On the analytic properties of a cubic Dirichlet series associated to a cubic metaplectic form

TL;DR

This paper investigates the analytic continuation of a cubic Dirichlet series linked to a metaplectic form over a cubic cover of GL_2, extending classical results and employing Langlands's beyond endoscopy approach.

Contribution

It establishes nontrivial analytic continuation of the cubic Dirichlet series using advanced number theory techniques, even without full Shimura correspondence.

Findings

Achieves analytic continuation to Re(s)>9/7+ε for the cubic series.

Provides an asymptotic formula for a spectral sum of the series.

Identifies a key identity relating cubic exponential sums to Kloosterman sums.

Abstract

In this paper we study the analytic properties of a certain cubic Dirichlet series associated to a metaplectic form $f$ over the cubic cover of $GL_2.$ Such a sum generalizes the work of Shimura in studying a similar quadratic Dirichlet series for a half-weight modular form $f.$ Shimura connects the analytic properties of his Dirichlet series to the L-function of a holomorphic modular form via a converse theorem. This connection, and its higher cover generalizations, has been given the name: Shimura's correspondence. Even assuming Shimura's correspondence for the cubic cover of $GL_2,$ the analytic properties of our cubic Dirichlet series are intractable. However, using Langlands's beyond endoscopy idea and analytic number theory, we get nontrivial analytic continuation of the series. Specifically, we obtain an asymptotic for a spectral sum of these cubic Dirichlet series plus an error term. Assuming a certain uniformity hypothesis we can get analytic properties of an individual cubic Dirichlet series of a metaplectic form. In particular we show the cubic series has analytic continuation to $\Re(s)>\frac{9}{7}+\epsilon,$ for any $\epsilon$ with at most a pole at $s=\frac{3}{2}$ if the metaplectic form $f$ is the residual Eisenstein series. A key tool needed in studying this series is an identity relating cubic exponential sums to Kloosterman sums. While we do not make a traditional trace formula comparison in this paper, this very same identity is crucial to the fundamental lemma in work of Mao and Rallis.