Le Bois Marie
35, route de Chartres
A classical result of Kronecker and Weber states that the value of the elliptic j-function at a point of complex multiplication (i.e. a point lying in the intersection of the upper half-plain and some imaginary quadratic field) is algebraic. B. Gross and D. Zagier have conjectured that a similar phenomenon also holds for certain modular eigenfunctions of the hyperbolic Laplace operator. Namely, the higher Green's functions are real-valued functions of two variables on the upper half-plane which are bi-invariant under the action of SL2(Z), have a logarithmic singularity along the diagonal and are eigenfunctions of the hyperbolic Laplace operator with eigenvalue k(1-k) for some positive integer k. The conjecture formulated in "Heegner points and derivatives of L-series'' (1986) predicts when the value of a higher Green's function at a pair of points of complex multiplication is equal to the logarithm of an algebraic number. In this talk I would like to present a proof of this conjecture for a pair of points both lying in the same imaginary quadratic field.