\(\mathcal L\)-invariant of symmetric powers of Tate curves (Q1016581)

In [Contemp. Math. 165, 149--174 (1994; Zbl 0838.11070)], \textit{R. Greenberg} defined a new invariant \({\mathcal L}(F_ \infty)\) for the \({\mathbb Z}_ p\)-extension \(F_ \infty\) of the field of \(p\)-adic numbers \({\mathbb Q}_ p\). In [Pure Appl. Math. Q. 5, No. 1, 81--125 (2009; Zbl 1252.11089)] the author made explicit a conjectural formula of the \(\mathcal L\)-invariant of symmetric powers of a Tate curve over a totally real field. In the paper under review, the formula for Greenberg's \(\mathcal L\)-invariant is proved when the symmetric power is of adjoint type, assuming a conjecture (Conjecture 0.1 in this paper) on the ring structure of a Galois deformation ring of the symmetric powers. The case \(n=1\) and \(F={\mathbb Q}\) of this conjecture, via the solution of the Shimura-Taniyama conjecture by Wiles and Taylor-Wiles, follows from \textit{M. Kisin}'s work [Invent. Math. 153, No. 2, 373--454 (2003; Zbl 1045.11029); ibid. 157, No. 2, 275--328 (2004; Zbl 1150.11020)]. With additional assumptions the conjecture is proved here for \(n=1\) and \(F\) totally real; namely, for an odd prime \(p\) and \(F\) a totally real field, it is considered an elliptic curve \(E_ {/F}\) over the ring of integers \(\mathcal O\) of \(F\) such that \(E_ {/F}\) has ordinary good reduction at every \(p\)-adic place of \(F\) and \(E\) does not have complex multiplication. Conjecture 0.1 is proved for \(n=1\) assuming Hilbert modularity of \(E\) over \(F\) and that: \begin{itemize} \item[{\(\bullet\)}] The \({\mathbb F}_ p\)-lineal Galois representation \(\bar{\rho}=(T_ p E \bmod p\)) is absolutely irreducible over \({\text{Gal}}(\bar{F}/F[\mu_ p])\). \item [{\(\bullet\)}] The semi simplification of \(\bar{\rho}\) restricted to the decomposition group \(D_ {\mathfrak p}\) is non-scalar for \({\mathfrak p}\mid p\).\end{itemize}