Sommes de Dedekind et généralisations de lʹéquation diophantienne de Markoff Original Research Article

The article uses a generalization of the structure of sequences appearing on the continued fraction development of algebraic numbers given by the markoff theory. Thanks to Dedekind sums and their reciprocity law, these sequences give birth to new diophantine equations and a solution for them. Their sets of solutions can then be described with the same methods as used in the classical theory. In the general case, the transposition of the conjecture of Cassels and Zagier is not valid. Some complement are given for the modular invariant of the group SL(2, Z). It is shown how to extend to GL(2, Z).