Fonctions polylogarithmes, nombres polyzêtas et groupes pro-unipotents
For an eigenfunction of the Laplacian on a hyperbolic Riemann surface, the coefficients of the Fourier expansion are described as intertwining functionals. All intertwiners are classified. A refined growth estimate for the coefficients is given and a summation formula is proved.
We give a geometric interpretation of an arithmetic rule to generate explicit formulas for the Fourier coefficients of elliptic modular forms and their associated Jacobi forms. We discuss applications of these formulas and derive as an example a criterion similar to Tunnel's criterion for a number to be a congruent number.
For a cocompact group of we fix a real non-zero harmonic -form . We study the asymptotics of the hyperbolic lattice-counting problem for under restrictions imposed by the modular symbols . We prove that the normalized values of the modular symbols, when ordered according to this counting, have a Gaussian distribution.