Explore the intricacies of arithmetic field theory for elliptic curves in this comprehensive lecture by Minhyong Kim. Delve into the fascinating world of rational points on modular curves, a key area of interest for arithmetic geometers. Gain advanced insights into the geometry of modular curves, their Q-rational points, and both classical and non-abelian Chabauty methods. Learn about the theoretical and computational aspects of determining K-rational points on modular curves XH(K) for various fields and subgroups. Discover the connections between elliptic curves, modular forms, and modular curves as central objects in arithmetic geometry. Benefit from a balanced approach that covers both advanced topics and practical examples, making this lecture suitable for researchers at various levels of expertise.