Beyond endoscopy for the Symmetric Cube L-function

TL;DR

This paper explores advanced trace formula techniques beyond traditional endoscopy to analyze the symmetric cube L-function associated with $GL_2$ automorphic forms, introducing new identities linking exponential sums to Kloosterman sums.

Contribution

It presents a novel approach to studying symmetric power representations using a trace formula beyond endoscopy, including a new identity connecting exponential sums to Kloosterman sums.

Findings

Establishes a new identity relating cubic exponential sums to Kloosterman sums.

Provides initial insights into the symmetric cube L-function via a non-endoscopic trace formula.

Extends techniques for automorphic forms over $Q$ with cube roots of unity.

Abstract

This paper is a first attempt at getting information on a symmetric power representation of a $GL_2$ automorphic form via a trace formula that is beyond endoscopic techniques. In particular, we study the symmetric third power representation for all forms over $\Q$ adjoined the cube roots of unity. A key tool needed in the study is an identity relating cubic exponential sums to Kloosterman sums. This very same identity is crucial to the fundamental lemma of a trace formula comparison in work of Mao and Rallis.