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The Hurwitz problem for abelian differentials

Julien Boulanger, Rodolfo Gutiérrez-Romo, Erwan Lanneau

TL;DR

The work advances the Hurwitz problem for abelian differentials by refining the maximal translation-group bound $\mathsf{t}(g)$ and demarcating when genus-$g$ regular origamis realize the bound. It develops a precise group-theoretic framework (p-groups, semidirect products, Frobenius complements) to classify genera via the slope $\mathsf{c}(g)$ with $\mathsf{t}(g)=\mathsf{c}(g)(g-1)$, proving emptiness results for $\mathsf{G}(2)$ and $\mathsf{G}(2^\alpha-1)$ and constructing infinite families of genera using $\mathrm{PSL}(2,p)$-type translation groups. The paper also characterizes when regular origamis can exist in specific strata, and shows, in many cases, the nonexistence of regular origamis in infinite families of genera, while linking to Sophie Germain primes to produce explicit reg origami examples in certain genera. Collectively, the results illuminate the arithmetic and geometric structure of translation surfaces and provide infinite families of genera with and without genus-$g$ regular origamis, with broad implications for automorphism groups of Riemann surfaces endowed with abelian differentials.

Abstract

Fix $g \geq 2$. Let $\mathsf{t}(g)$ be the maximal order of the translation group among all genus-$g$ abelian differentials. By work of Schlage-Puchta and Weitze-Schmithüsen, $\mathsf{t}(g) \leq 4(g - 1)$. They also classify the $g$ attaining this bound. We assume $g$ is outside this class. We first prove that either $\mathsf{t}(g) = (2(m + 1) / m) (g - 1)$ for some $m \in \mathbb{N} \setminus \{0\}$, when regular genus-$g$ origamis exist, or $\mathsf{t}(g) = 2(g - 1)$, when they do not exist. In the former case, only some values of $m > 1$ are realizable; $m = 5$ is the smallest. The resulting set of genera, those satisfying $\mathsf{t}(g) = (12/5)(g - 1)$, contains infinitely long arithmetic progressions. The same holds for any odd prime $m$ congruent to $2$ modulo $3$. In the latter case, "many" strata of the form $\mathcal{H}(g - 1, g - 1)$, $\mathcal{H}(2k^q)$ or $\mathcal{H}(k^{2q})$, where $k \geq 1$ is an integer and $q$ is prime, contain no regular origamis; we derive a complete classification. As an application, we exhibit infinite families of genera $g$ for which $\mathsf{t}(g) = 2(g - 1)$: $g = p + 1$ for prime $p \geq 5$; $g = p^2 + 1$ for prime, but not Sophie Germain prime, $p$; and $g = pq + 1$, for distinct primes $p, q \geq 5$.

The Hurwitz problem for abelian differentials

TL;DR

The work advances the Hurwitz problem for abelian differentials by refining the maximal translation-group bound and demarcating when genus- regular origamis realize the bound. It develops a precise group-theoretic framework (p-groups, semidirect products, Frobenius complements) to classify genera via the slope with , proving emptiness results for and and constructing infinite families of genera using -type translation groups. The paper also characterizes when regular origamis can exist in specific strata, and shows, in many cases, the nonexistence of regular origamis in infinite families of genera, while linking to Sophie Germain primes to produce explicit reg origami examples in certain genera. Collectively, the results illuminate the arithmetic and geometric structure of translation surfaces and provide infinite families of genera with and without genus- regular origamis, with broad implications for automorphism groups of Riemann surfaces endowed with abelian differentials.

Abstract

Fix . Let be the maximal order of the translation group among all genus- abelian differentials. By work of Schlage-Puchta and Weitze-Schmithüsen, . They also classify the attaining this bound. We assume is outside this class. We first prove that either for some , when regular genus- origamis exist, or , when they do not exist. In the former case, only some values of are realizable; is the smallest. The resulting set of genera, those satisfying , contains infinitely long arithmetic progressions. The same holds for any odd prime congruent to modulo . In the latter case, "many" strata of the form , or , where is an integer and is prime, contain no regular origamis; we derive a complete classification. As an application, we exhibit infinite families of genera for which : for prime ; for prime, but not Sophie Germain prime, ; and , for distinct primes .

Paper Structure

This paper contains 29 sections, 71 theorems, 101 equations, 1 figure, 3 tables.

Table of Contents

  1. Introduction
  2. Context and motivations
  3. Background and preliminaries
  4. Basic facts about groups
  5. Semidirect products and their commutator subgroup.
  6. Basic facts about p-groups
  7. Frobenius p-complement theorem.
  8. Basic facts about regular origamis and t(g)
  9. Riemann--Hurwitz formula
  10. Proof of Theorem \ref{['thm:not_regular']}
  11. Building regular origamis
  12. Direct products
  13. Examples of regular origamis.
  14. The sets G(2) and G(2^ alpha - 1) for alpha >= 2 are empty
  15. The set G(2)
  16. ...and 14 more sections

Key Result

Theorem A

Let $g \geq 2$. The number $\mathsf{t}(g)$ is always of the form where the "slope" $\mathsf{c}(g)$ is either $2$, or has the form $2(m+1)/m$ for some integer $m = \mathsf{m}(g) \geq 1$ such that $m \mid 2(g - 1)$, $3 \nmid m$, and $4 \nmid m$. Finally, the case $\mathsf{c}(g)=2$ arises if and only if no genus-$g$ regular origamis exist.

Figures (1)

  • Figure 3.1: An origami in the stratum $\mathcal{H}_3(1^4)$ constructed as a regular cover over a torus with two marked points and possessing translation group $\mathbb{Z}/4\mathbb{Z}$.

Theorems & Definitions (132)

  • Theorem A: \ref{['thm:not_regular']}
  • Theorem B: \ref{['thm:G(2)_empty', 'thm:G(2^k-1)_empty']}
  • Theorem C: \ref{['thm:G_m_general']}
  • Remark 1.1
  • Theorem D: \ref{['thm:no_regular_origamis']}
  • Remark 1.2
  • Theorem E: \ref{['thm:no_regular_origamis', 'cor:p^2l+1']}
  • Remark 1.4
  • Theorem F
  • Remark 1.6
  • ...and 122 more