Beyond two criteria for supersingularity: coefficients of division polynomials

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Let E: y² = x³ + Ax + B be an elliptic curve defined over a finite field of characteristic p≥ 3. In this paper we prove that the coefficient at xᵖ⁽ᵖ⁻¹⁾/² in the p-th division polynomial ψₚ(x) of E equals the coefficient at xᵖ⁻¹ in (x³ + 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.