A Specialisation of the Bump–Friedberg L-function

Matringe, Nadir

Canadian mathematical bulletin, Tome 58 (2015) no. 3, pp. 580-595

Résumé

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

Détail

DOI : 10.4153/CMB-2015-014-1

Mots-clés : 11F70, 11F66, automorphic L functions