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

(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

Fhas at least a double root in , in other words in the decomposition above there existsi>1such that is not equal to a constant. SinceFis of degree 3, this means that or else with , and of degree 1. It follows in particular that all the roots ofFmodulopare in itself.(2) We have just seen that if then all the roots of

Then one finds that Now sinceFare in and there is at least a double root. Hence writeFis primitive, and cannot both be zero modulop, henceOn the other hand, if then we cannot have otherwise so , absurd.

(3) From now on we assume that . Replacing

by definition of .Fby an equivalent formG, we may assume that the triple root ofGmodulopis at ,i.ethat for some . This implies that . SinceGis primitive, we have .We could also have written for an integral form , from which we obtain , which immediately implies the result.

Now assume

p=3and that , or equivalently that . Then , hence , so , hence , so finally .(4) Assume

and sincep=3and that . Then since , we see that and . It follows thatFis primitive we cannot have and , so is a root ofFmodulo 3, hencetheroot ofFmodulo 3. Hence if and only if . Sincebandcare divisible by 3, the value of modulo 9 depends only onxandymodulo 3, hence and the result follows.