A certain Dirichlet series of Rankin-Selberg type associated with the Ikeda lift of half-integral weight

TL;DR

This paper derives an explicit formula for a Rankin-Selberg type Dirichlet series linked to Siegel cusp forms of half-integral weight, using advanced relations and maps in the theory of automorphic forms.

Contribution

It extends the explicit formula for Rankin-Selberg series to half-integral weight Siegel cusp forms via novel methods involving Maass relations and Jacobi form maps.

Findings

Explicit formula involving infinite sums over fundamental discriminants

Generalization of known integral weight results to half-integral weight case

Use of generalized Maass relations and index-shift maps in calculations

Abstract

In this article we obtain an explicit formula for certain Rankin-Selberg type Dirichlet series associated to certain Siegel cusp forms of half-integral weight. Here these Siegel cusp forms of half-integral weight are obtained from the composition of the Ikeda lift and the Eichler-Zagier-Ibukiyama correspondence. The integral weight version of the main theorem had been obtained by Katsurada and Kawamura. The result of the integral weight case is a product of $L$-function and Riemann zeta functions, while half-integral weight case is a infinite summation over negative fundamental discriminants with certain infinite products. To calculate explicit formula of such Rankin-Selberg type Dirichlet series, we use a generalized Maass relation and adjoint maps of index-shift maps of Jacobi forms.