The Spin $L$-function on ${\mathrm{GSp}}_6$ via a non-unique model

Abstract

We give two global integrals that unfold to a non-unique model and represent the partial Spin $L$-function on ${\mathrm{GSp}}_6$. We deduce that for a wide class of cuspidal automorphic representations $\ddot{I}€,$ the partial Spin $L$-function is holomorphic except for a possible simple pole at $s=1$, and that the presence of such a pole indicates that $\ddot{I}€$ is an exceptional theta lift from ${\mathrm{G}}_2$. These results utilize and extend previous work of Gan and Gurevich, who introduced one of the global integrals and proved these facts for a special subclass of these $\ddot{I}€$ upon which the aforementioned model becomes unique. The other integral can be regarded as a higher rank analogue of the integral of Kohnen-Skoruppa on ${\mathrm{GSp}}_4$.

final version, to appear in American Journal of Mathematics