Algorithmic Characterization of the set U

We will now introduce a set U of cubic forms, and study some of its properties. In the next section, we will prove that U is the sought after image I of the Davenport-Heilbronn correspondence.

We first need some notation. For a prime p, we let be the set of such that if or or if p=2. In other words, if with a fundamental discriminant, if and only if .

In particular is the set of fundamental discriminants, i.e of discriminants of quadratic fields.

Furthermore, let be the set of such that either , or else

We will summarize the condition for some by saying that F has three identical roots in , and we will write .

Finally, we set . The Davenport-Heilbronn theorem states that U=I, the image of the map that we are looking for. Before proving this theorem, we must study in detail the set U.

Fot this, a very useful notion is that of the Hessian of a form F.

Definition 3.1 Let be a cubic form. We define the Hessian of F by the formula

Proposition 3.2

Let be a cubic form and its Hessian.

• For any we have .
• .
• .
• 3dP-cQ+bR=3aR-bQ+cP=0.

[Proof]

We are now ready to give an algorithmic description of the set U. We need two propositions.

Proposition 3.3

Let be a primitive form, and let its Hessian. Recall that we write if F has a triple root in . Then

• if and only if F has at least a double root in , and if this is the case all the roots of F are in fact in itself.
• if and only if .
• If and , then if and only if . If then .
• If then we have the following: If , then . If but , then . If and , then there exists such that , and then .

Proposition 3.4

Let F be a primitive form. We can write with a fundamental discriminant. Then if and only if either , or if

[Proof]

col3.5

Let F be a primitive cubic form and p be a prime. Then if and only if F has at least a double root modulo p, and .

[Proof]