HTML and/or PDF files in the folder deflist-cov
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B(ranch) C(ycle) L(emma): A function on a compact projective curve is a map to projective one space. Many problems in arithmetic geometry translate to finding curves with particular types of functions over a given field K. The BCL formula constrains the possibility of such maps. It gives the exact definition field of various types of Hurwitz spaces and is a tool to decide what branch points might allow maps from curves to exist over K: Branch-Cycle-Lem.html

The R(egular) I(nverse) G(alois) P(roblem): An RIGP realization for a group G over Q produces a realization of G as a Galois group over every number field. The RIGP has a formulation entirely using Nielsen classes for inner and absolute Hurwitz spaces. For some fields making this result effective requires only knowing about Hurwitz space components. That is a first step in results toward the RIGP over the rationals. Debes' B(eckman)-B(lack) conjecture generalization is supported by results like Fried-Voelklein for PAC fields, and it enhances the role of Hilbert's Irreducibility Theorem in the RIGP. This file explains the most refined version of the RIGP – the Nielsen-RIGP – using the example of dihedral groups, contrasting the classically motivated case of involution realizations with the 1st year algebra production of their regular realizations. RIGP.html

Hurwitz Spaces: A Nielsen class is a generalization of the notion of the genus. Given that, a Hurwitz space is variety attached to a Nielsen class that generalizes the notion of the moduli of curves of a given genus. This definition is preliminary to that of Nielsen classes and braid group actions on them. It includes how a compact Riemann surface, and a nonconstant analytic function on it, produces a Hurwitz space component.
1. Why the abelian case is not a good model
2. A gamut of connectedness applications
3. The RIGP interpretation
4. Cover Notation
5. Grabbing a cover by its Branch Points
6. What more do you need to know?
Hurwitz-Spaces.html

Nielsen Classes: A Nielsen class is defined by two things:
1. a finite group G; and
2. a collection of conjugacy classes in G.
This gives a notion of sphere covers in the Nielsen class. An equivalence on covers – absolute and inner and their reduced versions are the most common – gives an equivalence on Nielsen classes and an attached Hurwitz space. Nielsen-Classes.html

Nielsen Classes Continued: This page continues Topics I and II of Nielsen-Classes.html with topics/examples on how Nielsen classes define families of covers.
1. Families of covers and the introduction of braids
2. Interpreting Hurwitz space components as braid orbits on Nielsen classes
3. Examples from dihedral groups and pure-cycles
4. Why do we need (the complication of) Nielsen classes?
Nielsen-ClassesCont.html

C(onway)F(ried)P(arker)V(oelklein) connectivity Theorem : The version of the CFPV in the Fried-Voelklein '91 paper (Mathematische Annalen) Appendix provides, for any finite group G, a covering group G*, and infinitely many G* c(onjugacy) c(lass) c(ollection)s (each with nonexplicit high-multiplicity) so the attached Hurwitz spaces are irreducible and defined over the rationals. Recent versions identify Hurwitz space components corresponding to cccs, without using the mysterious high-multiplicity and covering group assumptions. CFPV-Thm.html

sh(ift)-in(cidence)-matrix: True, any reduced Hurwitz space of four branch point covers is a Dessin d'enfants: An upper half-plane quotient and cover of the j-line branched at 0 (of ram. index 3), 1 (of ram. index 2) and infinity. Several things distinguish it from a general d'enfants.

First: It is a significant moduli space for regular realizations of finite groups as Galois groups.
Second: it has a natural pairing on its cusps.

This pairing produces the sh(ift)-in(cidence) matrix which graphically displays the components, their genuses, the nature of its cusps, and much more. Modular curves are the very special case where the groups are just dihedral groups. sh-Inc-Mat.html

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