\(\mathcal L\)-invariant for Siegel-Hilbert forms (Q273551)

Summary: We prove a formula for the Greenberg-Benois \(\mathcal L\)-invariant of the spin, standard and adjoint Galois representations associated with Siegel-Hilbert modular forms. In order to simplify the calculation, we give a new definition of the \(\mathcal L\)-invariant for a Galois representation \(V\) of a number field \(F\neq \mathbb Q\); we also check that it is compatible with Benois' definition for \(\mathrm{Ind}_{F}^{\mathbb Q}(V)\).