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...

In [12] and [13], Silverman discusses the problem of bounding the Mordell-Weil ranks of elliptic curves over towers of function fields. We first prove generalizations of the theorems of those two papers by a different method, allowing non-abelian Galois groups and removing the dependence on Tate’s conjectures. We then prove some theorems about the growth of Mordell-Weil ranks in towers of funct...

We present a Mordell-Lang statement for Drinfeld modules. In the case of Drinfeld modules of finite characteristic, we show the connection between our Mordell-Lang statement and the study of the ring of quasi-endomorphisms for a certain minimal group in the theory of separably closed fields associated to the Drinfeld module.

We combine various well-known techniques from the theory of heights, the theory of “noncritical Belyi maps”, and classical analytic number theory to conclude that the “ABC Conjecture”, or, equivalently, the so-called “Effective Mordell Conjecture”, holds for arbitrary rational points of the projective line minus three points if and only if it holds for rational points which are in “sufficiently...

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 ...

Suppose E is an elliptic curve over Q and χ a Dirichlet character. We use statistical properties of modular symbols to estimate heuristically the probability that L(E,χ,1)=0. Via Birch Swinnerton-Dyer conjecture, this gives heuristic Mordell–Weil rank grows in abelian extensions Q. Using heuristic, we find large class infinite F where expect E(F) be finitely generated. Our work was inspired by ...