On Greenberg's $L$-invariant of the symmetric sixth power of an ordinary cusp form

TL;DR

This paper derives a formula for Greenberg's $L$-invariant of the symmetric sixth power of an ordinary cusp form, connecting Galois deformation theory, automorphic lifts, and Hida theory to facilitate potential computations of these invariants.

Contribution

It provides a new explicit formula for Greenberg's $L$-invariant of symmetric sixth powers using advanced automorphic and Galois deformation techniques.

Findings

Expresses $L$-invariant in terms of derivatives of Frobenius eigenvalues.

Uses symmetric cube lift to $ ext{GSp}(4)$ and Hida theory for the derivation.

Suggests a method to compute $L$-invariants for all symmetric powers.

Abstract

We derive a formula for Greenberg's $L$-invariant of Tate twists of the symmetric sixth power of an ordinary non-CM cuspidal newform of weight $\geq4$, under some technical assumptions. This requires a "sufficiently rich" Galois deformation of the symmetric cube which we obtain from the symmetric cube lift to $\GSp(4)_{/\QQ}$ of Ramakrishnan--Shahidi and the Hida theory of this group developed by Tilouine--Urban. The $L$-invariant is expressed in terms of derivatives of Frobenius eigenvalues varying in the Hida family. Our result suggests that one could compute Greenberg's $L$-invariant of all symmetric powers by using appropriate functorial transfers and Hida theory on higher rank groups.