The Other Galois Representation of an Elliptic Curve

Let E be an elliptic curve defined over \Q.  The \Q¯-points of E form an abelian group on which the Galois group G\Q=\Gal(\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(\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.