Explore a 46-minute lecture on the arithmetic of quadratic twists of elliptic curves, presented by Ye Tian for the International Mathematical Union. Delve into the behavior of Selmer groups and L-values of elliptic curves under quadratic twists, using the congruent number problem as a fundamental example. Gain insights into recent progress towards the BSD conjecture. Examine topics such as the Goldfeld Conjecture for CM Families, equivalence relations for quadratic twist families, and the distribution of 2-Selmer groups. Investigate Kolyvagin's Question, modularity of Heegner cycles, and the Arithmetic Rallis inner product formula. Access accompanying slides for visual support and deeper understanding of this advanced mathematical discourse.