We give explicit generators for subgroups of finite index of the Mordell-Weil groups of several families of elliptic surfaces introduced by Masato Kuwata.

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

Let p be a prime, let r and q powers of p, b relatively prime integers not divisible by p. $$C/{\mathbb {F}}_{r}(t)$$ the superelliptic curve with affine equation $$y^b+x^a=t^q-t$$ , J Jacobian C. By work Pries Ulmer (Trans Am Math Soc 368(12):8553–8595, 2016), satisfies Birch Swinnerton-Dyer conjecture (BSD). Generalizing Griffon (Pacific 305(2):597–640, 2020) we compute L-function in terms ce...

A simplified method of descent via isogeny is given for Jacobians of curves of genus 2. This method is then used to give applications of a theorem of Coleman for computing all the rational points on certain curves of genus 2. 0 Introduction The following classical result of Chabauty [3] is a curiosity of the literature in that there has been a 50 year period during which applications have been ...

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 ...

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...

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...