2006/11/20 by Michael D. Fried · 1 citation
Mathematics · #math.NT #msc:11F32 #msc:11G18 #msc:11R58 #msc:20B05 #msc:20C25 #msc:20D25 #msc:20E18 #msc:20F34
published as in Groupes de Galois arithmetiques et (Luminy 2004; eds. D. Bertrand and P. Dèbes), Sem. et Congres, Vol. 13 (2006), 165--230
arxiv created 2006/11/20 · arxiv updated 2009/12/01
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 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 says, if G is p-perfect, there are no rational points at high levels of a component branch. When r=4, MT levels (minus their cusps) are upper half plane quotients covering the j-line. Our topics. * Identifying component branches on a MT from g-p', p and Weigel cusp branches using the MT generalization of spin structures. * Listing cusp branch properties that imply the (weak) Main Conjecture and extracting the small list of towers that could possibly fail the conjecture. * Formulating a (strong) Main Conjecture for higher rank MTs (with examples): almost all primes produce a modular curve-like system.