On symmetric power L-invariants of Iwahori level Hilbert modular forms

We compute the arithmetic L-invariants (of Greenberg-Benois) of twists of symmetric powers of p-adic Galois representations attached to Iwahori level Hilbert modular forms (under some technical conditions). Our method uses the automorphy of symmetric powers and the study of analytic Galois representations on p-adic families of automorphic forms over symplectic and unitary groups. Combining these families with some explicit plethysm in the representation theory of GL(2), we construct global Galois cohomology classes with coefficients in the symmetric powers and provide formulae for the L-invariants in terms of logarithmic derivatives of Hecke eigenvalues.