On the structure of Selmer groups of $p$-ordinary modular forms over $\mathbf{Z}_p$-extensions

Abstract

We prove analogues of the major algebraic results of Greenberg-Vatsal for Selmer groups of $p$-ordinary newforms over $\mathbf{Z}_p$-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 $\mathbf{Z}_p$-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 $\Lambda$-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 used by Greenberg-Vatsal work in our more general setting.