On the $\mathcal L$-invariant of the adjoint of a weight one modular form

TL;DR

This paper proves the equality of two different $\\mathcal{L}$-invariants associated with the adjoint of a weight one modular form, linking algebraic and analytic definitions through class field theory and explicit regulators.

Contribution

It establishes the equality of algebraic and analytic $\mathcal{L}$-invariants for the adjoint of a weight one modular form, connecting different perspectives.

Findings

Algebraic and analytic $\mathcal{L}$-invariants are equal.

The algebraic invariant is expressed via a regulator of global units.

The analytic invariant matches the regulator through class field theory.

Abstract

The purpose of this article is proving the equality of two natural $\mathcal L$-invariants attached to the adjoint representation of a weigth one cusp form, each defined by purely analytic, respectively algebraic means. The proof departs from Greenberg's definition of the algebraic $\mathcal L$-invariant as a universal norm of a canonical $\mathbb{Z}_p$-extension of $\mathbb{Q}_p$ associated to the representation. We relate it to a certain $2\times 2$ regulator of $p$-adic logarithms of global units by means of class field theory, which we then show to be equal to the analytic $\mathcal L$-invariant computed by Rivero and the second author.