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Dessins and Arithmetic (DessArith)
Date du début: 1 sept. 2012, Date de fin: 31 août 2014 PROJET  TERMINÉ 

"A dessin d’enfant is a type of graph drawing used to study algebraic groups and to provide combinatorial invariants for the action of the absolute Galois group. This project aims to develop new algorithms for computing dessins d’enfants and use these and other methods to attack Diophantine equations. In particular, we are interested in the Generalized Fermat Equation, which has been described by Darmon as the ""new holy grail of number theory"" in the sense that it replaces Fermat's Last Theorem as the major unsolved Diophantine problem. Our aim is to provide a practical method of translating any specific case of the generalized Fermat equation to a problem of determining rational points on curves. The field of curves is a rich one with many eminent practitioners and this translation is the surest way to provide progress for the generalized Fermat equation. The tool for this translation will be our dessins computed by our new algorithm."