Control Theorems for Fine Selmer Groups

Abstract

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.