On completely faithful Selmer groups of elliptic curves and Hida deformations

In this paper, we study completely faithful torsion Zp〚G〛-modules with applications to the study of Selmer groups. Namely, if G is a nonabelian group belonging to certain classes of polycyclic pro-p group, we establish the abundance of faithful torsion Zp〚G〛-modules, i.e., non-trivial torsion modules whose global annihilator ideal is zero. We then show that such Zp〚G〛-modules occur naturally in arithmetic, namely in the form of Selmer groups of elliptic curves and Selmer groups of Hida deformations. It is interesting to note that faithful Selmer groups of Hida deformations do not seem to have appeared in literature before. We will also show that faithful Selmer groups have various arithmetic properties. Namely, we show that faithfulness is an isogeny invariant, and we will prove “control theorem” results on the faithfulness of Selmer groups over a general strongly admissible p-adic Lie extension.