On Deligne's conjecture for symmetric fifth $L$-functions of modular forms

TL;DR

This paper proves Deligne's conjecture for symmetric fifth L-functions of certain elliptic modular forms, establishing important period relations and advancing understanding of special values of these L-functions.

Contribution

It provides the first proof of Deligne's conjecture for symmetric fifth L-functions of elliptic newforms of weight greater than 5, linking motivic and Betti-Whittaker periods.

Findings

Proved Deligne's conjecture for symmetric fifth L-functions of elliptic newforms.

Established period relations between motivic and Betti-Whittaker periods.

Enhanced understanding of special values of symmetric power L-functions.

Abstract

We prove Deligne's conjecture for symmetric fifth $L$-functions of elliptic newforms of weight greater than $5$. As a consequence, we establish period relations between motivic periods associated to an elliptic newform and the Betti-Whittaker periods of its symmetric cube functorial lift to ${\rm GL}_4$.