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

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 πcomes from GSO(4,A), then π? is the Weil transfer of Π_?1 \otimes Π_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 ?Π_?1 \otimes Π_2 on GSO(4) transfers to Π_?1 \boxplus Π_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).