Cubic Rings, Forms : Delone-Faddeev Correspondence

Look at the beautiful book The Theory of Irrationalities of Third Degree by Delone and Faddeev.

The correspondence is between cubic rings and integral binary cubic forms up to equivalences. This correspondence is crucial to understand statistics of cubic rings (and fields). For instance, how do we count them by discriminants, how are the lattices distributed in the space of lattices, how do we count them by Galois groups etc etc.

Cubic Rings are commutative rings which are of rank 3 as free modules over $\displaystyle \mathbb Z .$ That is they are rings which look like $\displaystyle \mathbb Ze_1 + \mathbb Ze_2 +\mathbb Ze_3$ as modules. Some natural examples of such rings are $\displaystyle \mathbb Z[\alpha] ,$ where $\displaystyle \alpha$ satisfies a minimal polynomial of degree 3, and more generally rings of integers of cubic fields. These examples are integral domains: $\displaystyle ab=0 \implies a=0 \lor b=0.$ Delone-Fadeev deal with integral domains, but Gan-Gross-Savin generalized the correspondence to arbitrary cubic rings.

We consider cubic rings up to isomorphism (as rings). That is the presentation/generators of the rings might look different, but you can find a map that preserve the ring structures.

Binary cubic forms are expressions of the form

$\displaystyle f(x, y)=a x^{3}+b x^{2} y+c x y^{2}+d y^{3} , a,b,c,d \in \mathbb Z .$

There is a “twisted” action of $\displaystyle GL_2(\mathbb Z)$ given by

$\displaystyle \gamma(f)(x, y)=\frac{f((x, y) \cdot \gamma)}{\det(\gamma)} \quad \forall \gamma \in \mathrm{GL}_{2}(\mathbb{Z})$

We now state the correspondence.

Delone-Fadeev, Gan-Gross-Savin: $\displaystyle {\text { cubic rings } } / (\sim \text{isomorphism}) \quad \stackrel{1-1}{\longleftrightarrow} \quad{\text { integral binary cubic forms }} / \mathrm{GL}_{2}(\mathbb{Z}) \text { -equivalence }$

The correspondence preserves the discriminants.

Discriminant of the cubic ring: Discriminant of the cubic ring $R$ can defined as the determinant of the bilinear form given by trace form.

$\displaystyle R \times R \to \mathbb Z , (x,y) \to \text{tr}(xy)..$

That is if $\displaystyle \{\alpha_i\} .$ is basis for $\displaystyle R .$ , then $\displaystyle \text{disc}(R) = \det{(\alpha_i\alpha_j)} .$

Note that given the multiplication map $\displaystyle m_{\alpha}: R \to R, x \to \alpha x .$
Trace of an element $\displaystyle \alpha \in R$ is the trace of the map $\displaystyle m_{\alpha} .$ and the norm of an element is the determinant of $\displaystyle m_{\alpha} .$

Discriminant of the binary cubic form: For a cubic form $\displaystyle f(x,y) .$ the discriminant is the discriminant of the polynomial $\displaystyle f(x,1) .$ and can be explicitly given by

$\displaystyle \text{disc}(f)=b^{2} c^{2}-4 a c^{3}-4 d b^{3}-27 a^{2} d^{2}+18 a b c d .$

The correspondence can be stated in several equivalent forms. Thinking in these different ways helps us to understand various aspects. Some formulations are betters to understand the independence on the choice of basis, some formulations give explicit way to write down the cubic form given the cubic ring, some help us to see the action of the groups (under changes of basis) clearly etc.

Method 1 (Wedge product): The equivalence can be given by the following map

$\displaystyle R/\mathbb Z \to \Lambda^3 R, \quad \alpha \to 1 \land \alpha \land \alpha^2 .$

This is a cubic map and if we write $\alpha = Xe_1 + Ye_2 +Z.1 .$ , the map can seen as taking $(X,Y) \to f(X,Y) .$ for some binary cubic form $f .$ because $\Lambda^3 R .$ is one dimensional.

