Related papers: Elliptic Fermat numbers and elliptic divisibility …

If $E$ is an elliptic curve over $\mathbb{Q}$, then it follows from work of Serre and Hooley that, under the assumption of the Generalized Riemann Hypothesis, the density of primes $p$ such that the group of $\mathbb{F}_p$-rational points…

In this paper, we investigate extreme values of $\omega(E(\mathbb{F}_p))$, where $E/\mathbb{Q}$ is an elliptic curve with complex multiplication and $\omega$ is the number-of-distinct-prime-divisors function. For fixed $\gamma > 1$, we…

For an elliptic curve $E$ over a finite field $\F_q$, where $q$ is a prime power, we propose new algorithms for testing the supersingularity of $E$. Our algorithms are based on the Polynomial Identity Testing (PIT) problem for the $p$-th…

The Fermat quotient $q_p(a):=(a^{p-1}-1)/p$, for prime $p\nmid a$, and the Wilson quotient $w_p:=((p-1)!+1)/p$ are integers. If $p\mid w_p,$ then $p$ is a Wilson prime. For odd $p,$ Lerch proved that $(\sum_{a=1}^{p-1} q_p(a) - w_p)/p$ is…

Let $C$ be a curve defined over a number field $K$. A point $P\in C(\overline{\mathbb{Q}})$ is called $K$-quadratic if $[K(P):K]=2$. Let $K$ be a number field such that the rank of the elliptic curves $E_1:\,y^2= x^3 + 4x$ and $E_2:\,y^2=…

We study the existence of a Zsigmondy bound for a sequence of divisors associated to points on an elliptic curve over a function field. More precisely, let $k$ be an algebraically closed field, let $\mathcal{C}$ be a nonsingular projective…

This paper studies the task of two-sources randomness extractors for elliptic curves defined over finite fields $K$, where $K$ can be a prime or a binary field. In fact, we introduce new constructions of functions over elliptic curves which…

Consider elliptic curves $ E=E_\sigma: y^2 = x (x+\sigma p) (x+\sigma q), $ where$ \sigma =\pm 1, $ $p$ and $ q$ are prime numbers with $p+2=q$. (1) The Selmer groups $ S^{(2)}(E/{\mathbf{Q}}), S^{(\phi)}(E/{\mathbf{Q})}$, and $\…

It is shown that, under some mild technical conditions, representations of prime numbers by binary quadratic forms can be computed in polynomial complexity by exploiting Schoof's algorithm, which counts the number of $\mathbb F_q$-points of…

We present a general method for analytically factorizing the n-fold form factor integrals $f^{(n)}_{N,N}(t)$ for the correlation functions of the Ising model on the diagonal in terms of the hypergeometric functions…

An abelian variety admits only a finite number of isomorphism classes of principal polarizations. The paper gives an interpretation of this number in terms of class numbers of definite Hermitian forms in the case of a product of elliptic…

It is known, that for every elliptic curve over Q there exists a quadratic extension in which the rank does not go up. For a large class of elliptic curves, the same is known with the rank replaced by the 2-Selmer group. We show, however,…

Two rational primes p, q are called dual elliptic if there is an elliptic curve E mod p with q points. They were introduced as an interesting means for combining the strengths of the elliptic curve and cyclotomy primality proving…

Let $\ell$ be a prime number and let $E$ and $E'$ be $\ell$-isogenous elliptic curves defined over a finite field $k$ of characteristic $p \ne \ell$. Suppose the groups $E(k)$ and $E'(k)$ are isomorphic, but $E(K) \not \simeq E'(K)$, where…

Let $\ell$ and $p \geq 3$ be distinct prime numbers. Let $E/\mathbb{Q}_{\ell}$ be an elliptic curve with $p$-torsion module $E_p$. Let $\mathbb{Q}_{\ell}(E_p)$ be the $p$-torsion field of $E$. We provide a complete description of the degree…

For a given elliptic curve $\mathbf{E}$ over a finite field of odd characteristic and a rational function $f$ on $\mathbf{E}$ we first study the linear complexity profiles of the sequences $f(nG)$, $n=1,2,\dots$ which complements earlier…

Generalised Fermat equation (GFE) is the equation of the form $ax^p+by^q=cz^r$, where $a,b,c,p,q,r$ are positive integers. If $1/p+1/q+1/r<1$, GFE is known to have at most finitely many primitive integer solutions $(x,y,z)$. A large body of…

In this paper we will give a global description of the Frobenius for the division fields of an elliptic curve E which is strictly analogous to the cyclotomic case. This is then applied to determine the splitting of a prime p in subfields of…

Let $C$ be a smooth genus one curve described by a quartic polynomial equation over the rational field $\mathbb Q$ with $P\in C(\mathbb Q)$. We give an explicit criterion for the divisibility-by-$2$ of a rational point on the elliptic curve…

Mersenne numbers and Fermat numbers are two hot and difficult issues in number theory. This paper constructs a special group for every positive odd number other than 1, and discovers an algorithm for determining the multiplicative order of…