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Crossing the transcendental divide: from Schottky groups to algebraic curves

Samantha Fairchild, Ángel David Ríos Ortiz

Abstract

Though the uniformization theorem guarantees an equivalence of Riemann surfaces and smooth algebraic curves, moving between analytic and algebraic representations is inherently transcendental. Our analytic curves identify pairs of circles in the complex plane via free groups of Möbius transformations called Schottky groups. We construct a family of non-hyperelliptic surfaces of genus $g\geq 3$ where we know the Riemann surface as well as properties of the canonical embedding, including a nontrivial symmetry group and a real structure with the maximal number of connected components (an $M$-curve). We then numerically approximate the algebraic curve and Riemann matrices underlying our family of Riemann surfaces.

Crossing the transcendental divide: from Schottky groups to algebraic curves

Abstract

Though the uniformization theorem guarantees an equivalence of Riemann surfaces and smooth algebraic curves, moving between analytic and algebraic representations is inherently transcendental. Our analytic curves identify pairs of circles in the complex plane via free groups of Möbius transformations called Schottky groups. We construct a family of non-hyperelliptic surfaces of genus where we know the Riemann surface as well as properties of the canonical embedding, including a nontrivial symmetry group and a real structure with the maximal number of connected components (an -curve). We then numerically approximate the algebraic curve and Riemann matrices underlying our family of Riemann surfaces.
Paper Structure (23 sections, 21 theorems, 43 equations, 7 figures, 1 algorithm)

This paper contains 23 sections, 21 theorems, 43 equations, 7 figures, 1 algorithm.

Table of Contents

  1. Introduction
  2. A new family of examples
  3. Numerical approximations
  4. Connection to other works
  5. Outline
  6. Preliminaries
  7. From Riemann surfaces to algebraic curves
  8. Geometry of Schottky groups
  9. Proof of main theorems
  10. Non-Hyperelliptic proof
  11. Explicit Computations
  12. Genus 3
  13. Genus 4
  14. Explicit circles and mappings
  15. Numerical Experiments
  16. ...and 8 more sections

Key Result

Proposition 1.1

BK13 The following Poincaré theta series, when convergent, form a basis of holomorphic differentials for $n=1,\ldots, g$ where $G/G_n$ are cosets with representatives given by all elements $f_{i_1}^{j_1}\cdots f_{i_k}^{j_k}$ where $i_k \neq n$ and $A_n, B_n$ are the fixed points of $f_n$.

Figures (7)

  • Figure 1: A genus 2 curve by gluing the circles $\mathsf{C}_j$ to $\mathsf{C}_j'$ for $j=1,2$ under Möbius transformations $f_1,f_2$.
  • Figure 2: Examples of the curves from \ref{['def:family']} for $g=2,3$ on the left and right respectively with fixed points $A_j$ and $B_j$ the attracting (resp. repelling) fixed points of $f_j$. Also on the left see \ref{['thm:maing2']} for the Weierstrass points $x_i$.
  • Figure 3: From left to right, three different limits sets $\Lambda$ for Fuchsian, classical, and general Schottky groups of genus 2. We represent $\Lambda$ in dark blue and the light blue circles are images of the isometric circles under the elements in $G$.
  • Figure 4: On the left is a fundamental domain for the action of the group $D_g = \langle r,h \rangle$. On the right, the transformations $f_j$ are formed by the reflections $\sigma, \sigma_1, \sigma_2, \sigma_3$ defined in \ref{['sec:maintheorems']}.
  • Figure 5: A representation of the real points of the genus 2 curve with $\eta = \frac{\pi}{12}$ and $\theta = \frac{\pi}{4}$ on the left. In the middle we zoom in near $x=0$, and on the right, we zoom further in near $x=-.09$ to see these curves create 3 separate smooth curves.
  • ...and 2 more figures

Theorems & Definitions (40)

  • Proposition 1.1
  • Definition 1.2
  • Theorem 1.3
  • Theorem 1.4
  • Theorem 1.5
  • Example 2.1
  • Definition 2.2
  • Theorem 2.3
  • Remark 2.4
  • Lemma 2.5
  • ...and 30 more