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- W1567205104 abstract "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) compo- nents from {H(Gp,k(G),C)} 1=0 , the (G,C,p) Modular Tower (MT). The classical modular curve towers {Y1(p k+1 )} 1=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." @default.
- W1567205104 created "2016-06-24" @default.
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- W1567205104 date "2006-11-20" @default.
- W1567205104 modified "2023-09-27" @default.
- W1567205104 title "THE MAIN CONJECTURE OF MODULAR TOWERS AND ITS HIGHER RANK GENERALIZATION" @default.
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