Choices of division sequences on complex elliptic curves

Choices of division sequences on complex elliptic curves

Mon, 02/03/2009
15:00 		Martin Bays (Oxford) Logic Seminar Add to calendar SR1
Choices of division sequences on complex elliptic curves
Let $ \mathbb{E} $ be an elliptic curve defined over a number field $ k $, and let $ a\in\mathbb{E}(\mathbb{C}) $ be a complex point. Among the possible choices of sequences of division points of $ a $, $ (a_n)_n $ such that $ a_1 = a $ and $ na_{nm} = a_m $, we can pick out those which converge in the complex topology to the identity. We show that the algebraic content of this effect of the complex topology is very small, in the sense that any set of division sequences which shares certain obvious algebraic properties with the set of those which converge to the identity is conjugated to it by a field automorphism of $ \mathbb{C} $ over $ k $. As stated, this is a result of algebra and number theory. However, in proving it we are led ineluctably to use model theoretic techniques - specifically the concept of "excellence" introduced by Shelah for the analysis of $ L_{\omega_1,\omega} $ categoricity, which reduces the question to that of proving certain unusual versions of the theorems of Mordell-Weil and Kummer-Bashmakov. I will discuss this and other aspects of the proof, without assuming any model- or number-theoretic knowledge on the part of my audience.