Related papers: Constructing elliptic curves over finite fields wi…
We present a very efficient algorithm to construct an elliptic curve E and a finite field F such that the order of the point group E(F) is a given prime number N. Heuristically, this algorithm only takes polynomial time Otilde((\log N)^3),…
Elliptic curves over finite fields with predefined conditions in the order are practically constructed using the theory of complex multiplication. The stage with longest calculations in this method reconstructs some polynomial with integer…
We present an algorithm that, on input of a positive integer N together with its prime factorization, constructs a finite field F and an elliptic curve E over F for which E(F) has order N. Although it is unproved that this can be done for…
In this paper, we determine all tetraelliptic modular curves $X_1(N)$ over $\mathbb Q$, and find some tetraelliptic maps $\phi_N$ from $X_1(N)$ to elliptic curves for those tetraelliptic $X_1(N)$. Also we will construct $\phi_N$ explicitly…
In this paper, we present several methods for construction of elliptic curves with large torsion group and positive rank over number fields of small degree. We also discuss potential applications of such curves in the elliptic curve…
In this note is we exhibit an elementary method to construct explicitly curves over finite fields with many points. Despite its elementary character the method is very efficient and can be regarded as a partial substitute for the use of…
Let \( E \) be a complex elliptic curve with conductor \( N \) and modular invariant \( j(E) \in \mathbb{Q} \). We construct a class of modular polynomials $F_N(x,j)$ that relate the modular function $x$ on $X_0(N)$ to the $j$-invariant…
We propose a randomized algorithm to compute isomorphisms between finite fields using elliptic curves. To compute an isomorphism between two fields of cardinality $q^n$, our algorithm takes $$n^{1+o(1)} \log^{1+o(1)}q + \max_{\ell}…
Given a prime power q, for every pair of positive integers m and n with m dividing the GCD of n and q-1, we construct a modular curve over F_q that parametrizes elliptic curves over F_q along with F_q-defined points P and Q of order m and…
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…
For a square-free integer $N$, we present a procedure to compute $\mathbb{Q}$-curves parametrized by rational points of the modular curve $X_0^*(N)$ when this is hyperelliptic.
In this paper we study the problem of how to determine all elliptic curves defined over an arbitrary number field $K$ with good reduction outside a given finite set of primes $S$ of $K$ by solving $S$-unit equations. We give examples of…
Let O be a maximal order in the quaternion algebra B_p over Q ramified at p and infinity. The paper is about the computational problem: Construct a supersingular elliptic curve E over F_p such that End(E) = O. We present an algorithm that…
We construct pairs of elliptic curves over number fields with large intersection of projective torsion points.
We construct isotrivial and non-isotrivial elliptic curves over $\mathbb{F}_q(t)$ with an arbitrarily large set of separable integral points. As an application of this construction, we prove that there are isotrivial log-general type…
Elliptic curves with a known number of points over a given prime field with n elements are often needed for use in cryptography. In the context of primality proving, Atkin and Morain suggested the use of the theory of complex multiplication…
We look for elliptic curves featuring rational points whose coordinates form two arithmetic progressions, one for each coordinate. A constructive method for creating such curves is shown, for lengths up to 5.
We give several new constructions for moderate rank elliptic curves over $\mathbb{Q}(T)$. In particular we construct infinitely many rational elliptic surfaces (not in Weierstrass form) of rank 6 over $\mathbb{Q}$ using polynomials of…
In this paper, we describe the construction of superelliptic curves with a rational point of prescribed order on their jacobians. The construction is based on Hensel's Lemma and produces for a given integer $N$ a superelliptic curve of…
We compute the Mordell-Weil groups of the modular Jacobian varieties of hyperelliptic modular curves $X_1(M, MN)$ over every number field which is the composition of quadratic fields. Also we prove criteria for the existence of elliptic…