From a basic understanding of algebraic geometry (on the level of Gathmann's old lecture notes), algebraic number theory (on the level of the first chapter of Neukirch's Algebraic Number Theory), and some basics of elliptic curves including their moduli space (on the level of the first chapter of Silverman's Advanced Topics in the Arithmetic of Elliptic Curves), how does one get to the "Lego Teichmuller" theory described in the first few parts of Grothendieck's Esquisse d'un Programme and related research as described in this survey Braids, Galois Groups, and Some Arithmetic Functions by Ihara? (Another related survey is Schneps' Grothendieck’s “Long March through Galois theory”.) What subjects should be studied, i.e. what are the prerequisites for understanding this research?

P.S. I am not entirely sure that the subject I am referring to is in fact the one called "Lego Teichmuller". I am referring to the relations between the absolute Galois group $\text{Gal}(\bar{\mathbb{Q}}/\mathbb{Q})$ and geometric objects like Riemann surfaces, covering spaces, moduli spaces, and Teichmuller spaces. The projective line with three points removed $\mathbb{P}-\{0,1,\infty\}$ seems to occupy a special place in this subject. I believe the linked articles should convey better what I am looking for.

  • $\begingroup$ If I understand your question correctly, then I would recommend the paper by Guillot -- arxiv.org/abs/1309.1968 -- and would suggest you have a look at other introductory texts on dessins d'enfants (Jones-Wolfart, Girondo-Gonzalez Diaz are obvious suggestions). $\endgroup$ – Nick Gill Oct 13 '16 at 8:50

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