CONTENTS We analyze the 128-dimensional Mordell-Weil lattice of a cer1 . Introduction j -a jn elliptic curve over the rational function field k(t)f where k is 2. Statement of Results a finite field of 2 elements. By proving that the elliptic curve 3. Proof of Rank, Discriminant and Tate-Safarevic Group has trivial Tate-Safarevic group and nonzero rational points of 4. Proof of Minimal Norm, Den...

Mazur and Tate proposed a conjecture which compares the Mordell-Weil rank of an elliptic curve overQwith the order of vanishing of Mazur-Tate elements, which are analogues of Stickelberger elements. Under some relatively mild assumptions, we prove this conjecture. Our strategy of the proof is to study divisibility of certain derivatives of Kato’s Euler system. CONTENTS

We prove that the triviality of the Galois action on the suitably twisted odd-dimensional étale cohomogy group with finite coefficients of an absolutely irreducible smooth projective variety implies the existence of certain primitive roots of unity in the field of definition of the variety. This text was inspired by an exercise in Serre’s Lectures on the Mordell–Weil theorem.

We shall show that the Picard number of the generic fiber of an abelian fibered hyperkähler manifold over the projective space is always one. We then give a few applications for the Mordell-Weil group. In particular, by deforming O’Grady’s 10dimensional manifold, we construct an abelian fibered hyperkähler manifold of MordellWeil rank 20, which is the maximum possible among all known ones.

In this paper, we construct a point on the Jacobian of a non-hyperelliptic genus four curve which is defined over a quadratic extension of the base field. We then show that this point generates the Mordell–Weil group of the Jacobian of the universal genus four curve.

We present the details of a model theoretic proof of an analogue of the Manin-Mumford conjecture for semiabelian varieties in positive characteristic. As a by-product of the proof we reduce the general positive characteristic Mordell-Lang problem to a question about purely inseparable points on subvarieties of semiabelian varieties.