Derivatives and exceptional poles of the local exterior square L-function for $$GL_m$$

Abstract

Let $$\pi $$ be an irreducible admissible representation of $$GL_m(F)$$, where F is a non-archimedean local field of characteristic zero. In 1990’s Jacquet and Shalika established an integral representation for the exterior square L-function. We complete, following the method developed by Cogdell and Piatetski-Shapiro, the computation of the local exterior square L-function $$L(s,\pi ,\wedge ^2)$$ via the integral representation in terms of L-functions of supercuspidal representations by a purely local argument. With this result, we show the equality of the local analytic L-functions $$L(s,\pi ,\wedge ^2)$$ via the integral representation for the irreducible admissible representation $$\pi $$ for $$GL_m(F)$$ and the local arithmetic L-functions $$L(s, \wedge ^2(\phi (\pi )))$$ of its Langlands parameter $$\phi (\pi )$$ through local Langlands correspondence.