Explore a lecture on the local-global principle for quadratic twists of abelian varieties presented by Francesc Fite at the International Centre for Theoretical Sciences. Delve into advanced topics in arithmetic geometry as part of the "Rational Points on Modular Curves" program. Gain insights into the study of rational points on varieties, with a focus on modular curves. Examine the relationship between elliptic curves, modular forms, and modular curves in arithmetic geometry. Learn about theoretical and computational aspects of determining K-rational points on modular curves XH(K) for various fields and subgroups. Discover advanced concepts including the geometry of modular curves, Q-rational points, and Chabauty methods. Benefit from a balance of advanced topics and practical examples in this 1-hour 9-minute presentation, suitable for researchers and graduate students in mathematics.