Derivatives of Frobenius and Derivatives of Hodge weights

TL;DR

This paper investigates derivatives of Frobenius and Hodge weights in Galois representations, generalizing key invariants and formulas, and establishing new auxiliary properties to support these generalizations.

Contribution

It introduces a generalized Fontaine-Mazur L-invariant and extends the Greenberg-Stevens-Colmez formula for families of Galois representations with triangulations.

Findings

Generalized Fontaine-Mazur L-invariant

Extended Greenberg-Stevens-Colmez formula

Proved projection vanishing properties

Abstract

In this paper we study the derivatives of Frobenius and the derivatives of Hodge weights for families of Galois representations with triangulations. We generalize the Fontaine-Mazur L-invariant and use it to build a formula which is a generalization of the Greenberg-Stevens-Colmez formula. For the purpose of proving this formula we show two auxiliary results called projection vanishing property and "projection vanishing implying L-invariants" property.