Class groups, Selmer groups and Cassels--Tate pairings

In this largely expository talk we will give an introduction to certain Selmer groups associated to (compatible systems of) finite Galois modules, and describe an analogue of the Cassels--Tate pairing in this setting, due to Flach. We will then discuss the effect of twisting these modules by quadratic characters and show how class groups of quadratic fields, and Selmer groups of quadratic twist families of elliptic curves, naturally fit into this framework. If time permits we will discuss additional objects of arithmetic interest which can be understood from this viewpoint.