Z\'eros des fonctions L et formes toro\"idales

An algebraic number field $K$ defines a maximal torus $T$ of the linear group $G=G{L}_n$. Let $\chi$ be a character of the idele class group of $K$, satisfying suitable assumptions. The $\chi$-toroidal forms are the functions defined on $G(\mathbf{Q})Z(\mathbf{A})\mathrm{\backslash }G(\mathbf{A})$ such that the Fourier coefficient corresponding to $\chi$ with respect to the subgroup induced by $T$ is zero. The Riemann hypothesis is equivalent to certain conditions concerning some spaces of toroidal forms, constructed from Eisenstein series. Furthermore, we define a Hilbert space and a self-adjoint operator on this space, whose spectrum equals the set of zeroes of $L(s,\chi )$ on the critical line.