# Beyond two criteria for supersingularity: coefficients of division polynomials

Christophe Debry^{[1]}

- [1] KU Leuven and Universiteit van Amsterdam Departement Wiskunde, Celestijnenlaan 200B 3001 Leuven, Belgium

Journal de Théorie des Nombres de Bordeaux (2014)

- Volume: 26, Issue: 3, page 595-605
- ISSN: 1246-7405

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topDebry, 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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