Ideal class groups of division fields of elliptic curves and everywhere unramified rational points (Q6585297)

Denote the ideal class group of a number field \(F\) by \(\mathrm{Cl}(F)\). Fix a prime \(p\) and its power \(q = p^n\) for some positive integer \(n\). Let \[A = \mathrm{Cl}(\mathbb{Q}(\mu_p)) \otimes \mathbb{Z}_p,\] where \(\mu_p\) is the group of \(p\)-th roots of unity. The structure of \(A\) is a well studied issue (cf. [\textit{K. Ribet}, Invent. Math. 34, 151--162 (1976; Zbl 0338.12003)]).\N\NLet \(E\) be an elliptic curve over \(\mathbb{Q}\). Define \[A(E) = \mathrm{Cl}(\mathbb{Q}_E) \otimes \mathbb{Z}_p\] where \(\mathbb{Q}_E = \mathbb{Q}(E[p])\). Let the semisimplification of \(A(E)\) have the following decomposition \[A(E)^{\mathrm{ss}} = \bigoplus_M M^{\oplus \rho(M)}\] as irreducible \(\mathrm{Gal}(\mathbb{Q}_E/ \mathbb{Q})\)-modules. Motivated by [\textit{D. Prasad} and \textit{S. Shekhar}, Pac. J. Math. 312, No. 1, 203--218 (2021; Zbl 1487.11057)], the author asks the following question:\N\N\textbf{Question.} How large is the multiplicity \(\rho(M)\) for each irreducible component \(M\)?\N\NDefine a subgroup of \(E(\mathbb{Q})\) by \[E(\mathbb{Q})_{\mathrm{ur}} := \mathrm{Ker} \left( E(\mathbb{Q}) \to \prod_{\ell \text{ prime}} \frac{E(\mathbb{Q}_\ell^{\mathrm{ur}})}{q E(\mathbb{Q}_{\ell}^{\mathrm{ur}})} \right)\] where \(\mathbb{Q}_{\ell}^{\mathrm{ur}}\) is the maximal unramified extension of \(\mathbb{Q}_\ell\). Denote lenght of a \(\mathbb{Z}_p\)-module \(M\) by \(\mathrm{length}(M)\). The main result of the article states:\N\NTheorem. Put \(K = \mathbb{Q}(E[q])\) and \(G = \mathrm{Gal}(K / \mathbb{Q})\). If \(H^1 (G, E[q]) = 0\), then \[\mathrm{length} \left(\mathrm{Hom}_G (\mathrm{Cl}(K), E[q]) \right) \ge r\] where \(r = \mathrm{length} (E(\mathbb{Q})_{\mathrm{ur}} / q E(\mathbb{Q}))\). Moreover, assume that \(E[p]\) is an irreducible \(\mathrm{Gal}(\mathbb{Q}_E / \mathbb{Q})\)-module. Then \[\mathrm{ord}_p (h) \ge 2r,\] where \(h\) is the order of the group \(\mathrm{Cl}(K)\).