Recovering Cusp forms on GL(2) from Symmetric Cubes

TL;DR

This paper investigates the uniqueness of cusp forms on GL(2) based on their symmetric cube lifts, establishing conditions under which forms are twist equivalent or related through special L-functions, with implications for automorphy and Galois representations.

Contribution

It proves that cusp forms with the same symmetric cube are either twist equivalent or linked by a pole in a specific L-function, advancing understanding of automorphic forms and their symmetric powers.

Findings

If two cusp forms share the same symmetric cube, they are either twist equivalent or linked by a pole in a degree 36 L-function.

Assuming automorphy of the symmetric fifth power, the forms are icosahedral in a specific sense.

The work connects symmetric power lifts to automorphy and Galois representations, providing criteria for form equivalence.

Abstract

Suppose $\pi$, $\pi'$ are cusp forms on GL$(2)$, not of solvable polyhedral type, such that they have the same symmetric cubes. Then we show that either $\pi$, $\pi'$ are twist equivalent, or else a certain degree $36$ $L$-function associated to the pair has a pole at $s=1$. If we further assume that the symmetric fifth power of $\pi$ is automorphic, then in the latter case, $\pi$ is icosahedral in a suitable sense, agreeing with the usual notion when there is an associated Galois representation.