Diophantine equations: the modular method and more
Since the proof of Fermat’s Last Theorem, many Diophantine equations have been solved using deep results about elliptic curves, modular forms, and associated Galois representations. For this so-called modular method, both the underlying tools, as well as the establishment of new applications, are under active development. In many interesting cases, the tools and applications can greatly benefit from combining them with other methods, e.g. from transcendence theory or rational points on curves. The purpose of this talk is to discuss some techniques from the modular method and explain how these, sometimes in combination with other methods, can be applied to explicitly solve certain Diophantine equations.