François Brunault (École normale supérieure de Lyon)
An abelian variety defined over a number field is called strongly modular when its L-function is the product of L-functions of modular forms of weight 2. In this talk, we will show a weak version of Beilinson's conjectures for non-critical L-values of strongly modular abelian varieties. We will explain the interest of formulating an equivariant version of these conjectures (after Burns and Flach), as well as the main ingredients of the proof: a Hecke-equivariant version of Beilinson's theorem on modular curves, and a modularity result for endomorphism algebras. As an application, we deduce a weak version of Zagier's conjecture on L(E,2) when E is a Q-curve without complex multiplication which is completely defined over a quadratic field.