Josep M. Miret, Daniel Sadornil, Juan Tena, Javier Valera
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A note on factorisation patterns of division polynomials of elliptic curves over finite fields
Let $E$ be an elliptic curve defined over a finite field $\mathbf{F}_{q}$, $q = p^{d}$, $p > 3$, and a prime number $\ell > 3$ such that $q \equiv 1 \pmod{\ell}$ and $\ell \mid \# E(\mathbf{F}_{q})$. In this paper we study the possible factorisation patterns over $\mathbf{F}_{q}[x]$ of the $\ell^{k}$-division polynomials associated to $E$ with $k \geq 2$, extending the work of Verdure [6] for $k=1$.
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The aim of the Proceedings of the Japan Academy, Series A, is the rapid publication of original papers in mathematical sciences. The paper should be written in English or French (preferably in English), and at most 6 pages long when published. A paper that is a résumé or an announcement (i.e. one whose details are to be published elsewhere) can also be submitted.
The paper is published promptly if once communicated by a Member of the Academy at its General Meeting, which is held monthly except in July and in August.
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