Arithmetic coherent states and quantization theory

Abstract

Let f ( z ) = q κ ∑ m ⩾ 0 a m q m , with q = e 2 i π z and 0 < κ ⩽ 1 , be a holomorphic modular form of real weight τ + 1 , τ > − 1 , for the group Γ = SL ( 2 , Z ) and for an arbitrary multiplier; let s τ be the distribution on the half-line such that s τ ( t ) = ∑ m ⩾ 0 a m δ ( t − m − κ ) . Let D τ + 1 be the usual realization, in a Hilbert space H τ + 1 of functions on the half–line, of a representation from the projective discrete series of G = SL ( 2 , R ) (or the prolongation thereof in the case when − 1 < τ ⩽ 0 ). Then, the set of transforms s τ g = D τ + 1 ( g −1 ) s τ , g describing any set of representatives of G mod Γ, can be regarded as a set of coherent states for the representation under study. Analyzing appropriate operators in H τ + 1 by means of their diagonal matrix elements against the distributions s τ g brings to light, as a spectral-theoretic density, the convolution L-function L ( f ¯ ⊗ f , s ) . Much more can, and will, be said in a forecoming Note in the cases when τ = ± 1 2 . To cite this article: A. Unterberger, C. R. Acad. Sci. Paris, Ser. I 346 (2008).