Beyond two criteria for supersingularity: coefficients of division polynomials

2013/03/20 by Christophe Debry, Debry, Christophe
Computer Science · Mathematics · #Algebraic Geometry and Number Theory #Coding theory and cryptography #Cryptography and Residue Arithmetic #FOS: Mathematics #Number Theory (math.NT) #math.NT

paper · pdf · doi:10.48550/arxiv.1303.5002

8 pages

arxiv created 2013/03/20 · openalex publication_date 2013/03/20 · arxiv updated 2013/03/21 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

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 x^p(p-1)/2 in the p-th division polynomial ψ_p(x) of E equals the coefficient at x^p-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.

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