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Computing the Field of moduli of some non-hyperelliptic pseudo-real curves

Ruben A. Hidalgo

Abstract

The explicit computation of the field of moduli of a closed Riemann surface is, in general, a difficult task. In this paper, for each even integer $k \geq 2$, we consider a suitable $2$-real parameter family of non-hyperelliptic pseudo-real Riemann surfaces of genus $g=1+(2k-3)k^{4}$. For each of them, we compute its field of moduli and also a minimal field of definition.

Computing the Field of moduli of some non-hyperelliptic pseudo-real curves

Abstract

The explicit computation of the field of moduli of a closed Riemann surface is, in general, a difficult task. In this paper, for each even integer , we consider a suitable -real parameter family of non-hyperelliptic pseudo-real Riemann surfaces of genus . For each of them, we compute its field of moduli and also a minimal field of definition.
Paper Structure (20 sections, 8 theorems, 41 equations)

This paper contains 20 sections, 8 theorems, 41 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Field of moduli and fields of definition of curves
  4. The field of moduli of $C$
  5. Fields of definitions of $C$
  6. Weil's Galois descent theorem
  7. Generalized circles and cross-ratio
  8. A parameter space
  9. Generalized Fermat curves of type $(k,5)$
  10. The two-real-dimensional family $C^{(k)}_{r,\theta}$
  11. Computing the field of moduli of $C^{(k)}_{r,\theta}$
  12. A preliminary result
  13. Explicit computation of the field of moduli
  14. Some remarks
  15. The case $r=r_{\theta}$, $e^{i\theta} \notin \{\pm 1, \pm i\}$
  16. ...and 5 more sections

Key Result

Theorem 2.2

Let $C$ be a smooth and irreducible complex projective algebraic curve defined over a finite Galois extension ${\mathbb L}$ of a field ${\mathcal{M}}$. If for every $\sigma \in {\rm Aut}({\mathbb L}/{\mathcal{M}})$ there is an isomorphism $f_{\sigma}:C \to C^{\sigma}$, defined over ${\mathbb L}$, su

Theorems & Definitions (13)

  • Theorem 2.2: Weil's Galois descent theorem Weil
  • Theorem 2.3
  • proof
  • Lemma 3.2
  • proof
  • Lemma 3.5
  • proof
  • Lemma 3.6
  • proof
  • Theorem 3.7
  • ...and 3 more