Co-periods and central symmetric cube L-values

TL;DR

This paper investigates co-period integrals related to automorphic forms on $ ext{GL}(2)$ and exceptional theta series on a cubic cover, establishing local uniqueness, global Euler decompositions, and conjectures linking these integrals to symmetric cube $L$-values.

Contribution

It introduces a new framework connecting co-period integrals with symmetric cube $L$-functions, including local multiplicity results and a conjectural global formula.

Findings

Hom-space is always one-dimensional in the local setting

Provides Euler decomposition for global co-period integrals

Proposes an Ichino-Ikeda type conjecture relating co-periods to symmetric cube $L$-values

Abstract

In this article, we study the co-period integral attached to an automorphic form on $\GL(2)$ and two exceptional theta series on the cubic Kazhdan-Patterson cover of $\GL(2)$. In the local aspect, we show the $\Hom$-space is always of one dimension and conduct the unramified calculations. In the global aspect, we give the Euler decomposition for the co-period integrals of Eisenstein series and propose an Ichino-Ikeda type conjecture relating the co-period integrals of cuspidal forms to the central critical value of symmetric cube $L$-functions. We also deduce from the local multiplicity one result that there exist cuspidal automorphic forms with prescribed local components and non-vanishing central symmetric cube $L$-values.