Computing a lower bound for the canonical height is a crucial step in determining a Mordell–Weil basis for elliptic curves. This paper presents an algorithm for computing such a lower bound for elliptic curves over number fields without searching for points. The algorithm is illustrated by some examples.

In this paper, we investigate a local to global principle for Galois cohomology of number fields with coefficients in the Tate module an abelian variety. [G. Banaszak and P. Krasoń, On étale K-groups curves, J. K-Theory Appl. Algebra Geom. Topol. 12 (2013) 183–201], G. author obtained sufficient condition validity [Formula: see text]-theory curve. This fact has been established by means analysi...

In this paper, we discuss the parity result for multiple Dirichlet series which contains some special values of zeta functions as cases, such Mordell-Tornheim type values, root systems and so on. Moreover, can give an explicit expression in terms lower by using main theorem.

Probably most mathematicians would have agreed with Weil (certainly I would have), until earlier last year, when a German mathematician, Gerd Faltings, proved the Mordell conjecture, opening thereby a new chapter in number theory. In fact, his paper also establishes two other important conjectures, due to Tate and Shafarevich, and these ach ievements m a y well prove to be equally significant. ...

We give a proof of the geometric Mordell-Lang conjecture, in any characteristic. Our method involves a model-theoretic analysis of the kernel of Manin’s homomorphism and of a certain analog in characteristic p. Department of Mathematics, Massachusetts Institute of Technology, 2-277, Cambridge, Massachusetts 02139 Current address: Department of Mathematics, Hebrew University, Jerusalem, Israel E...

Let Ep be an elliptic curve over a prime finite field Fp, p ≥ 5, and Pp, Qp ∈ Ep(Fp). The elliptic curve discrete logarithm problem, ECDLP, on Ep is to find mp ∈ Fp such that Qp = mpPp if Qp ∈ 〈Pp〉. We propose an algorithm to attack the ECDLP relying on a Hasse principle detecting linear dependence in Mordell-Weil groups of elliptic curves via a finite number of reductions.

Watkins conjectured that for an elliptic curve $E$ over $\mathbb {Q}$ of Mordell-Weil rank $r$, the modular degree is divisible by $2^r$. If has non-trivial rational $2$-torsion, we prove conjecture all quadratic twists squarefree integers with sufficiently many prime factors.

We show that elliptic curves whose Mordell-Weil groups are finitely generated over some infinite extensions of Q, can be used to show the Diophantine undecidability of the rings of integers and bigger rings contained in some infinite extensions of rational numbers.

By showing that the elliptic curve (x2 13)(y2 13) = 48 has infinitely many rational points, we prove that Letac's construction produces infinitely many genuinely different ideal 9th-order multigrades. We give one (not very small) new example, and, by finding the Mordell-Weil group of the curve, show how to find all examples obtainable by Letac's method.

These notes are based on lectures given at the “Arithmetic of Hyperelliptic Curves” workshop, Ohrid, Macedonia, 28 August–5 September 2014. They offer a brief (if somewhat imprecise) sketch of various methods for computing the set of rational points on a curve, focusing on Chabauty and the Mordell–Weil sieve.