Explore the intricacies of Eisenstein cycles and Manin-Drinfeld properties in this comprehensive lecture by Debargha Banerjee, presented as part of the "Rational Points on Modular Curves" program at the International Centre for Theoretical Sciences. Delve into advanced topics in arithmetic geometry, focusing on the study of rational points on modular curves. Gain insights into the theoretical and computational aspects of determining K-rational points on modular curves XH(K) for various fields and subgroups. Examine the geometry of modular curves, their Q-rational points, and both classical and non-abelian Chabauty methods. Benefit from a balanced approach that combines advanced topics with practical examples, making this lecture suitable for researchers and students alike in the field of arithmetic geometry.