### 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 .

### Proof

(1) Assume that . We know that any polynomial in one variable over can be written as where are pairwise coprime and squarefree polynomials, and essentially in a unique manner (up to multiplication of each by suitable constants). This result can be homogenized and transformed into an identical one for homogeneous polynomials in 2 variables.

Since , by definition of the discriminant this means that F has at least a double root in , in other words in the decomposition above there exists i>1 such that is not equal to a constant. Since F is of degree 3, this means that or else with , and of degree 1. It follows in particular that all the roots of F modulo p are in itself.

(2) We have just seen that if then all the roots of F are in and there is at least a double root. Hence write

Then one finds that

Now since F is primitive, and cannot both be zero modulo p, hence

On the other hand, if then we cannot have otherwise so , absurd.

(3) From now on we assume that . Replacing F by an equivalent form G, we may assume that the triple root of G modulo p is at , i.e that for some . This implies that . Since G is primitive, we have .

Assume first . We thus have

by definition of .

We could also have written for an integral form , from which we obtain , which immediately implies the result.

Now assume p=3 and that , or equivalently that . Then , hence , so , hence , so finally .

(4) Assume p=3 and that . Then since , we see that and . It follows that

and since F is primitive we cannot have and , so is a root of F modulo 3, hence the root of F modulo 3. Hence if and only if . Since b and c are divisible by 3, the value of modulo 9 depends only on x and y modulo 3, hence

and the result follows.