Beyond two criteria for supersingularity: coefficients of division polynomials
Journal de Théorie des Nombres de Bordeaux, Volume 26 (2014) no. 3, pp. 595-605.

Let f(x) be a cubic, monic and separable polynomial over a field of characteristic p3 and let E be the elliptic curve given by y 2 =f(x). In this paper we prove that the coefficient at x 1 2p(p-1) in the p–th division polynomial of E equals the coefficient at x p-1 in f(x) 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.

Soit f(x) un polynôme cubique, unitaire et séparable avec coefficients dans un corps de caractéristique p3, et soit E la courbe elliptique donnée par l’équation y 2 =f(x). Dans cet article on démontre que le coefficient du monôme x 1 2p(p-1) dans le p–ième polynôme de division de E est égal au coefficient du monôme x p-1 dans f(x) 1 2(p-1) . Lorsque le corps de base est fini, le premier coefficient est nul si et seulement si E est supersingulière, ce qui, par un critère classique de Deuring (1941), est équivalent à la nullité du deuxième coefficient. Donc les zéros des coefficients sont les mêmes. L’égalité des coefficients qu’on démontre dans cet article entraîne clairement cette égalité de zéros.

Received:
Revised:
Accepted:
Published online:
DOI: 10.5802/jtnb.881
Christophe Debry 1

1 KU Leuven and Universiteit van Amsterdam Departement Wiskunde, Celestijnenlaan 200B 3001 Leuven, Belgium
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Christophe Debry. Beyond two criteria for supersingularity:  coefficients of division polynomials. Journal de Théorie des Nombres de Bordeaux, Volume 26 (2014) no. 3, pp. 595-605. doi : 10.5802/jtnb.881. https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.881/

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