Les formes automorphes feuilletées

Abstract

We introduce a family $\{\mathcal {B}_s(\Gamma )\}_{s\in \mathbb C}$ of line bundles over the unit tangent bundle $\text {T}^1X_\Gamma$ of a hyperbolic surface $X_\Gamma$. These bundles vary in an holomorphic way when restricted to the leaves of the stable foliation and to the leaves of the unstable foliation. A stable (resp. unstable) foliated automorphic form of weight s on $\text {T}^1X_\Gamma$ is defined as a continuous section of $\mathcal {B}_s(\Gamma )\rightarrow \text {T}^1X_\Gamma$ which is holomorphic along the leaves of the stable (resp. unstable) foliation. We study mainly the case when $X_\Gamma$ is compact. In that situation we construct, from any $s(1-s)$-eigenfunction of the Laplace operator on $X_\Gamma$, two foliated automorphic forms, one of weight s and one of weight $1-s$. We give some general properties of the foliated automorphic forms ; in particular, we construct when $0<\mathfrak {R}s<1$, an isomorphism between the spaces $\mathcal {A}_s (\Gamma )$ et $\mathcal {A}_{1-s}(\Gamma )$ of the stable automorphic forms of respective weights s and $1-s$.