**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.