Related papers: Finding rational points on elliptic curves using 6…

We give a method for the computation of integral points on a hyperelliptic curve of odd degree over the rationals whose genus equals the Mordell-Weil rank of its Jacobian. Our approach consists of a combination of the $p$-adic approximation…

In the present paper, we prove, for a large class of elliptic curves defined over $\mathbb{Q}$, the existence of an explicit infinite family of quadratic twists with analytic rank $0$. In addition, we establish the $2$-part of the…

Let K be a field of characteristic different from 2 and C an elliptic curve over K given by a Weierstrass equation. To divide an element of the group C by 2, one must solve a certain quartic equation. We characterise the quartics arising…

We describe a tabulation of (conjecturally) modular elliptic curves over the field Q(sqrt(5)) up to the first curve of rank 2. Using an efficient implementation of an algorithm of Lassina Dembele, we computed tables of Hilbert modular forms…

We give a formula for the component at p of the p-adic height pairing of a divisor of degree 0 on a hyperelliptic curve. We use this to give a Chabauty-like method for finding p-adic approximations to p-integral points on such curves when…

We compute a lower bound of the canonical height on quadratic twists of certain elliptic curves. Also we show a simple method for constructing families of quadratic twists with an explicit rational point. % from cubic polynomials. Using the…

Julia Robinson has given a first-order definition of the rational integers Z in the rational numbers Q by a formula (\forall \exists \forall \exists)(F=0) where the \forall-quantifiers run over a total of 8 variables, and where F is a…

We complete the computation of all $\mathbb{Q}$-rational points on all the $64$ maximal Atkin-Lehner quotients $X_0(N)^*$ such that the quotient is hyperelliptic. To achieve this, we use a combination of various methods, namely the…

Let $p$ be a prime number and $E_{p}$ denote the elliptic curve $y^2=x^3+px$. It is known that for $p$ which is congruent to $1, 9$ modulo $16$, the rank of $E_{p}$ over $\mathbb{Q}$ is equal to $0, 2$. Under the condition that the Birch…

We study the family of elliptic curves $y^2=x(x-a^2)(x-b^2)$ parametrized by Pythagorean triples $(a,b,c)$. We prove that for a generic triple the lower bound of the rank of the Mordell-Weil group over $\mathbb{Q}$ is 1, and for some…

Let $ p $ and $ q $ be odd prime numbers with $ q - p = 2, $ the $\varphi -$Selmer groups, Shafarevich-Tate groups ($ \varphi - $ and $ 2-$part) and their dual ones as well the Mordell-Weil groups of elliptic curves $ y^{2} = x (x \pm p) (x…

Let $E/\mathbb{Q}$ be a semistable elliptic curve such that $\mathrm{ord}_{s=1}L(E,s) = 1$. We prove the $p$-part of the Birch and Swinnerton-Dyer formula for $E/\mathbb{Q}$ for each prime $p \geq 5$ of good reduction such that $E[p]$ is…

We compute the rational points on certain members of the following family of hyperelliptic curves \[C_a \colon y^2 = x^8 + (4-4a^4) x^6 + (8a^4 + 6)x^4 + (4-4a^4)x^2 + 1\] via the method first developed by Dem'yanenko \cite{dem1966rational}…

We give refined methods for proving finiteness of the Chabauty--Coleman--Kim set $X(\mathbb{Q}_2 )_2 $, when $X$ is a hyperelliptic curve with a rational Weierstrass point. The main developments are methods for computing Selmer conditions…

We investigate the properties of a family of approximations of the Hasse-Weil $L$-function associated to an elliptic curve $E$ over $\mathbb{Q}$. We give a precise expression for the error of the approximations, and provide a visual…

In this work, we consider the rational points on elliptic curves over finite fields F_{p}. We give results concerning the number of points on the elliptic curve y^2{\equiv}x^3+a^3(mod p)where p is a prime congruent to 1 modulo 6. Also some…

Let $\ell$ and $p \geq 3$ be different primes. Let $E/\mathbb{Q}_\ell$ and $E'/\mathbb{Q}_\ell$ be elliptic curves with isomorphic $p$-torsion. Assume that $E$ has potentially multiplicative reduction. We classify when all…

I provide methods of constructing elliptic and hyperelliptic curves over global fields with interesting rational points over the given fields or over large field extensions. I also provide a elliptic curves defined over any given number…

It is well-known that every elliptic curve over the rationals admits a parametrization by means of modular functions. In this short note, we show that only finitely many elliptic curves over $\mathbf{Q}$ can be parametrized by modular…

For the integer $ D=pq$ of the product of two distinct odd primes, we construct an elliptic curve $E_{2rD}:y^2=x^3-2rDx$ over $\mathbb Q$, where $r$ is a parameter dependent on the classes of $p$ and $q$ modulo 8, and show, under the parity…