$\mathcal{L}$-invariant for Siegel-Hilbert forms

Abstract

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 $F\ne \mathbb{Q}$; we also check that it is compatible with Benois' definition for ${\mathrm{Ind}}_F^{\mathbb{Q}}(V)$.