The Main Conjecture of Modular Towers and its higher rank generalization
| dc.creator | Fried, Michael D. | |
| dc.date | 2006-11-20 | |
| dc.date.accessioned | 2026-07-07T07:33:07Z | |
| dc.date.available | 2026-07-07T07:33:07Z | |
| dc.description | The genus of projective curves discretely separates decidedly different two variable algebraic relations. So, we can focus on the connected moduli M_g 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 \ge 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(G_{p,k}(G),C)}_{k=0}^{\infty}, the (G,C,p) Modular Tower (MT). The classical modular curve towers {Y_1(p^{k+1})}_{k=0}^\infty (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. | |
| dc.identifier | https://arxiv.org/abs/math/0611594 | |
| dc.identifier | http://arxiv.org/abs/math/0611594 | |
| dc.identifier | in Groupes de Galois arithmetiques et (Luminy 2004; eds. D. Bertrand and P. Dèbes), Sem. et Congres, Vol. 13 (2006), 165--230 | |
| dc.identifier.uri | http://salesiana.dossiersoluciones.com/handle/123456789/119347 | |
| dc.subject | Number Theory | |
| dc.subject | Primary 11F32, 11G18, 11R58; Secondary 20B05, 20C25, 20D25, 20E18, 20F34 | |
| dc.title | The Main Conjecture of Modular Towers and its higher rank generalization | |
| dc.type | text |