Computing a lower bound for the canonical height is a crucial step in determining a Mordell–Weil basis of an elliptic curve. This paper presents a new algorithm for computing such lower bound, which can be applied to any elliptic curves over totally real number fields. The algorithm is illustrated via some examples.

In this paper we investigate the 2-Selmer rank in families of quadratic twists of elliptic curves over arbitrary number fields. We give sufficient conditions on an elliptic curve so that it has twists of arbitrary 2-Selmer rank, and we give lower bounds for the number of twists (with bounded conductor) that have a given 2-Selmer rank. As a consequence, under appropriate hypotheses we can find m...

Let C : Y 2 = anX n + · · · + a 0 be a hyperelliptic curve with the a i rational integers, n ≥ 5, and the polynomial on the right irreducible. Let J be its Jacobian. We give a completely explicit upper bound for the integral points on the model C, provided we know at least one rational point on C and a Mordell–Weil basis for J(Q). We also explain a powerful refinement of the Mordell–Weil sieve ...

Consider a smooth, geometrically irreducible, projective curve of genus $g\ge 2$ defined over number field degree $d \ge 1$. It has at most finitely many rational points by the Mordell Conjecture, theorem Faltings. We show that is bounded only in terms $g$, $d$ and Mordell–Weil rank curve's Jacobian, thereby answering affirmative question Mazur. In addition we obtain uniform bounds, $g$ $d$, fo...

We give a structure theorem for the $m$-torsion of Jacobian general superelliptic curve $y^m=F(x)$. study existence torsion on curves form $y^q=x^p-x+a$ over finite fields characteristic $p$. apply those results to bound from below Mordell-Weil ranks Jacobians certain $\mathbb Q$.

We study an infinite family of Mordell curves (i.e. the elliptic curves in the form y = x + n, n ∈ Z) over Q with three explicit integral points. We show that the points are independent in certain cases. We describe how to compute bounds of the canonical heights of the points. Using the result we show that any pair in the three points can always be a part of a basis of the free part of the Mord...

Abstract We prove that $164\, 634\, 913$ is the smallest positive integer a sum of two rational sixth powers, but not powers. If $C_{k}$ curve $x^{6} + y^{6} = k$ , we use existence morphisms from to elliptic curves, together with Mordell–Weil sieve, rule out points on for various k .

We prove a special case of a dynamical analogue of the classical Mordell-Lang conjecture. In particular, let φ be a rational function with no superattracting periodic points other than exceptional points. If the coefficients of φ are algebraic, we show that the orbit of a point outside the union of proper preperiodic subvarieties of (P) has only finite intersection with any curve contained in (...

We show that there exist genus one curves of every index over the rational numbers, answering affirmatively a question of Lang and Tate. The proof is “elementary” in the sense that it does not assume the finiteness of any Shafarevich-Tate group. On the other hand, using Kolyvagin’s construction of a rational elliptic curve whose Mordell-Weil and Shafarevich-Tate groups are both trivial, we show...