Machine-Learning Dessins d'Enfants: Explorations via Modular and Seiberg-Witten Curves

Having learnt of the remarkable theorem of Bely ˇ ı [1] which relates the existence of algebraic models of Riemann surfaces to that of analytic properties of rational functions thereon, Grothendieck launched an entire programme [2] by pictorially representing 1 this structure as bipartite graphs (the dessin) drawn on the Riemann surface. He hypothesised dessins d'enfants in their current form as a conceptual representation of the absolute Galois group over the rationals, one the most mysterious and least understood objects in number theory. Subsequently, he developed a generalisation of Bely ˇ ı's theorem which extends the surfaces considered in the mapping to more general Riemann surfaces. Properties of the mapping are identified with combinatorial invariants of the dessin d'enfant graphs [2] (q.v.