Members' Colloquium

Let $E$ be an elliptic curve defined over $\Q$.  The $\bar\Q$-points of $E$ form an abelian group on which the Galois group $G_{\Q} = \Gal(\bar\Q/\Q)$ acts.  The usual Galois representation associated to $E$ captures the action of $G_{\Q}$ on the points of finite order.  However, one could also look at the action of $G_{\Q}$ on the free part of $E(\bar\Q)$.  This infinite-dimensional representation encodes a great deal of interesting arithmetic information.  I will state a conjecture concerning this other Galois representation and present supporting evidence from probability theory, Ramsey theory, algebraic geometry, and number theory.