Refined conjectures of ‘Birch—Swinnerton-Dyer type’ and the theory of Euler systems

In the 1980s, Mazur and Tate proposed refinements of the Birch–Swinnerton-Dyer conjecture that also capture congruences between twists of Hasse–Weil L-series by Dirichlet characters. In this talk, I will report on new results towards these refined conjectures, obtained in joint work with Matthew Honnor. I will also outline how the results fit into a more general approach to refined conjectures on special values of L-series via an enhanced theory of Euler systems. This final part will touch upon joint work with David Burns.