showed it to be "essentially" correct (Conway and Guy 1996). Conway and Guy (1996) therefore call the nine values of having where is the binary quadratic form discriminant corresponding to an quadratic field (, , , , , , , , and ; OEIS A003173) the Heegner numbers. The Heegner numbers have a number of fascinating properties.Stark (1967) and Baker (1966) gave independent proofs of the fact that only nine such numbers exist; both proofs were accepted. Baker (1971) and Stark (1975) subsequently and independently solved the generalized class number problem completely for . Oesterlé (1985) solved the case , and Arno (1992) solved the case . Wagner (1996)..

Gauss's class number conjecture

In his monumental treatise Disquisitiones Arithmeticae, Gauss conjectured that the class number of an imaginary quadratic field with binary quadratic form discriminant tends to infinity with . A proof was finally given by Heilbronn (1934), and Siegel (1936) showed that for any , there exists a constant such thatas . However, these results were not effective in actually determining the values for a given of a complete list of fundamental discriminants such that , a problem known as Gauss's class number problem.Goldfeld (1976) showed that if there exists a "Weil curve" whose associated Dirichlet L-series has a zero of at least third order at , then for any , there exists an effectively computable constant such thatGross and Zaiger (1983) showed that certain curves must satisfy the condition of Goldfeld, and Goldfeld's proof was simplified by Oesterlé (1985)...

Fundamental theorem of genera

Consider proper equivalence classes of forms with discriminant equal to the field discriminant, then they can be subdivided equally into genera of forms which form a subgroup of the proper equivalence class group under composition (Cohn 1980, p. 224), where is the number of distinct prime divisors of . This theorem was proved by Gauss in 1801.

Dirichlet structure constant

(1)where is the fundamental unit and is the number of substitutions which leave the binary quadratic form unchanged, given by(2)

Class group

Let be a number field, then each fractional ideal of belongs to an equivalence class consisting of all fractional ideals satisfying for some nonzero element of . The number of equivalence classes of fractional ideals of is a finite number, known as the class number of . Multiplication of equivalence classes of fractional ideals is defined in the obvious way, i.e., by letting . It is easy to show that with this definition, the set of equivalence classes of fractional ideals form an Abelian multiplicative group, known as the class group of .

Class field theory

Take a number field and a divisor of . A congruence subgroup is defined as a subgroup of the group of all fractional ideals relative prime to () that contains all principal ideals that are generated by elements of that are equal to 1 (mod ). These principal ideals split completely in all Abelian extensions and are consequently part of the kernel of the Artin map for each Abelian extension .When there exists an Abelian extension such that contains all the primes that ramify in and such that equals the kernel of the Artin map, then is called the class field of .To formulate the main theorems, the equivalence relation on congruence subgroups is needed, namely that and are called equivalent if there exists a divisor such that .Class field theory consists of two basic theorems. The existence theorem states that to every equivalence class of congruence subgroups, there belongs a class field . The classification theorem states that for each number field , there..

Fundamental discriminant

An integer is a fundamental discriminant if it is not equal to 1, not divisible by any square of any odd prime, and satisfies or . The function FundamentalDiscriminantQ[d] in the Wolfram Language version 5.2 add-on package NumberTheory`NumberTheoryFunctions` tests if an integer is a fundamental discriminant.It can be implemented as: FundamentalDiscriminantQ[n_Integer] := n != 1&& (Mod[n, 4] == 1 \[Or] ! Unequal[Mod[n, 16], 8, 12])&& SquareFreeQ[n/2^IntegerExponent[n, 2]]The first few positive fundamental discriminants are 5, 8, 12, 13, 17, 21, 24, 28, 29, 33, ... (OEIS A003658). Similarly, the first few negative fundamental discriminants are , , , , , , , , , , , ... (OEIS A003657).

Class number

For any ideal in a Dedekind ring, there is an ideal such that(1)where is a principal ideal, (i.e., an ideal of rank 1). Moreover, for a Dedekind ring with a finite ideal class group, there is a finite list of ideals such that this equation may be satisfied for some . The size of this list is known as the class number.Class numbers are usually studied in the context of the orders of number fields. If this order is maximal, then it is the ring of integers of the number field, in which case the class number is equal to the order of the class group of the number field; otherwise it is equal to the order of the Picard group of the nonmaximal order in question.When the class number of a ring of integers in a number field is 1, the ring corresponding to a given ideal has unique factorization and, in a sense, the class number is a measure of the failure of unique factorization in that ring.A finite series giving exactly the class number of a ring is known as a class number formula...

Heegner number

The values of for which imaginary quadratic fields are uniquely factorable into factors of the form . Here, and are half-integers, except for and 2, in which case they are integers. The Heegner numbers therefore correspond to binary quadratic form discriminants which have class number equal to 1, except for Heegner numbers and , which correspond to and , respectively.The determination of these numbers is called Gauss's class number problem, and it is now known that there are only nine Heegner numbers: , , , , , , , , and (OEIS A003173), corresponding to discriminants , , , , , , , , and , respectively. This was proved by Heegner (1952)--although his proof was not accepted as complete at the time (Meyer 1970)--and subsequently established by Stark (1967).Heilbronn and Linfoot (1934) showed that if a larger existed, it must be . Heegner (1952) published a proof that only nine such numbers exist, but his proof was not accepted as complete at the time. Subsequent..

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