Many years ago, one of us was reading through L. J. Mordell’s “Diophantine Equations” and was struck by a curious statement—namely, that the curve C : y2 = x3 + 17 contains exactly sixteen points (x, y) with x and y integers (see [6, p. 250]). A list of the points followed. Many questions immediately came to mind. How did they find these points, called integer points? How did they prove that th...

Let K be a number field and let E/K be an elliptic curve. The Mordell-Weil Theorem states that E(K), the set of K-rational points on E, can be given the structure of a finitely generated abelian group. In this note we consider elliptic curves defined over the rationals Q and provide bounds for the size of E(K)Tors, where K is an algebraic Galois extension of Q. Theorem 1. Let E/Q be an elliptic...

In this paper we explain how to bound the p-Selmer group of an elliptic curve over K, a number eld. Our method is an algorithm which is relatively simple to implement, although it requires data such as units and class groups from number elds of degree at most p 1. Our method is practical for p = 3 but for larger values of p becomes impractical with current computing power. In the examples we ha...

Let X A be a reduced, irreducible, closed subvariety of a semiabelian variety A over an algebraically closed eld k. We would like to study the structure of such an X. Our point of departure is arithmetic: motivated by Lang's conjectures 15, 17, 16], as manifested by the theorems of Faltings 6, 7] and Vojta 27], we study the Mordell exceptional locus: that is, the union of translated positive di...

In this paper we establish a Hasse principle concerning the linear dependence over Z of nontorsion points in the Mordell-Weil group of an abelian variety over a number field.

The Birch and Swinnerton-Dyer conjecture states that the rank of the Mordell-Weil group of an elliptic curve E equals the order of vanishing at the central point of the associated L-function L(s, E). Previous investigations have focused on bounding how far we must go above the central point to be assured of finding a zero, bounding the rank of a fixed curve or on bounding the average rank in a ...

The canonical height ĥ on an abelian variety A defined over a global field k is an object of fundamental importance in the study of the arithmetic of A. For many applications it is required to compute ĥ(P ) for a given point P ∈ A(k). For instance, given generators of a subgroup of the Mordell-Weil group A(k) of finite index, this is necessary for most known approaches to the computation of gen...

The goal of this paper is to explain how a simple but apparently new fact of linear algebra together with the cohomological interpretation of L-functions allows one to produce many examples of L-functions over function fields vanishing to high order at the center point of their functional equation. Conjectures of Birch and Swinnerton-Dyer, Bloch, and Beilinson relate the orders of vanishing of ...