Debry: Beyond two criteria for supersingularity: coefficients of division polynomials

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Christophe Debry
Beyond two criteria for supersingularity: coefficients of division polynomials
Journal de théorie des nombres de Bordeaux, 26 no. 3 (2014), p. 595-605, doi: 10.5802/jtnb.881
Article PDF | Reviews MR 3320494

Résumé - Abstract

Let $f(x)$ be a cubic, monic and separable polynomial over a field of characteristic $p\ge 3$ and let $E$ be the elliptic curve given by $y^2 = f(x)$. In this paper we prove that the coefficient at $x^{\frac{1}{2}p(p-1)}$ in the $p$–th division polynomial of $E$ equals the coefficient at $x^{p-1}$ in $f(x)^{\frac{1}{2}(p-1)}$. For elliptic curves over a finite field of characteristic $p$, the first coefficient is zero if and only if $E$ is supersingular, which by a classical criterion of Deuring (1941) is also equivalent to the vanishing of the second coefficient. So the zero loci of the coefficients are equal; the main result in this paper is clearly stronger than this last statement.

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