Our plan is to try to understand Faltings’s proof of the Mordell conjecture. The focus will be on his first proof, which is more algebraic in nature, proves the Shafarevich and Tate conjectures, and also gives us a chance to learn about some nearby topics, such as the moduli space of abelian varieties or p-adic Hodge theory. The seminar will meet 4:10–5:30 every Thursday in 1360. Some relevant ...

Let $A$ be a non-zero abelian variety over field $F$ that is not algebraic finite field. We prove the rational rank of group $A(F)$ infinite when large in sense Pop (also called ample). The main ingredient deduction 1-dimensional case relative Mordell-Lang conjecture from result R\ossler.

Abstract We prove that $164\, 634\, 913$ is the smallest positive integer a sum of two rational sixth powers, but not powers. If $C_{k}$ curve $x^{6} + y^{6} = k$ , we use existence morphisms from to elliptic curves, together with Mordell–Weil sieve, rule out points on for various k .

We characterize all infinite-dimensional graded virtual modules for Thompson's sporadic simple group whose traces are weight 32 weakly holomorphic modular forms satisfying certain special properties. then use these to detect the non-triviality of Mordell–Weil, Selmer, and Tate-Shafarevich groups quadratic twists elliptic curves.

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 define the Mordell exceptional locus Z(V ) for affine varieties V ⊂ Ga with respect to the action of a product of Drinfeld modules on the coordinates of Ga. We show that Z(V ) is a closed subset of V . We also show that there are finitely many maximal algebraic φmodules whose translates lie in V . Our results are motivated by DenisMordell-Lang conjecture for Drinfeld modules.

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...

We use Hodge theory to prove a new upper bound on the ranks of Mordell-Weil groups for elliptic curves over function fields after regular geometrically Galois extensions of the base field, improving on previous results of Silverman and Ellenberg, when the base field has characteristic zero and the supports of the conductor of the elliptic curve and of the ramification divisor of the extension a...

In 1963 Manin proved the Mordell conjecture for function fields (see [Man63]): Let K be a function field and let X be a nonisotrivial curve of genus at least 2 defined over K. Then X has finitely many K-rational points. Some years later Parshin (in the case of a complete base, [Par68]) and Arakelov (in the general case, [Ara71]) proved the Shafarevich conjecture for function fields: Let B be a ...