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

TL;DR

This paper derives formulas for the Greenberg-Benois $4$-invariant associated with Galois representations from Siegel-Hilbert modular forms, extending the understanding of these invariants in number theory.

Contribution

It introduces a new definition of the $4$-invariant for Galois representations over number fields and verifies its compatibility with Benois' existing definition.

Findings

Formulas for the $4$-invariant in specific cases

A new, simplified definition of the $4$-invariant for number field representations

Compatibility check with Benois' definition for induced representations

Abstract

We prove in some cases 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 $\mathrm{Ind}_F^{\mathbb{Q}}(V)$.