— Publication: In Groupes de Galois arithmetiques et differentiels (Luminy 2004; eds. D. Bertrand and P. Dèbes), Sem. et Congres, Vol. 13 (2006), 165–230. The genus of projective curves discretely separates decidedly different two variable algebraic relations. So, we can focus on the connected moduli Mg of genus g curves. Yet, modern applications require a data variable (function) on such curve...