Henri Darmon (McGill)

The equivariant Birch and Swinnerton-Dyer conjecture

If E is an elliptic curve over Q and V is an Artin representation of Q, a natural equivariant refinement of the Birch and Swinnerton-Dyer conjecture relates the order of vanishing of the Hasse-Weil-Artin L-series L(E,V,s) at s=1 to the multiplicity with which V appears in the Mordell-Weil group of E over the algebraic closure of Q, viewed as a Galois representation. In particular, it implies that V does not appear when L(E,V,1) is non-zero. I will outline the ideas entering into the proof of this last statement for certain irreducible Artin representations V of dimensions 1, 2 and 4. The one-dimensional setting is a landmark result of Kato, and the two and four-dimensional settings are the object of ongoing joint projects with Bertolini-Rotger and Rotger respectively, building on Kato's fundamental insights.