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

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 ${\mathrm{GL}}_4$.