On the structure of Selmer groups of p-ordinary modular forms over Zp-extensions

We prove analogues of the major algebraic results of [GV00] for Selmer groups of p-ordinary newforms over Z(p)-extensions which may be neither cyclotomic nor anticyclotomic, under a number of technical hypotheses, including a cotorsion assumption on the Selmer groups. The main complication which arises in our work is the possible presence of finite primes which can split completely in the Z(p)-extension being considered, resulting in the local cohomology groups that appear in the definition of the Selmer groups being significantly larger than they are in the case of a finitely decomposed prime. We give a careful analysis of the A-module structure of these local cohomology groups and identify the relevant finiteness condition one must impose to make the proof of the key cohomological surjectivity result [GVOO, Proposition 2.1] work in our more general setting. (C) 2017 Elsevier Inc. All rights reserved.