On a conjecture of Hecke concerning elementary class number formulas

Let K be a totally real algebraic number field of class number h_K and L a totally imaginary quadratic extension of K of class number h_L. Hecke conjectured that there exists an elementary formula for the first factor h_L/h_K of the class number of L. The paper develops a theory which allows computation of h_L/h_K in terms of the periods of certain complex differential forms associated to a manifold defined in a natural way from K. Thus, Hecke's conjecture is reduced to the problem of finding elementary formulas for these periods. The essential idea of the proof consists of establishing a Kronecker limit formula for the non-analytic Eisenstein series for the Hilbert modular group for K.Research supported by NSF Grant GP 20538