This is an introductory exposition to background material useful to appreciate various formulations of the Mordell–Lang conjecture (now established by recent spectacular work due to Vojta, Faltings, Hrushovski, Buium, Voloch, and others). It gives an exposition of some of the elementary and standard constructions of algebro-geometric models (rather than model-theoretic ones) with applications (...

We will give an upper bound of Mordell-Weil rank r for relatively minimal brations of curves of genus g 1 on rational surfaces. Under the assumption that a bration is not locally trivial, we have r 4g+4. Moreover the maximal case (r = 4g + 4) will be studied in detail. We determine the structure of such brations and also the structure of their Mordell-Weil lattices introduced by Shioda.

We prove a dynamical version of the Mordell-Lang conjecture for étale endomorphisms of quasiprojective varieties. We use p-adic methods inspired by the work of Skolem, Mahler, and Lech, combined with methods from algebraic geometry. As special cases of our result we obtain a new proof of the classical Mordell-Lang conjecture for cyclic subgroups of a semiabelian variety, and we also answer posi...

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...

We prove versions of the Mordell-Lang conjecture for semiabelian varieties defined over fields of positive characteristic.

with equality if and only if the triangle is equilateral and P is its center. This inequality was conjectured by Erdős [1] and proved by Mordell and Barrow [2]. Oppenheim [3] established a number of additional inequalities relating the six distances p, q, r , x , y, and z. Such an inequality will be referred to as an Erdős-Mordell-type inequality. A survey of some of these inequalities can be f...

be a positive definite quadratic form with determinant D R and integer coefficients a ;Call it an even form if all a ; ; are even ., an odd form if at least one a ;; is odd . Then Í,, is called non-decomposable, if it cannot be expressed as a sum of two non-negative quadratic forms with integer coefficients . Mordell 1 ) proved that f, can always be decomposed into a sum of five squares of line...

Matsumoto et al. define the Mordell-Tornheim L-functions of depth k by LMT(s1, . . . , sk+1;χ1, . . . , χk+1) := ∞ ∑

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 the Mordell-Weil rank of the Jacobian equals the genus. In this case we get an explicit bound for the number of such p-integral points, and we are able to use...

the mordell-weil theorem states that the group of rational points‎ ‎on an elliptic curve over the rational numbers is a finitely‎ ‎generated abelian group‎. ‎in our previous paper, h‎. ‎daghigh‎, ‎and s‎. ‎didari‎, on the elliptic curves of the form $ y^2=x^3-3px$‎, ‎‎bull‎. ‎iranian math‎. ‎soc‎.‎‎ 40 (2014)‎, no‎. ‎5‎, ‎1119--1133‎.‎, ‎using selmer groups‎, ‎we have shown that for a prime $p$...