There are two definitions of the supersingular primes: one group-theoretic, and the other number-theoretic. Group-theoretically, let Γ_0(N) be the modular group Γ_0, and let X_0(N) be the compactification (by adding cusps) of Y_0(N) = Γ_0(N) H, where H is the upper half-plane. Also define w_N to be the Fricke involution defined by the block matrix [[0, -1], [N, 0]]. For p a prime, define X_0^+(p) = X_0(p)/(w_p). Then p is a supersingular prime if the genus of X_0^+(p) = 0. The number-theoretic definition involves supersingular elliptic curves defined over the algebraic closure of the finite field F_p. They have their j-invariant in F_p.
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