Endoscopy and the Transfer from GSp(4) to GL(4)

TL;DR

This paper investigates the transfer of automorphic representations from GSp(4) to GL(4), demonstrating how L-functions behave and confirming the Langlands functoriality principle for endoscopic groups.

Contribution

It establishes the existence of specific L-functions with particular poles and clarifies the transfer process from GSO(4) to GSp(4) and then to GL(4), aligning with Langlands' conjectures.

Findings

L-functions of cuspidal GSp(4) representations can have a pole of order 2 at s=1

GSO(4) representations transfer to GSp(4) via Weil transfer

Data from GSO(4) transfers to GL(4) through endoscopic and twisted endoscopic transfers

Abstract

We show the existence of an L-functions of a cuspidal representation of GSp(4,A)*GSp(4,A) which has a pole of order 2 at s = 1, even for globally generic representations. However if \pi comes from GSO(4,A), then \pi? is the Weil transfer of \Pi_?1 \otimes \Pi_2 realized as a representation of GSO(4,A). This agrees with Langlands Functoriality principle as GSO(4) is an endoscopic group for GSp(4) and shows that data ?\Pi_?1 \otimes \Pi_2 on GSO(4) transfers to \Pi_?1 \boxplus \Pi_2 through the composite of the endoscopic transfer from GSO(4) to GSp(4) and the twisted endoscopic transfer from GSp(4) to GL(4).