Courant-sharp eigenvalues of a two-dimensional torus

In this paper, we determine, in the case of the Laplacian on the flat two-dimensional square torus, all the eigenvalues having an eigenfunction which satisfies Courant's theorem with equality (Courant-sharp situation). Following the strategy of Pleijel (1956), the proof is a combination of a lower bound (à la Weyl) of the counting function, with an explicit remainder term, and of a Faber-Krahn inequality for domains on the torus (deduced as in Bérard-Meyer from an isoperimetric inequality), with an explicit upper bound on the area.