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Zbl 0688.14015
Voloch, J.F.
A note on elliptic curves over finite fields. (English)
[J] Bull. Soc. Math. Fr. 116, No.4, 455-458 (1988). ISSN 0037-9484

Let ${\bbfF}\sb q$ denote the finite field of $q\quad elements.$ {\it W. C. Waterhouse} [Ann. Sci. Éc. Norm. Supér. 521-560 (1969; Zbl 0188.530)] showed that for given $t\in {\bbfZ}$, $\vert t\vert \le 2\sqrt{q}$, there corresponds an elliptic curve over ${\bbfF}\sb q$ if and only if one of the following conditions is satisfied $(q=p\sp h$ with p prime):\par (1) $(t,q)=1;$ \par (2) $t=0$, h odd or $p\not\equiv 1$ (mod 4); \par (3) $t=\pm q$, h even or $p\not\equiv 1$ (mod 3); \par (4) $t=\pm 2\sqrt{q}$, h even; \par (5) $t=\pm \sqrt{2q}$, h odd and $p=2;$ \par (6) $t=\pm \sqrt{3q}$, h odd and $p=3.$ \par {\it R. Schoof} [``Non-singular plane cubic curves over finite fields'', Ph.D. Thesis (Univ. Utrecht 1985); see also J. Comb. Theory, Ser. A 46, 183-11 (1987; Zbl 0632.14021)] has determined all the possible structures for the group of rational points on an elliptic curve over ${\bbfF}\sb q$ in the $cases\quad (2)-(6).\quad -$ The main result of this paper is to describe all the possible group structures for the case (1). \par Theorem. Let t be an integer with $\vert t\vert \le 2\sqrt{q}$ and $(t,q)=1$. Then the possible groups that an elliptic curve over ${\bbfF}\sb q$ with $N=q+1-t$ rational points can be are $$ {\bbfZ}/p\sp{\nu\sb p(N)}\oplus \oplus\sb{\ell \ne p}({\bbfZ}/\ell\sp{r\sb{\ell}}\oplus {\bbfZ}/\ell\sp{s\sb{\ell}}) $$ with $r\sb{\ell}+s\sb{\ell}=\nu\sb{\ell}(N)$ and $\min (r\sb{\ell},s\sb{\ell})\le \nu\sb{\ell}(q-1)$. Here, for a prime $\ell$, $\nu\sb{\ell}(n)$ denotes the largest integer such that $\ell\sp{\nu\sb{\ell}(n)}$ $divides\quad n.$ \par The proof rests on the following two lemmas. \par Lemma 1: Given $N\not\equiv 1$ (mod p) such that there exists an elliptic curve with N points over ${\bbfF}\sb q$, there exists at least one such elliptic curve with its group of rational points being cyclic. - Lemma 2: If E is an elliptic curve over ${\bbfF}\sb q$ and $\ell \ne p$ is a prime such that E has a cyclic subgroup of order $\ell\sp n$, then for any $r\le s$ with $r+s=n$ and $\ell\sp r\vert q-1$, there exists an elliptic curve over ${\bbfF}\sb q $ $\ell\sp{\infty}$-isogenous to E and containing a subgroup isomorphic to ${\bbfZ}/\ell\sp r\oplus {\bbfZ}/\ell\sp s.$ \par The same result has been obtained by {\it H. G. Rück} [Math. Comput. 49, 301-304 (1987; Zbl 0628.14019)] independently with a different method.
[N.Yui]

MSC 2000:: *14G05 Rationality questions, rational points
14G15 Finite ground fields
14H52 Elliptic curves
14H45 Special curves and curves of low genus

Keywords: structure of group rational points; isogeny; elliptic curve over finite field

Citations: Zbl 0188.530; Zbl 0632.14021; Zbl 0628.14019

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