Control Theorems for Fine Selmer Groups

We study the growth of the p-primary fine Selmer group, R(E/F ') , of an elliptic curve over an intermediate sub-extension F ' of a p-adic Lie extension, L/F. We estimate the Z(p)-corank of the kernel and cokernel of the restriction map r(L/F ') : R(E/F ')-> R(E /L)(Gal(L/F ')) with F ' a finite extension of F contained in L. We show that the growth of the fine Selmer groups in these intermediate sub-extension is related to the structure of the fine Selmer group over the infinite level. On specializing to certain (possibly non-commutative) p-adic Lie extensions, we prove finiteness of the kernel and cokernel and provide growth estimates on their orders.