On the cyclicity of elliptic curves over finite field extensions
Let \(E\) be an elliptic curve over the finite field \(\mathbb{F}_q\) such that \(E(\mathbb{F}_q)\), its group of \(\mathbb{F}_q\)-rational points, is cyclic. Put \(C(E)=\{n: E(\mathbb{F}_{q^n})\) is cyclic\}. The author obtains some general properties of the sets \(C(E)\) and gives a complete description if \(E\) is supersingular. If \(T\) is a positive integer, let \(c(E,T)\) denote the number of elements in \(C(E)\) that are at most \(T\). The author also obtains results on the asymptotic value of \(c(E,T)/T\). In the last section, the author considers the related question of the size of the \(N\)-division field of \(E\) and obtains bounds for this number if \(E\) is supersingular. The problems studied here were raised by \textit{I. E. Shparlinski} in Chapter 6 of his book [Computational and algorithmic problems in finite fields. Kluwer (1992; Zbl 0780.11064)].
scientific article; zbMATH DE number 1369826
scientific article