In this paper we give a much shorter proof for a result of B.M.M de Weger. For this purpose we use the theory of linear forms in complex and $p$-adic elliptic logarithms. To obtain an upper bound for these linear forms we compare the results of Hajdu and Herendi and Rémond and Urfels.

The rational solutions with as denominators powers of $2$ to the elliptic diophantine equation $y^2=x^3-228x+848$ are determined. An idea of Yuri Bilu is applied, which avoids Thue and Thue-Mahler equations, and deduces four-term ($S$-) unit equations with special properties, that are solved by linear forms in real and $p$-adic logarithms.

Let $F$ be a function field of characteristic $p\>0$, $ℱ/F$ a ${\mathbb{Z}}_l^d$-extension (for some prime $l\ne p$) and $E/F$ a non-isotrivial elliptic curve. We study the behaviour of the $r$-parts of the Selmer groups ($r$ any prime) in the subextensions of $ℱ$ via appropriate versions of Mazur’s Control Theorem. As a consequence we prove that the limit of the Selmer groups is a cofinitely generated (in some cases cotorsion) module over the Iwasawa algebra of $ℱ/F$.

In this article, we formalize a set of points on an elliptic curve over GF(p). Elliptic curve cryptography [10], whose security is based on a difficulty of discrete logarithm problem of elliptic curves, is important for information security.

Let $E$ be an elliptic curve over $\mathbb{Q}$ with good supersingular reduction at a prime $p\ge 3$ and $a_p=0$. We generalise the definition of Kobayashi’s plus/minus Selmer groups over $\mathbb{Q}\left({\mu }_{p^{\infty }}\right)$ to $p$-adic Lie extensions $K_{\infty }$ of $\mathbb{Q}$ containing $\mathbb{Q}\left({\mu }_{p^{\infty }}\right)$, using the theory of $(\varphi ,\Gamma )$-modules and Berger’s comparison isomorphisms. We show that these Selmer groups can be equally described using Kobayashi’s conditions via the theory of overconvergent power series. Moreover, we show that such an approach gives the usual Selmer groups in the ordinary case....

We give some easy necessary and sufficient criteria for twists of abelian varieties by Artin representations to be simple.

Let K be any quadratic field with ${}_K$ its ring of integers. We study the solutions of cubic equations, which represent elliptic curves defined over ℚ, in quadratic fields and prove some interesting results regarding the solutions by using elementary tools. As an application we consider the Diophantine equation r+s+t = rst = 1 in ${}_K$. This Diophantine equation gives an elliptic curve defined over ℚ with finite Mordell-Weil group. Using our study of the solutions of cubic equations in quadratic fields...

Let k ∈ ℤ be such that ${|}_k\left(\mathbb{Q}\right)|$ is finite, where ${}_k:y²=1-2kx+k²x²-4x³$. We complete the determination of all solutions to xyz = 1 and x + y + z = k in integers of number fields of degree at most four over ℚ.

This article concerns the problem of solving diophantine equations in rational numbers. It traces the way in which the 19th century broke from the centuries-old tradition of the purely algebraic treatment of this problem. Special attention is paid to Sylvester’s work “On Certain Ternary Cubic-Form Equations” (1879–1880), in which the algebraico-geometrical approach was applied to the study of an indeterminate equation of third degree.

We perform descent calculations for the families of elliptic curves over $Q$ with a rational point of order $n=5$ or 7. These calculations give an estimate for the Mordell-Weil rank which we relate to the parity conjecture. We exhibit explicit elements of the Tate-Shafarevich group of order 5 and 7, and show that the 5-torsion of the Tate-Shafarevich group of an elliptic curve over $Q$ may become arbitrarily large.

For a commutative algebraic group $G$ over a perfect field $k$, Ribet defined the set of almost rational torsion points $G_{tors,k}^{ar}$ of $G$ over $k$. For positive integers $d$, $g,$ we show there is an integer $U_{d,g}$ such that for all tori $T$ of dimension at most $d$ over number fields of degree at most $g$, $T_{tors,k}^{ar}\subseteq T\left[U_{d,g}\right]$. We show the corresponding result for abelian varieties with complex multiplication, and under an additional hypothesis, for elliptic curves without complex multiplication. Finally, we show that except for an explicit finite...