equivalence of form class group and class group


There are only a finite number of reduced primitive positive integral binary quadratic forms of a given negative http://planetmath.org/node/IntegralBinaryQuadraticFormsdiscriminantPlanetmathPlanetmathPlanetmath Δ. Given Δ, call this number hΔ, the form of Δ.

Thus, for example, since there is only one reduced form of discriminant -163, we have that h-163=1.

It turns out that the set of reduced forms of a given negative discriminant can be turned into an abelian groupMathworldPlanetmath, called the , 𝒞Δ, by defining a “multiplicationPlanetmathPlanetmath” on forms that is based on generalizationsPlanetmathPlanetmath of identitiesPlanetmathPlanetmathPlanetmath such as

(2⁢x2+2⁢x⁢y+3⁢y2)⁢(2⁢z2+2⁢z⁢w+3⁢w2)=(2⁢x⁢z+x⁢w+y⁢z+3⁢y⁢w)2+5⁢(x⁢w+y⁢z)2

where all of these forms have discriminant -20.

Now, given an algebraic extensionMathworldPlanetmath K of ℚ, ideal classes of 𝒪K also form an abelian group, called the http://planetmath.org/node/IdealClassideal class group of K, 𝒞K. The order of 𝒞K is called the class number of K and is denoted hK. See the ideal class entry for more detail.

For an algebraic extension K/ℚ, one also defines the http://planetmath.org/node/DiscriminantOfANumberFielddiscriminant of the extensionPlanetmathPlanetmathPlanetmathPlanetmath, dK. For quadratic extensions K=ℚ⁢[n], where n is assumed squarefreeMathworldPlanetmath, the discriminant can be explicitly computed to be

dK={4⁢nif ⁢n≡2,3(mod4)nif ⁢n≡1(mod4)

For imaginary quadratic extensions, the form class group and the class group turn out to be the same!

Theorem 1.

Let K=Q⁢(n),n<0 squarefree, be a quadratic extension. Then CK, the class group of K, is isomorphicPlanetmathPlanetmathPlanetmath to the group of reduced forms of discriminant dK, CdK.

One can in fact exhibit an explicit correspondence 𝒞dK→𝒞K:

a⁢x2+b⁢x⁢y+c⁢y2↦(a,b+dK2)

Note in particular that the simplest, or principal, form of discriminant dK (x2-dK⁢y2 or x2+x⁢y+1-dK4⁢y2) maps to the ideal (1)=𝒪K; these forms are the identities in 𝒞dK. Showing that the map is 1-1 and onto is not difficult; showing that it is a group isomorphism is more difficult but nevertheless essentially amounts to a computation.

This theorem allows us to simply compute at least the size of the class group for quadratic extensions K by computing the number of reduced forms of discriminant dK. For example, suppose K=ℚ⁢(-23). Since -23≡1(4), 𝒪K=ℤ⁢[1+-232] and dK=-23.

What are the forms of discriminant -23? |b|≤a≤233<8<3, and b is odd, so b=±1. 4⁢a⁢c-b2=23, so a⁢c=6. We thus get three reduced forms:

(1,1,6)
(2,1,3)
(2,-1,3) reduced since |b|≠a,a≠c

Note that (1,-1,6) is not reduced, since b<0 but |b|=a.

So we know that the order of the class group 𝒞K is 3, so 𝒞K≅ℤ/3⁢ℤ.

We can use the explicit correspondence above to find representatives of the three elements of the class group using the map from forms to ideals.

(1,1,6) →(1,1+-232)=(1)
(2,1,3) →(2,1+-232)
(2,-1,3) →(2,-1+-232)

In fact, a more general form of Theorem 1 is true. If K is an algebraic number fieldMathworldPlanetmath, A⊂𝒪K, then A is not a Dedekind domainMathworldPlanetmath unless A=𝒪K. But even in this case, if one considers only those ideals that are invertible in A, one can define a group structureMathworldPlanetmath in a similar way; this is once again called the class group of A. In the case that K is a quadratic extension, these subrings of 𝒪K are called orders of K.

It is the case that each discriminant Δ<0,Δ≡0,1(mod4) corresponds to a unique order in a quadratic extension of ℚ. Specifically,

Theorem 2.

Let Δ<0,Δ≡0,1(mod4). Write Δ=m2⁢Δ′ where Δ′ is squarefree. Let K=Q⁢(Δ′). Then

𝒪Δ={ℤ⁢[m2⁢Δ′],Δ′≡2,3(mod4)ℤ⁢[m⁢1+Δ′2],Δ′≡1(mod4)

is a subring of OK, and CΔ≅COΔ. (Note that if Δ′≡2,3(mod4), then m must be even. For otherwise, m2≡1(mod4) and thus Δ≡2,3(mod4), which is impossible. Thus m/2 is an integer in this case)

This reduces to the first theorem in the event that Δ=dK.

Thus there is a 1-1 correspondence between discriminants Δ<0 and orders of quadratic fields; in particular, the ring of algebraic integers of any quadratic field corresponds to the forms of discriminant equal to the discriminant of the field.

Title equivalence of form class group and class group
Canonical name EquivalenceOfFormClassGroupAndClassGroup
Date of creation 2013-03-22 16:56:27
Last modified on 2013-03-22 16:56:27
Owner rm50 (10146)
Last modified by rm50 (10146)
Numerical id 5
Author rm50 (10146)
Entry type Theorem
Classification msc 11E12
Classification msc 11E16
Classification msc 11R29