L-functions for GSp(4)xGL(2)in the case of high GL(2) conductor

TL;DR

This paper extends the integral representation of the GSp(4) x GL(2) L-function to cases where the GL(2) representation has high conductor, confirming it matches the local Euler factor in all cases and deriving a global special value result.

Contribution

It generalizes Furusawa's integral representation to high conductor GL(2) cases, providing new local and global L-function results.

Findings

The local zeta integral equals the local Euler factor for all conductors.

The work confirms the integral representation's validity in high conductor cases.

A global special value result for the L-function is obtained.

Abstract

Furusawa has given an integral representation for the degree 8 L-function of GSp(4) x GL(2) and has carried out the unramified calculation. The local p-adic zeta integrals were calculated in our earlier work under the assumption that the GSp(4) representation \pi is unramified and the GL(2) representation \tau has conductor p. In the present work we generalize to the case where the GL(2) representation has arbitrarily high conductor. The result is that the zeta integral represents the local Euler factor L(s,\pi \times \tau) in all cases. As a global application we obtain a special value result for a GSp(4) x GL(2) global L-function coming from classical holomorphic cusp forms with arbitrarily high level for the elliptic modular form.