Method 2 (quadratic resolvent): There is a unique quadratic ring $\displaystyle S .$ whose discriminant matches with $\displaystyle R .$

Quadratic rings up to isomorphism are determined by their discriminants!

$\displaystyle \begin{aligned} {\text { Quadratic Rings }} / \text { isom } & \stackrel{1: 1}{\longleftrightarrow} {\text{Discriminants }}{D \in \mathbb{Z}: D \equiv 0,1(\bmod 4)} \\ R & \mapsto \text{Disc}(R) \\ \mathbb{Z}[\tau] /\left(\tau^{2}-D \tau+\frac{D^{2}-D}{4}\right) & \leftarrow D \end{aligned} .$

Now consider the map $\displaystyle \phi_{3,2}: R\to S .$ given by

$\displaystyle r \mapsto \frac{\left(r-r^{\prime}\right)^{2}\left(r^{\prime}-r^{\prime \prime}\right)^{2}\left(r^{\prime \prime}-r\right)^{2}+\left(r-r^{\prime}\right)\left(r^{\prime}-r^{\prime \prime}\right)\left(r^{\prime \prime}-r\right)}{2} .$

This map reduces to a map on $\displaystyle R/\mathbb Z .$ and $\displaystyle \phi_{3,2}(\alpha) .$ for $\displaystyle \alpha =Xe_1+Ye_2+Z.1 .$ turns out to be $\displaystyle A(X,Y)+ B(X,Y)\tau .$ . $\displaystyle B(X,Y) .$ is the required cubic form.

Method 3 (Index of subring): For an element $\displaystyle \alpha \in R .$ consider the index of the subring $\mathbb Z[\alpha] .$ in $R .$ . That is we are comparing the covolumes of $R .$ and $\mathbb Z[\alpha] .$ , and the ratio $\displaystyle [R:\mathbb Z[\alpha]] .$ is a cubic form on the $R/\mathbb Z .$ . That if $\displaystyle \alpha = Xe_1 + Ye_2 +Z.1 .$ , the index $\displaystyle [R:\mathbb Z[\alpha] = f(X,Y)$ for a binary cubic form $\displaystyle f. .$

Method 4 (Discriminants): Using the fact that the discriminant is the square of covolume, we can see that

$\displaystyle f(X,Y) =[R:\mathbb Z[\alpha] =\sqrt\frac{\text{disc}(\alpha)}{\text{disc}(R)} .$

Method 5 (Geometry): A cubic form $\displaystyle f(x,y) = a x^{3}+b x^{2} y+c x y^{2}+d y^{3} .$ defines a projective variety $\displaystyle V_f$ over $\displaystyle \mathbb Z .$ given by $\displaystyle f(x,y) =0 \in \mathbb P^{1}(\mathbb Z) .$ . The ring of functions on this variety is the cubic ring we want!

If we have the cubic ring, we can determine the cubic forms also by geometry (we need to find an equation defining the variety inside $\displaystyle P^{1} .$ , so intrinsically in terms of the ring/scheme, we need to map to $P^{1} .$ which can be produced using trace zero elements of an ideal class.

Method 6 (Explicit in terms of a basis): Assume that the cubic ring is generated by $\displaystyle 1, e_1, e_2 .$ . Then $e_1e_2 .$ should be of the form $\displaystyle k+le_1+me_2 .$ . Shifting the basis to $\displaystyle e_1-m, e_2-l .$ one can assume that we have a new basis $\displaystyle 1, f_1, f_2 .$ such that $\displaystyle f_1f_2 =n \in \mathbb Z .$

So we have

$\displaystyle \begin{aligned} f_1^{2} &=p+b e_1-a e_2 \\ f_2^{2} &=q+de_1 -c e_2 \\ f_1 f_2 &=n \end{aligned} .$

Computing $\displaystyle e_1^2e_2, e_1e_2^2$ in two different ways (associativity) we get more relations

$\displaystyle p=-ac, q=-bd, n =-ad .$

Thus we have $\displaystyle (a,b,c,d) .$ which determines the multiplication table for the basis $1, f_1, f_2 .$ with the above equations.

