Let E be an elliptic curve defined over ℚ. The ℚ̅-points of E form an abelian group on which the Galois group Gℚ=Gal(ℚ̅/ℚ) acts. The usual Galois representation associated to E captures the action of Gℚ on the points of finite order. However, one could also look at the action of Gℚ on the free part of E(ℚ̅). 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.
This video is part of the Institute for Advanced Study‘s Members’ colloquium.
