Bounded-Depth Cassels–Tate Detection and the Structure of Selmer Groups

This paper studies the inverse-limit structure of p-primary Selmer groups and Tate–Shafarevich groups of elliptic curves over the rational numbers in the cyclotomic Z_p-extension. We isolate a precise obstruction to uniform local–global compatibility, termed defect amplification, and show that its presence forces unbounded growth of Selmer groups along the cyclotomic tower. Under standard analytic boundedness hypotheses arising in Iwasawa theory, such growth is impossible. Assuming analytic boundedness, we prove the existence of bounded-degree annihilators in the Iwasawa algebra that induce bounded-depth Cassels–Tate detection uniformly across finite layers. This excludes defect amplification and establishes a rigidity principle for inverse limits of p-primary Tate–Shafarevich groups. As a consequence, when the Iwasawa mu-invariant vanishes, the Selmer group admits a height-compatible filtration whose length equals the Mordell–Weil rank. In analytically controlled cases, this recovers the Birch–Swinnerton–Dyer predicted formula for algebraic rank.