Abstract
An algebraic number field K defines a maximal torus T of the linear group G=GL n . Let χ be a character of the idele class group of K, satisfying suitable assumptions. The χ-toric form are the functions defined on G Q Z A ⧹G A such that the Fourier coefficient corresponding to χ with respect to the subgroup induced by T is zero. The Riemann hypothesis is equivalent to certain conditions concerning some spaces of toric 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, χ) on the critical line. To cite this article: G. Lachaud, C. R. Acad. Sci. Paris, Ser. I 335 (2002) 219–222.
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