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
for some and x, y in not both zero, and in addition .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
Then we have , Q=bc-9ad and .Let be a cubic form and its Hessian.
We are now ready to give an algorithmic description of the set U. We need two propositions.
Let be a primitive form, and let its Hessian. Recall that we write if F has a triple root in . Then
Let F be a primitive form. We can write with a fundamental discriminant. Then if and only if either , or if
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 .