On the structure of Selmer groups of \(p\)-ordinary modular forms over \(\mathbb{Z}_{p}\)-extensions (Q1747230)

Let \(F\) be a number field and \(F_\infty/F\) be a \(\mathbb{Z}_p\)-extension (\(p\) an odd prime) with Galois group \(\Gamma:=\mathrm{Gal}(F_\infty/F)\). Let \(f=\sum_{n\geqslant 1} a_nq^n\) be a normalized newform of level \(N\) and weight \(k\geqslant 2\), with Hecke eigenvalues \(a_n\) belonging to the ring of integers \(\mathcal{O}\) of a finite extension \(K\) of \(\mathbb{Q}_p\). Let \(V\) be a 2-dimensional \(K\) vector space on which \(G_{\mathbb{Q}}:=\mathrm{Gal}(\overline{\mathbb{Q}}/\mathbb{Q})\) acts via \(\rho_f:G_{\mathbb{Q}}\rightarrow\mathrm{GL}_2(K)\) (the \(p\)-adic Galois representation associated to \(f\)) and consider a \(\mathrm{Gal}(\overline{\mathbb{Q}}/F)\)-stable lattice \(T\) in \(V\) to define an \(\mathcal{O}\)-module \(A:=V/T\). Main example: take \(f\) associated to an elliptic curve \(E\) defined over \(\mathbb{Q}\) and \(A=E[p^\infty]\). The paper deals with Selmer groups (or \textit{modified} Selmer groups) for this setting, defined via the usual local cohomology conditions: ordinary at primes dividing \(p\) and unramified at all other primes (\(p\) odd helps in avoiding Archimedean primes, while modified or \textit{minimal} Selmer groups have trivial conditions at a finite set of auxiliary primes, usually the ones dividing \(N\)). The author (with a few natural technical hypotheses on the behaviour of primes dividing \(p\) in \(F_\infty/F\), on the irreducibility of \(\rho_f\) and on \(\mathrm{Sel}(F_\infty,A)\) being a cotorsion \(\mathcal{O}[[\Gamma]]\)-module) studies the structure of \textit{minimal} Selmer groups to provide bounds for the corank and to prove they have no proper \(\mathcal{O}[[\Gamma]]\)-submodules of finite index (thus generalizing some of the results of \textit{R. Greenberg} and \textit{V. Vatsal} [Invent. Math. 142, No. 1, 17--63 (2000; Zbl 1032.11046)]). The main application is a comparison formula between Iwasawa invariants of (Selmer groups associated to) newforms \(f_1\) and \(f_2\) with the same residual representation (i.e., with \(\rho_{f_1}\) and \(\rho_{f_2}\) isomorphic modulo the uniformizer of \(\mathcal{O}\)). The main tool used in the proofs is a careful analysis of local cohomology groups (which has its effects on the exactness of the classical short sequence involving Selmer groups) in particular for primes \(v\nmid p\) splitting completely in \(F_\infty/F\) (a phenomenon which was not present in the cyclotomic \(\mathbb{Z}_p\)-extensions considered in the paper by Greenberg-Vatsal mentioned above). The author manages to provide a full description for these local cohomology groups as well, but then he has to impose a sort of vanishing condition for totally split primes to make the proofs of the main results work (roughly speaking the contribution of totally split primes is either 0 or way too big to be controlled). The paper ends with two nice appendices on the \(\mathcal{O}[[\Gamma]]\)-modulo structure of Iwasawa modules and of their duals.