Iwasawa invariants for the false-Tate extension and congruences between modular forms
Let \(F=\mathbb{Q}(\mu_p)\) and let \(F^{\text{cyc}}\) be its cyclotomic \(\mathbb{Z}_p\)-extension (\(p\) an odd prime). Let \(f\) be a newform of weight \(k\geq 2\), level \(Np^r\) (with \((N,p)=1\)) and nebentypus character \(\chi\). Associated to \(f\) one has a two dimensional representation \(\rho_f: G_{\mathbb{Q}}\rightarrow \text{GL}_2(K_{f,\mathfrak{p}})\) (where \(K_{f,\mathfrak{p}}\) is the completion at a prime \(\mathfrak{p}|p\) of a number field \(K_f\) containing all the Fourier coefficients of \(f\)). Let \(T_f\) be a lattice in the representation space \(V_f\) of \(\rho_f\) and define \(A_f:=V_f/T_f\,\). For any pro-\(p\), \(p\)-adic Lie extension \(L/F\) one defines Selmer groups \(\text{Sel}(L,A_f)\) as subgroups of \(H^1(L,A_f)\) satisfying certain local conditions (as usual) and the goal of the paper is to study their structure as modules over the Iwasawa algebra \(\mathbb{Z}_p[[\text{Gal}(L/F)]]\) as \(f\) varies in a Hida family \({\mathbf H}(\overline{\rho})\) (where \(\overline{\rho}: G_{\mathbb{Q}}\rightarrow\text{GL}_2(\mathbb{F})\) is an absolutely irreducible residual representation into a finite field, note that in particular one has that \(f\in {\mathbf H}(\overline{\rho})\) yields \(\overline{\rho}_f=\overline{\rho}\)). For \(L=F^{\text{cyc}}\) using a deep result of \textit{K. Kato} [Astérisque 295, 117--290 (2004; Zbl 1142.11336)] which shows that the \(\text{Sel}(F^{\text{cyc}},A_f)\) are cotorsion modules, the author proves that if \(f,g\in {\mathbf H}(\overline{\rho})\) then \(\mu_f=0\) implies \(\mu_g=0\) and that \(\lambda_f=\lambda_g\) if they are in the same branch (where the \(\mu_*\) and \(\lambda_*\) are the classical Iwasawa invariants associated to the module \(\text{Sel}(F^{\text{cyc}},A_*)\,\)). For \(L=F_\infty:=\bigcup \mathbb{Q}(\mu_{p^n},m^{1/p^m})\) (i.e. the false-Tate extension) under the assumption that \(\text{Sel}(F_\infty,A_f)\) is cofinitely generated for some \(f\in {\mathbf H}(\overline{\rho})\), the author proves a ``control theorem'' which leads to the computation of the corank (over an appropriate Iwasawa algebra) of \(\text{Sel}(F_\infty,A_f)\) (which, for \(F_\infty\,\), plays the role of the \(\lambda\) invariant for \(F^{\text{cyc}}\)) in terms of \(\lambda_f\) and of the coranks of certain local cohomology groups. Then, by means of Galois cohomology and some deep theorems on the restriction of \(\rho_f\) to the decomposition groups \(G_l\) (\(l\) any prime) in \(G_{\mathbb{Q}}\,\), the author computes all those coranks and shows that if \(f,g\in {\mathbf H}(\overline{\rho})\), \(f\) and \(g\) are in the same branch and \(\mu_f=0\), then \[ \text{corank}\, \text{Sel}(F_\infty,A_f)=\text{corank}\, \text{Sel}(F_\infty, A_g)\;. \] The final section provides an example with a nice application to the proof of the Main Conjecture for the elliptic curve \(E=X_0(11)\) and the cusp form \(f_E\) associated to it.
scientific article; zbMATH DE number 5569030
scientific article