Click on any of the [ 10] items below.

1. Table of Contents for the Introduction and Chapters from Riemann's Existence Theorem: An elementary approach to moduli. Commissioned by Cambridge University Press. To Table of Contents! tableofcontents.pdf

2. CollectionChaps1-4.add: An Elementary approach to Moduli These chapters cover Riemann's existence Theorem to describe covers of the sphere, based on Nielsen classes prepped for braid action. They include many exercises, and descriptions of classical constructions of differentials, vector bundles, and deformation data for construction of Hurwitz spaces of covers in a Nielsen class. It is self-contained starting from chapter/verse quotes of Ahlfors graduate text on Complex variables. This text provides all necessary background for reading my book

Monodromy, l-adic Representations & the Regular Inverse Galois Problem.

That book is moving to completion nicely, and it has a published preliminary description available at In "Teichmuller theory and its impact," in the Nankai Series in Pure, Applied Mathematics and Theoretical Physics, the World Scientific Company (2018).

Below I have left a file of separated chapters, each with its own abstract. The differences: Chap. 4 is more complete here. Also Chapters 1, 2 and 3 below have their separate html files (at the end of their abstracts) giving ways to approach the chapters.

The chapter titles:
  1. Chapter 1: Prelude
  2. Chapter 2: Analytic Continuation
  3. Chapter 3: Complex Manifolds and Covers
  4. Chapter 4: Riemann's Existence Theorem
Unlike Helmut Volklein's book, which aims far more for the group theorist who wants to tackle the specifics of the Regular Inverse Galois Problem, these books aim for geometer/number theorists who need to understand the moduli of specific Hurwitz spaces. CollectionChaps1-4.pdf

3. Chapter 1: Scope of the Existence Theorem: An over view of the whole book: Joining Modular Towers and the Grothendieck-Teichmuller group for their ability to create useful moduli spaces. Includes the relation between classical theta functions, Eisenstein series and the Inverse Galois problem. Last revision: 10/10/02: prelude.html %-%-% prelude.pdf

4. Chapter 2: Analytic Continuation: The first text chapter, on Analytic continuation and an introduction to algebraic functions. Rooted solidly in quotations from Ahlfors. Latest Revision: 02/05/03. chpanal.html %-%-% chpanal.pdf

5. Chapter 3: Complex Manifolds and Covers: Introduces coordinates on a Riemann surface, and sufficient algebraic geometry to consider manifold compactifications of common Riemann surfaces. Aims directly at introducing Riemann's favorite subject -- necessary for his solution to the Jacobi Inversion Problem -- half-canonical classes. The detail on covering spaces, Galois covers and flat bundles goes beyond what is usual for a truly graduate level book. Latest Revision: 02/05/03. chpfund.pdf

6. Chapter 4: Riemann's Existence Theorem: I temporarily divided this into two parts: chpret.pdf the first part is in good shape, while the second part chpret2.pdf is still undergoing revision. The proof, combinatorics of its use (including Braid and Hurwitz monodromy group manipulations), and the algebra of coordinates attached to Riemann's Existence Theorem. We give a non-traditional approach to Abel's Theorem for genus 1 curves. This treatment of the j(τ) and λ(τ) functions and modular curves of complex variables motivates Chap. 5: Hurwitz monodromy and the development of Modular Towers. chpret4-firsthalf.pdf

7. A three page bibliography. bib.pdf

8. The slides for the talk: What Gauss told Riemann about Abel's Theorem! Talk given at the Florida Mathematics History lecture, during a semester in Honor of John Thompson's 70th birthday in 2003. flortalk.pdf

9. SiegelsTheoremPartI: Let C be a projective nonsingular (normal) curve, with C0 an affine open subset of it. This and Part II are notes I wrote up from reading Siegel's Theorem – end of my 3rd year of graduate school. I presented this in a seminar at a Bowdoin conference on Algebraic Geometry that summer, just before my 2-year post-doctoral at the Institute for Advanced Study. It is not verbatim Siegel's version – in German – of his famous finiteness theorem on quasi-integral points on C0. There were two pieces: if the genus, gC, of C exceeds 0; and a precise statement of a similar kind describing the exceptions if gC=0. I was aided by a partial verbatim translation into English by William LeVeque. My revisions were part of my initial budding research programs. Especially how I applied Weil's distribution Theorem to versions of Hilbert's Irreducibility Theorem.

SiegelsTheoremPartI.pdf

10. SiegelsTheoremPartII: Proof of Siegel's Theorem on a curve, of genus greater than 1, over a number field: Section Contents.

  1. Siegel's Fundamental Inequalities; modulo Statement A the proof of Siegel's Theorem
  2. Abel's Theorem, Jacobi Inversion
  3. Theta Functions, Riemann's Inversion Theorem
  4. Assuming Mordell-Weil, the proof of Statement A
SiegelsTheoremPartII.pdf