The Main Conjecture of Modular Towers and its higher rank generalization

TL;DR

This paper explores the structure of Modular Towers, their connection to algebraic curves, and formulates conjectures about the absence of rational points at high levels, extending classical modular curve concepts to higher ranks.

Contribution

It introduces a higher rank generalization of the Main Conjecture for Modular Towers and develops methods to identify component branches and cusp properties related to the conjecture.

Findings

Identified criteria for component branches on Modular Towers.

Listed cusp branch properties that support the Main Conjecture.

Formulated a strong Main Conjecture for higher rank Modular Towers.

Abstract

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.