Beyond two criteria for supersingularity: coefficients of division polynomials

Let E: y^2 = x^3 + Ax + B be an elliptic curve defined over a finite field of characteristic p\geq 3. In this paper we prove that the coefficient at x^{p(p-1)/2} in the p-th division polynomial \psi_p(x) of E equals the coefficient at x^{p-1} in (x^3 + Ax + B)^{(p-1)/2}. The first coefficient is zero if and only if the division polynomial has no roots, which is equivalent to E being supersingular. Deuring (1941) proved that this supersingularity is also equivalent to the vanishing of the second coefficient. So the zero loci of the coefficients (as functions of A and B) are equal; the main result in this paper is clearly stronger than this last statement.