$\mathscr{L}$-invariants of Artin motives

TL;DR

This paper computes Benois $ ext{L}$-invariants for weight 1 cuspforms and their adjoint representations, extending Gross' $p$-adic regulator to non-critical Artin motives and exploring the role of regular submodules in $p$-irregular cases.

Contribution

It extends the computation of Benois $ ext{L}$-invariants to non-critical Artin motives and analyzes the dependence on regular submodules in $p$-irregular cases using geometric and Hida theory insights.

Findings

Computed Benois $ ext{L}$-invariants for specific Artin motives.

Connected the choice of regular submodules to geometric parameters and $p$-refinements.

Identified how Hida theory and eigencurve geometry influence invariant detection.

Abstract

We compute Benois $\mathscr{L}$-invariants of weight $1$ cuspforms and of their adjoint representations and show how this extends Gross' $p$-adic regulator to Artin motives which are not critical in the sense of Deligne. Benois' construction depends on the choice of a regular submodule which is well understood when the representation is $p$-regular, as it then amounts to the choice of a ``motivic'' $p$-refinement. The situation is dramatically different in the $p$-irregular case, where the regular submodules are parametrized by a flag variety and thus depend on continuous parameters. We are nevertheless able to show in some examples, how Hida theory and the geometry of the eigencurve can be used to detect a finite number of choices of arithmetic and ``mixed-motivic'' significance.