Limit Tate-Shafarevich groups (1/2)

Analyzing elementary relations between $U(p)$ operators and
Picard functoriality of the Jacobians of each tower of modular
curves of $p$-power level, we get fairly exact control of the
ordinary part of the limit Barsotti-Tate groups and the
($p$-adically completed) ind-limit Mordell-Weil groups with
respect to the weight Iwasawa algebra. Computing Galois cohomology
of these controlled Galois modules, we obtain good control of the
(ordinary part of) limit Selmer groups and limit Tate-Shafarevich groups.