On a conjecture by Beilinson, Bloch and Kato

The prediction by Birch-Swinnerton-Dyer that certain algebraic and analytic invariants of an elliptic curve coincide
to higher even weight modular forms. This is the object of conjectures by Beilinson, Bloch and Kato which p-adic
analogue was formulated by Perrin-Riou. In this talk, we survey some of the results available that provide evidence
towards these conjectures and examine an algebraic invariant associated to certain twists of modular forms under
varying (Heegner) hypotheses.