Intégrales-fibres pour une formule de Chowla - Selberg

The Chowla-Selberg formula links the class number of negative fundamental quadratic discriminant to values of the Dedekind eta function evaluated over representatives of the class group. It enables the explicit calculation of a large class of elliptic integrals, especially the complete elliptic integrals of the first kind. In this work, we use the integral-fibre theory to offer a new proof of a significant case of this fundamental formula. The method we present here seems susceptible of getting a direct analytic access to a generalization of the explicit instances of the Chowla-Selberg formula in higher dimensions.