We will discuss our approach to global analytic geometry, based on overconvergent power series and functors of functions. We will explain how slight modifications of it allow us to develop a derived version of global analytic geometry. We will finish by discussing applications to the cohomological study of arithmetic varieties.

I will talk in down to earth terms about several main features of this theory.

We describe how p-adic height pairings allow us to find integral points on hyperelliptic curves, in the spirit of Kim's nonabelian Chabauty program. In particular, we discuss how to carry out this ``quadratic Chabauty'' method over quadratic number fields (joint work with Amnon Besser and Steffen Mueller) and present related ideas to find rational points on bielliptic genus 2 curves (joint work with Netan Dogra).