Mordell-Weil groups in procyclic extensions of a function field
Let \(K\) be a number field and let \(L\) be its cyclotomic \({\mathbb{Z}}_p\)-extension. Let \(E/K\) be an elliptic curve with good reduction at \(p\). Then Mazur has conjectured that the group \(E(L)\) is finitely generated. The author considers an analogue of this conjecture. The number field \(K\) is replaced by \(\mathbb{C}(T)\) (the rational function field in one variable over the complex numbers), and \(L\) is replaced by its procyclic extension \(\bigcup_{r \geq 1} \mathbb{C}(T^{1/r})\). Let \(E/K\) be an elliptic curve that has good or multiplicative reduction outside \(\{0,\infty\}\). Then the author proves that \(E(L)\) is finitely generated, provided one or the other of two technical conditions on the reduction behaviour of \(E\) is satisfied. The proof is split into two parts. First it is shown that the rank of \(E(K(T^{1/r}))\) is bounded independently of \(r\). This part of the proof follows a method due to \textit{P. F. Stiller} [Pac. J. Math. 128, No. 1, 157-189 (1987; Zbl 0591.14022)] and takes up most of the paper. This is also the point where the assumption on the reduction behaviour comes into play. The second part is to control the torsion part in the quotient of the Mordell-Weil groups at different levels.
procyclic extension of rational function field
scientific article; zbMATH DE number 1061773
scientific article