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

TL;DR

This paper computes the Fontaine--Mazur $ ext{L}$-invariant for a specific family of $( ext{phi}, ext{Gamma})$-modules, linking it to the logarithmic derivative of the trianguline parameter, extending previous Galois representation results.

Contribution

It generalizes the calculation of the Fontaine--Mazur $ ext{L}$-invariant to rank two trianguline families with noncrystalline semistable members.

Findings

Explicit formula for the $ ext{L}$-invariant in terms of the logarithmic derivative.

Extension of prior Galois representation results to $( ext{phi}, ext{Gamma})$-modules.

Provides a new tool for understanding the structure of trianguline 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.