On twisted modular curves

Modular curves are moduli spaces of elliptic curves equipped with certain level structures. This talk will be concerned with how the attendant theory has been used to answer questions about the modularity of elliptic curves over $\mathbb{Q}$ and over quadratic fields. In particular, we will outline two instances of the modularity switching technique over totally real fields: the 3-5 trick of Wiles and the 3-7 trick of Freitas, Le Hung and Siksek. The recent work of Caraiani and Newton over imaginary quadratic fields naturally leads one to consider the descent theory of 'twisted' modular curves, and this will be the focus of the final part of the talk.