Explore a comprehensive lecture on 2-Selmer groups, 2-class groups, and the arithmetic of binary forms delivered by Ashvin A. Swaminathan from Princeton University at the Fields Institute. Delve into advanced number theory topics, including the definition and properties of R, fractional ideals, Hecke's theorem, orbit parametrization, and the parametrize-and-count strategy. Examine the 2-torsion in the class group of R, integral solutions to superelliptic equations, and 2-Selmer groups of hyperelliptic Jacobians. Gain insights into elliptic curves, their Selmer groups, and progress toward related conjectures. Investigate the second moment and class group applications in this 57-minute seminar, part of the Fields Number Theory Seminar series.