$\mathscr{L}$-invariants and logarithm derivatives of eigenvalues of Frobenius

TL;DR

This paper investigates special $p$-adic Galois representations over local fields, extending Colmez's formula to a new context and including a degenerated version, contributing to understanding eigenvalues of Frobenius.

Contribution

It generalizes Colmez's formula to a class of $p$-adic Galois representations related to the exceptional zero conjecture, including a degenerated case.

Findings

Verification of the generalized Colmez's formula

Inclusion of a degenerated version of the formula

Insights into eigenvalues of Frobenius in the studied representations

Abstract

Let $K$ be a $p$-adic local field. In this work we study a special kind of $p$-adic Galois representations of it. These representations are similar to the Galois representations occurred in the exceptional zero conjecture for modular forms. In particular, we verify that a formula of Colmez can be generalized to our case. We also include a degenerated version of Colmez's formula.