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1. CMB Online first
A specialisation of the Bump-Friedberg $L$-function We study the restriction of the Bump-Friedberg integrals to affine
lines $\{(s+\alpha,2s),s\in\mathbb{C}\}$.
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 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 $L(\alpha+1/2,\pi)\neq
0$ and $L(1,\Lambda^2,\pi)=\infty$.
When $k$ is odd, the partial $L$-function is holmorphic in a
neighbourhood of $Re(s)\geq 1/2$ when $Re(\alpha)$ is
$\geq 0$.
Keywords:automorphic L functions Categories:11F70, 11F66 |
2. CMB 1999 (vol 42 pp. 263)
Mellin Transforms of Mixed Cusp Forms We define generalized Mellin transforms of mixed cusp forms, show
their convergence, and prove that the function obtained by such a
Mellin transform of a mixed cusp form satisfies a certain
functional equation. We also prove that a mixed cusp form can be
identified with a holomorphic form of the highest degree on an
elliptic variety.
Categories:11F12, 11F66, 11M06, 14K05 |