By the Mordell-Weil theorem the group of Q(z)-rational points of an elliptic curve is finitely generated. It is not known whether the rank of this group can get arbitrary large as the curve varies. Mestre and Nagao have constructed examples of elliptic curves E with rank at least 13. In this paper a method is explained for finding a 14th independent point on E, which is defined over k(z), with ...

Let V be a plane non–singular geometrically irreducible cubic curve over a finitely generated field k. The Mordell–Weil theorem for V can be restated in the following geometric form: there is a finite subset B ⊂ V (k) such that the whole V (k) can be obtained from B by drawing secants (and tangents) through pairs of previously constructed points and consecutively adding their new intersection p...

Let V be a plane smooth cubic curve over a finitely generated field k. The Mordell–Weil theorem for V states that there is a finite subset P ⊂ V (k) such that the whole V (k) can be obtained from P by drawing secants and tangents through pairs of previously constructed points and consecutively adding their new intersection points with V. Equivalently, the group of birational transformations of ...

The Mordell-Lang conjecture and its variants assert subvarieties of algebraic groups can meet certain subgroups only in a finite set of translates of subgroups. In most cases the number of such translates is unknown and it is not even known whether this number may be bounded by a function of the geometric data. In this note we show that some uniformity follows immediately from the finiteness re...

We introduce a double zeta function of Mordell–Tornheim type and compute its values at nonpositive integer points. then discuss possible generalization the Kaneko–Zagier conjecture for all

We develop an algorithm computing the transcendental lattice and the Mordell–Weil group of an extremal elliptic surface. As an example, we compute the lattices of four exponentially large series of surfaces

Let $K$ be a number field, and let $E/K$ an elliptic curve over $K$. The Mordell--Weil theorem asserts that the $K$-rational points $E(K)$ of $E$ form finitely generated abelian group. In this work, we complete classification finite groups which appear as torsion subgroup for cubic field. To do so, determine on modular curves $X_1(N)$ \[N = 21, 22, 24, 25, 26, 28, 30, 32, 33, 35, 36, 39, 45, 65...

We present the first examples of smooth elliptic Calabi-Yau threefolds with Mordell-Weil rank 10, highest currently known value. They are given by Schoen introduced Namikawa. explicitly compute Shioda homomorphism for generators group and their induced height pairing. There six isolated fibers Kodaira Type IV. Compactification F-theory on these gives an effective theory in dimensions which cont...

We make several observations about the category C of compact complex manifolds, considered as a many-sorted structure of finite Morley rank. We also point out that the Mordell-Lang conjecture holds for complex tori: if A is a complex torus, Γ a finitely generated subgroup of A, and X an analytic subvariety of A, then X ∩ Γ is a finite union of translates of subgroups of A. This is implicit in t...