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

Canadian mathematical bulletin

Nadir Matringe¹ 

[1] Université de Poitiers, Laboratoire de Mathématiques et Applications, , Téléport 2 - BP 30179, Boulevard Marie et Pierre Curie, , 86962, Futuroscope Chasseneuil Cedex

关键词: automorphic L functions;   

DOI  :  10.4153/CMB-2015-014-1

学科分类：数学（综合）

来源: University of Toronto Press * Journals Division

【 摘 要 】

We study the restriction of the Bump-Friedberg integrals to affinelines ${(s+alpha,2s),sinmathbb{C}}$. 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 $GL(k,mathbb{A})$ for $mathbb{A}$ the adeles of a number field. When $k$ is even, we showthat for a cuspidal automorphic representation $pi$, the partial $L$-function $L^{lin,S}(s,pi,alpha)$ has a poleat $1/2$, 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 $L(alpha+1/2,pi)eq0$ and $L(1,Lambda^2,pi)=infty$.When $k$ is odd, the partial $L$-function is holmorphic in aneighbourhood of $Re(s)geq 1/2$ when $Re(alpha)$ is $geq 0$.

【 授权许可】

Unknown