$\displaystyle f(x,y) = a x^{3}+b x^{2} y+c x y^{2}+d y^{3} ~~$ is the corresponding cubic form.

Example: If we take $\displaystyle R=\mathbb Z[\sqrt[3]{2}].$ with basis $1, \sqrt[3]{2}, \sqrt[3]{4} .$ , we get $\displaystyle(a,b,c,d) =(1,0,0,-2) .$ and hence the cubic form is $X^3-2Y^3 .$ .

On the otherhand the zero form $(a,b,c,d) =(0,0,0,0)$ corresponds to the cubic ring $\displaystyle \mathbb Z[X,Y]/ (XY, X^2, Y^2) .$

Remarks: If $R .$ is a order in a number field, then the discriminant is an irreducible polynomial and $\mathbb Q [X]/(f(X,1))$ is the number field. $R .$ is a maximal order if and only if $f(X,1) .$ satisfies some conditions $\mod p^2.$ In that case, a prime $p .$ splits in the field iff $f(X,1) \mod p .$ splits.

This concludes the description of the correspondence– now how do we use the correspondence to count things? The corresponding helped us to see cubic rings in terms of orbits of $GL_2(\mathbb Z)$ on a vector space of the cubic forms. Studying the fundamental domains and orbit counting methods, we can hope to get a handle on the quantities. For instance, in the quadratic case we get correspondences between ideal classes and Heegner points (geodesics in indefinite case) on the modular surface which helps us to study statistics/distribution of the ideal classes.

Bhargava’s work: If we want to generalize to quartic, quintic one can carryout the resolvent method as above. That is, we need to find a map to cubic, sextic ring respectively that preserves the discriminants. But there could be multiple such maps to different cubic (sextic) rings! Look at Bhargava’s series of papers on Higher Composition Laws to see see how to find different resolvent maps, and parameterize the pairs $(R,S)$ with orbits in different vector spaces of forms.

$\displaystyle \phi_{4,3} R\to S$ and $\phi_{5,4} R \to S.$ .

Reducing $\mod \mathbb Z$ , we get maps $\phi_{4,3} R/\mathbb Z \to S/\mathbb Z$ and $\phi_{5,4} R/\mathbb Z \to S/\mathbb Z$ .

Quartic case: The fundamental resolvent map is

$\displaystyle \phi_{4,3}(\alpha)=\alpha^{(1)} \alpha^{(2)}+\alpha^{(3)} \alpha^{(4)}$

$\phi_{4,3} (X,Y,Z) = A(X,Y,Z) \tau_1+B(X,Y,Z)\tau_2$ , thus we have correspondence

$\displaystyle {\text { (quartic ring R, cubic resolvent S) } } / \sim \quad {\longleftrightarrow} \quad{\text { pairs of integral ternary quadratic forms (A,B) }} / \mathrm{GL}_{2}(\mathbb{Z}) \times \mathrm{GL}_{3}(\mathbb{Z})\text { -equivalence }$

The quintic is more complicated. But the geometric picture is the same, we are trying to define the $n$ point schemes using hypersurfaces, the functions defining the hypersurfaces seem to be related to the resolvent maps.

Quintic case: The fundamental resolvent map is

$\displaystyle \phi_{5,4} (\alpha)=\Big(\alpha^{(1)} \alpha^{(2)}+\alpha^{(2)} \alpha^{(3)}+\alpha^{(3)} \alpha^{(4)}+\alpha^{(4)} \alpha^{(5)}+\alpha^{(5)} \alpha^{(1)}$

$\displaystyle -\alpha^{(1)} \alpha^{(3)}-\alpha^{(3)} \alpha^{(5)}-\alpha^{(5)} \alpha^{(2)}-\alpha^{(2)} \alpha^{(4)}-\alpha^{(4)} \alpha^{(1)}\Big)^{2}$

and we have the correspondence

$\displaystyle {\text { (quintic ring R, sextic resolvent S) } } / \sim \quad {\longleftrightarrow} \quad{\text { quadruples of alternating quintic forms}} / \text { -equivalence }$

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