The Main Conjecture of Modular Towers and Its Higher Rank Generalization

— 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 curves. The resulting spaces are versions, depending on our need from this data variable, of Hurwitz spaces. A Nielsen class (§1) is a set defined by r ≥ 3 conjugacy classes C in the data variable monodromy G. It gives a striking genus analog. Using Frattini covers of G, every Nielsen class produces a projective system of related Nielsen classes for any prime p dividing |G|. A nonempty (infinite) projective system of braid orbits in these Nielsen classes is an infinite (G,C) component (tree) branch. These correspond to projective systems of irreducible (dim r − 3) components from {H(Gp,k(G),C)} ∞ k=0 , the (G,C, p) Modular Tower (MT). The classical modular curve towers {Y1(pk+1)}∞k=0 (simplest case: G is dihedral, r = 4, C are involution classes) are an avatar. The (weak) Main Conjecture 1.2 says, if G is p-perfect, there are no rational points at high levels of a component branch. When r = 4, MTs (minus their cusps) are systems of upper half plane quotients covering the j-line. Our topics. – §3 and §4: Identifying component branches on a MT from g-p, p and Weigel cusp branches using the MT generalization of spin structures. – §5: Listing cusp branch properties that imply the (weak) Main Conjecture and extracting the small list of towers that could possibly fail the conjecture. – §6: Formulating a (strong) Main Conjecture for higher rank MTs (with examples): almost all primes produce a modular curve-like system.