arXiv subjects
Publications and source records attributed to Duc Van Huynh.
In this work, we study elliptic curves of the form $E_L: y^2 = x(x-1)(x-L)$, where $L^2-L+1 \in \left(\mathbb{Q}^{\times}\right)^2$ and $L \in \mathbb{Q} \setminus \{0,1\}$. We will show that for almost all quadratic extensions $K/\mathbb{Q}$ except one we have $E_L(K) \cong \mathbb{Z}/2\mathbb{Z} \oplus \mathbb{Z}/2\mathbb{Z}$.
We will describe an algorithm to construct an elliptic curve $E_{f_q}$ over some prime field $\mathbb{F}_p$ such that such that $|E_{f_q}(\mathbb{F}_p)| = f_q$, where $f_q$ is a probable Fibonacci prime for some prime index $q$. The algorithm is a variant of the efficient CM-construction by Br$\ddot{o}$ker and Stevenhagen, which is well suited for Fibonacci primes due to their arithmetic properties. The time complexity of our algorithm is expected to be lower than $\widetilde{O}(\log^3({f_q}))$. The construction process is a series of algorithms, where each is a test for primality.
Let K be a local field of characteristic p with perfect residue field k. In this paper we find a set of representatives for the k-isomorphism classes of totally ramified separable extensions L/K of degree p. This extends work of Klopsch, who found representatives for the k-isomorphism classes of totally ramified Galois extensions L/K of degree p.