The $\mathcal{L}$-invariant, the dual $\mathcal{L}$-invariant, and families

Abstract

Given a rank two trianguline family of $(\varphi,\Gamma)$-modules having a noncrystalline semistable member, we compute the Fontaine--Mazur $\mathcal{L}$-invariant of that member in terms of the logarithmic derivative, with respect to the Sen weight, of the value at p of the trianguline parameter. This generalizes prior work, in the case of Galois representations, due to Greenberg--Stevens and Colmez.