We give an elementary proof of the function field (or geometric) case of the Mordell-Lang conjecture in characteristic zero.

In one of their early works, Miranda and Persson have classified all possible configurations of singular fibers for semistable extremal elliptic fibrations on K3 surfaces. They also obtained the Mordell-Weil groups in terms of the singular fibers except for 17 cases where the determination and the uniqueness of the groups were not settled. In this paper, we settle these problems completely. We ...

Let C : Y 2 = anX + · · · + a0 be a hyperelliptic curve with the ai 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 whic...

Define Γ′ := Γ ′ +B = { γ + z | γ ∈ Γ′, h(z) < }. We may visualize Γ′ as a fattening of Γ ′, a “slab” in the height topology on A(k). Let X be a geometrically integral closed subvariety of A. Let Xk denote X ×k k, and so on. Define a semiabelian subvariety of a semiabelian variety A to be a subvariety B ⊆ A such that the group structure on A restricts to give B the structure of a semiabelian va...

By the Mordell-Weil theorem‎, ‎the group of rational points on an elliptic curve over a number field is a finitely generated abelian group‎. ‎There is no known algorithm for finding the rank of this group‎. ‎This paper computes the rank of the family $ E_p:y^2=x^3-3px $ of elliptic curves‎, ‎where p is a prime‎.

In this paper we prove several Lehmer type inequalities for Drinfeld modules which will enable us to prove certain Mordell-Weil type structure theorems for Drinfeld modules.

We present a visual proof of a lemma that reduces the proof of the Erdős-Mordell inequality to elementary algebra. In 1935, the following problem proposal appeared in the “Advanced Problems” section of the American Mathematical Monthly [5]: 3740. Proposed by Paul Erdős, The University, Manchester, England. From a point O inside a given triangle ABC the perpendiculars OP , OQ, OR are drawn to it...

We remark the density of the jumping loci of the Picard number of a hyperkähler manifold under small one-dimensional deformation and provide some applications for the Mordell-Weil groups of Jacobian K3 surfaces.