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Dessins and Arithmetic

Final Report Summary - DESSARITH (Dessins and Arithmetic)

Dessins d'enfants were invented by Alexander Grothendieck as a way to study the absolute Galois group of the rational numbers. This Galois group determines the structure of the field of algebraic numbers, and Grothendieck's idea was to understand this group by an action on certain drawings on surfaces. It is these drawings that were called dessins d'enfants.

The action of the Galois group has proved to be mysterious; while some sparse theoretical results are known, it is still difficult to find more subtle Galois invariants. In a particularly fruitful collaboration with Professor John Voight (Dartmouth), the researcher gives detailed methods for computing dessins using Groebner bases, complex analytic methods, p-adic methods, and modular forms. The researcher has also explored many aspects of curves of low genus connected to dessins, notably in collaboration with Professors Reynald Lercier (Rennes) and Christophe Ritzenthaler (Marseille), and this has led to several new results in the area.

The Marie Curie project has laid the necessary groundwork for a comprehensive database of dessins of low degree and genus. Realizing this database is the researcher's next goal. It will be an influential resource and will allow the formulation and testing of new conjectures in the subject.

The researcher's website, which contains his papers and programs is: