Raoul Bott has inspired many of us by the magnificence of his ideas, by the way he approaches and explains mathematics, and by his warmth, friendship, and humor. In celebration of Raoul’s eightieth birthday we offer this brief article in which we will explain how the recent cohomological ideas of Jan Nekovár̆ [N2] imply (under mild hypotheses plus the Shafarevich-Tate conjecture) systematic grow...

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.

We show that there exist genus one curves of every index over the rational numbers, answering affirmatively a question of Lang and Tate. The proof is “elementary” in the sense that it does not assume the finiteness of any Shafarevich-Tate group. On the other hand, using Kolyvagin’s construction of a rational elliptic curve whose Mordell-Weil and Shafarevich-Tate groups are both trivial, we show...

We give an algorithm to compute the canonical height on a Jacobian of a curve of genus 2. The computations involve only working with the Kummer surface and so lengthy computations with divisors in the Jacobian are avoided. We use this height algorithm to give an algorithm to perform the “infinite descent” stage of computing the Mordell-Weil group. This last stage is performed by a lattice enlar...

In this paper, we study the Mordell-Weil group of an elliptic curve as a Galois module. We consider an elliptic curve E defined over a number field K whose Mordell-Weil rank over a Galois extension F is 1, 2 or 3. We show that E acquires a point (points) of infinite order over a field whose Galois group is one of Cn×Cm (n = 1, 2, 3, 4, 6, m = 1, 2), Dn×Cm (n = 2, 3, 4, 6, m = 1, 2), A4×Cm (m = ...

In this article we would like to address the following situation. Let f : X → C be an elliptic surface with section S0. We further assume that X is relatively minimal and smooth, and that all singular fibers are semistable, that is, are cycles of P’s (type “Im” in Kodaira’s notation, see [K]). Given a section S of f , it will meet one and only one component in every singular fiber. The fundamen...

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...