Explore the profound research contributions of Jan Nekovář in this illuminating lecture by Olivier Fouquet from Université Paris-Saclay. Delve into the conceptual principles that unified Nekovář's work throughout his career, from pioneering constructions of Euler systems to groundbreaking parity results for L-functions of automorphic representations of GL2. Examine the intricate study of Galois action on quaternionic Shimura varieties and gain insights into the hidden symmetries that permeate Nekovář's mathematical legacy. This 70-minute talk, presented at the Institut des Hautes Etudes Scientifiques (IHES), offers a comprehensive overview of Nekovář's impactful work in number theory and algebraic geometry.