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One-dimensional strata of residueless meromorphic differentials

Myeongjae Lee, Guillaume Tahar

TL;DR

This work analyzes residueless genus-one meromorphic differentials on elliptic curves, showing the projectivized residueless locus has a canonical complex projective structure and is 1-dimensional. Using a multi-scale compactification, the authors compute the genus and Euler characteristics of connected components, and determine the degree of the forgetful map to $\mathcal{M}_{1,1}$, with explicit refinement by rotation number. They classify connected components of residueless loci, describe the associated cellular decomposition via an equatorial net, and count 0-,1-,2-cells to obtain the orbifold Euler characteristic; these methods yield precise formulas and prove the exceptional stratum $R_1(12,-3,-3,-3,-3)$ has exactly two non-hyperelliptic components with rotation number $3$, corresponding to cosets in $S_4/A_4$. The results generalize modular-curve phenomena to generalized strata, connect to Lamé-function geometry, and illuminate boundary combinatorics through a robust period-coordinate framework.

Abstract

In projectivized strata of meromorphic $1$-forms on elliptic curves with only one zero, the locus of residueless differentials is a complex curve endowed with a canonical complex projective structure. Drawing on the multi-scale compactification of strata, we provide formulas to compute the genus of these curves and the degree of their natural forgetful map to $\mathcal{M}_{1,1}$. Additionally, we distinguish two non-hyperelliptic components in the residueless locus of the exceptional stratum $\mathcal{H}_{1}(12,-3,-3,-3,-3)$ and hence complete the classification of the connected components of these loci.

One-dimensional strata of residueless meromorphic differentials

TL;DR

This work analyzes residueless genus-one meromorphic differentials on elliptic curves, showing the projectivized residueless locus has a canonical complex projective structure and is 1-dimensional. Using a multi-scale compactification, the authors compute the genus and Euler characteristics of connected components, and determine the degree of the forgetful map to , with explicit refinement by rotation number. They classify connected components of residueless loci, describe the associated cellular decomposition via an equatorial net, and count 0-,1-,2-cells to obtain the orbifold Euler characteristic; these methods yield precise formulas and prove the exceptional stratum has exactly two non-hyperelliptic components with rotation number , corresponding to cosets in . The results generalize modular-curve phenomena to generalized strata, connect to Lamé-function geometry, and illuminate boundary combinatorics through a robust period-coordinate framework.

Abstract

In projectivized strata of meromorphic -forms on elliptic curves with only one zero, the locus of residueless differentials is a complex curve endowed with a canonical complex projective structure. Drawing on the multi-scale compactification of strata, we provide formulas to compute the genus of these curves and the degree of their natural forgetful map to . Additionally, we distinguish two non-hyperelliptic components in the residueless locus of the exceptional stratum and hence complete the classification of the connected components of these loci.
Paper Structure (38 sections, 24 theorems, 47 equations, 10 figures)

This paper contains 38 sections, 24 theorems, 47 equations, 10 figures.

Table of Contents

  1. Introduction
  2. Main results
  3. Relations with other works
  4. Modular curves
  5. Generalized strata
  6. Lamé functions
  7. Isoresidual fibration
  8. Organization of the paper
  9. Acknowledgements.
  10. Geometry of translation surfaces
  11. Translation structures
  12. Local model of singularities
  13. Volume
  14. Rotation number of translation structures
  15. Hyperelliptic involutions
  16. ...and 23 more sections

Key Result

Theorem 1.1

The residueless stratum ${\mathcal{R}}_1(12,-3,-3,-3,-3)$ has six connected components. More precisely, it has:

Figures (10)

  • Figure 1: Types of cells
  • Figure 2: An isomorphism of a surface in the cell of cylinder type
  • Figure 3: The translation surface corresponding to an orbifold point of order 3 in a cell of 1-triangle / 3-degenerate type
  • Figure 4: The translation surface corresponding to an orbifold point of order 2 in a cell of 4-degenerate type
  • Figure 5: Dual graph of $\overline{X}$
  • ...and 5 more figures

Theorems & Definitions (52)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Theorem 1.4
  • Remark 1.5
  • Remark 2.1
  • Definition 2.2
  • Definition 2.3
  • Definition 2.4
  • Definition 2.5
  • ...and 42 more