When the trace of the Frobenius homomorphism is zero?

Question

Let's consider an elliptic curve over a finite field $\mathbb F_p$.

The trace of the Frobenius homomorphism is defined as:

$$a_p=p+1-\#E(\mathbb F_p)$$

See for example here.

I read that this value is zero if and only if the elliptic curve is supersingular. Unfortunately there is no proof give. So I am looking for a proof of this. (For example a good reference).

Answer 1

This is part (a) Exercise 5.10 in Chapter V of Silverman's "The Arithmetic of Elliptic Curves". The proof of part (a), however, is implicit in the proof of Theorem 4.1 of the same chapter.