Supersingular elliptic curves

In the previous chapter, we showed that Brandt matrices for an order in a definite quaternion algebra B contain a wealth of arithmetic. In the special case where \({{\,\mathrm{disc}\,}}B=p\) is prime, there is a further beautiful connection between Brandt matrices and the theory of supersingular elliptic curves, arising from the following important result: there is an equivalence of categories between supersingular elliptic curves over \(\overline{\mathbb{F }}_p\) and right ideals in a (fixed) maximal order \(\mathcal {O}\subset B\). We pursue this important connection in this chapter for the reader who has a bit more background in algebraic curves.