A specialisation of the Bump-Friedberg $L$-function

Abstract

We study the restriction of the Bump-Friedberg integrals to affine lines $\{(s+α,2s),s\in\C\}$. It has a simple theory, very close to that of the Asai $L$-function. It is an integral representation of the product $L(s+\alpha ,\pi )L(2s,{\Lambda }^2,\pi )$ which we denote by $L^{lin}(s,\pi ,\alpha )$ for this abstract, when $\pi$ is a cuspidal automorphic representation of $GL(k,A)$ for $A$ the adeles of a number field. When $k$ is even, we show that for a cuspidal automorphic representation $\pi$, the partial $L$-function $L^{lin,S}(s,\pi ,\alpha )$ has a pole at 1/2, if and only if $\pi$ admits a (twisted) global period, this gives a more direct proof of a theorem of Jacquet and Friedberg, asserting that $\pi$ has a twisted global period if and only if $L(\alpha +1/2,\pi )\ne 0$ and $L(1,{\Lambda }^2,\pi )=\infty$. When $k$ is odd, the partial $L$-function is holmorphic in a neighbourhood of $Re(s)\ge 1/2$ when $Re(\alpha )$ is $\ge 0$.

The paper is under the form that will appear in Canad. Math. Bull

References in corpus (1)

• On the local and global exterior square L-functions

Cited by in corpus (2)

• Model transition for representations of metaplectic type
• Generic representations of GL(n) over a p-adic field distinguished by a maximal Levi subgroup