# Beyond two criteria for supersingularity: coefficients of division polynomials

• [1] KU Leuven and Universiteit van Amsterdam Departement Wiskunde, Celestijnenlaan 200B 3001 Leuven, Belgium
• Volume: 26, Issue: 3, page 595-605
• ISSN: 1246-7405

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## Abstract

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Let $f\left(x\right)$ be a cubic, monic and separable polynomial over a field of characteristic $p\ge 3$ and let $E$ be the elliptic curve given by ${y}^{2}=f\left(x\right)$. In this paper we prove that the coefficient at ${x}^{\frac{1}{2}p\left(p-1\right)}$ in the $p$–th division polynomial of $E$ equals the coefficient at ${x}^{p-1}$ in $f{\left(x\right)}^{\frac{1}{2}\left(p-1\right)}$. 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.

## How to cite

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Debry, Christophe. "Beyond two criteria for supersingularity: coefficients of division polynomials." Journal de Théorie des Nombres de Bordeaux 26.3 (2014): 595-605. <http://eudml.org/doc/275765>.

@article{Debry2014,
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.},
affiliation = {KU Leuven and Universiteit van Amsterdam Departement Wiskunde, Celestijnenlaan 200B 3001 Leuven, Belgium},
author = {Debry, Christophe},
journal = {Journal de Théorie des Nombres de Bordeaux},
language = {eng},
month = {12},
number = {3},
pages = {595-605},
publisher = {Société Arithmétique de Bordeaux},
title = {Beyond two criteria for supersingularity: coefficients of division polynomials},
url = {http://eudml.org/doc/275765},
volume = {26},
year = {2014},
}

TY - JOUR
AU - Debry, Christophe
TI - Beyond two criteria for supersingularity: coefficients of division polynomials
JO - Journal de Théorie des Nombres de Bordeaux
DA - 2014/12//
PB - Société Arithmétique de Bordeaux
VL - 26
IS - 3
SP - 595
EP - 605
AB - 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.
LA - eng
UR - http://eudml.org/doc/275765
ER -

## References

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1. J. W. S. Cassels, A note on the division values of $\wp \left(u\right)$, Mathematical Proceedings of the Cambridge Philosophical Society 45, (1949), 167–172. Zbl0032.26103MR28358
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3. J. Cheon, S. Hahn, Division polynomials of elliptic curves over finite fields, Proc. Japan Acad. Ser. A Math. Sci. 72, 10, (1996), 226–227. Zbl0957.11031MR1435722
4. M. Deuring, Die Typen der Multiplikatorringe Elliptischer Funktionenkörper, Abh. Math., Sem. Univ. Hamburg 14, (1941), 197–272. Zbl0025.02003MR3069722
5. A. Enge, Elliptic curves and their applications to cryptography: An introduction, Kluwer Academic Publishers, (1999). Zbl1335.11002
6. H. Gunji, The Hasse invariant and $p$–division points of an elliptic curve, Arch. Math. 27, (1976), 148–158. Zbl0342.14008MR412198
7. J. McKee, Computing division polynomials, J. Math. Comp. 63, (1994), 767–771. Zbl0864.12007MR1248973
8. J. H. Silverman, The Arithmetic of Elliptic Curves, Graduate Texts in Mathematics 106, Springer–Verlag, New York, (2009). Zbl1194.11005MR2514094

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