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arXiv · 1804.04613

Derivatives and Exceptional Poles of the Local Exterior Square $L$-Function for $GL_m$

Yeongseong Jo
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Abstract

Let $\pi$ be an irreducible admissible representation of $GL_m(F)$, where $F$ is a non-archimedean local field of characteristic zero. We follow the method developed by Cogdell and Piatetski-Shapiro to complete the computation of the local exterior square $L$-function $L(s,\pi,\wedge^2)$ in terms of $L$-functions of supercuspidal representations via an integral representation established by Jacquet and Shalika in $1990$. We analyze the local exterior square $L$-functions via exceptional poles and Bernstein and Zelevinsky derivatives. With this result, we show the equality of the local analytic $L$-functions $L(s,\pi,\wedge^2)$ via integral integral representations 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 $\phi(\pi)$ via local Langlands correspondence.

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Yeongseong Jo. 2018-04-12. Derivatives and Exceptional Poles of the Local Exterior Square $L$-Function for $GL_m$. https://arxiv.org/abs/1804.04613

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