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Classifying compact Riemann surfaces by number of symmetries

Sebastián Reyes-Carocca, Pietro Speziali

TL;DR

The paper determines when a genus $g$ compact Riemann surface is uniquely determined by possessing an automorphism group of order $3g$ or $3g+3$, identifying explicit models $C_{g,2}$, $C_{g,4}$, and $C_{g,1}\cong C_{g,3}$ that realize these extremal symmetries. It analyzes the automorphism groups and full Jacobian decompositions of these surfaces, linking the group actions to precise signatures and to orientably-regular hypermaps. For odd $g\neq 21$, the $3g$-order case yields a unique surface up to isomorphism, while for even $g$ the $3g+3$-order case similarly yields a unique representative among certain congruence classes; Jacobians decompose into explicit products of abelian subvarieties with dimensions governed by Euler totients. The work also provides complete classifications in several subcases (e.g., $g=2p$ and $g=4p-1$ with prime $p$), presents isogeny decompositions tied to quotients by canonical automorphisms, and reformulates results in terms of orientably-regular hypermaps. The findings deepen the understanding of how symmetry order constrains the moduli of Riemann surfaces and their Jacobians, with implications for explicit algebraic models and for the theory of dessins d’enfants.

Abstract

In this article we consider compact Riemann surfaces that are uniquely determined by the property of possessing a group of automorphisms of a prescribed order, strengthening uniqueness results proved by Nakagawa. More precisely, we deal with the cases in which such an order is $3g$ and $3g+3,$ where $g$ is the genus. We prove that if $g$ is odd (respectively $g$ even and $g \not \equiv 2 \mbox{ mod } 3$) then there exists a unique Riemann surface of genus $g$ with a group of automorphisms of order $3g$ (respectively $3g+3$). A similar conclusion can be derived in terms of orientably-regular hypermaps. In addition, we determine the full automorphism group of such Riemann surfaces and provide decompositions of their Jacobians.

Classifying compact Riemann surfaces by number of symmetries

TL;DR

The paper determines when a genus compact Riemann surface is uniquely determined by possessing an automorphism group of order or , identifying explicit models , , and that realize these extremal symmetries. It analyzes the automorphism groups and full Jacobian decompositions of these surfaces, linking the group actions to precise signatures and to orientably-regular hypermaps. For odd , the -order case yields a unique surface up to isomorphism, while for even the -order case similarly yields a unique representative among certain congruence classes; Jacobians decompose into explicit products of abelian subvarieties with dimensions governed by Euler totients. The work also provides complete classifications in several subcases (e.g., and with prime ), presents isogeny decompositions tied to quotients by canonical automorphisms, and reformulates results in terms of orientably-regular hypermaps. The findings deepen the understanding of how symmetry order constrains the moduli of Riemann surfaces and their Jacobians, with implications for explicit algebraic models and for the theory of dessins d’enfants.

Abstract

In this article we consider compact Riemann surfaces that are uniquely determined by the property of possessing a group of automorphisms of a prescribed order, strengthening uniqueness results proved by Nakagawa. More precisely, we deal with the cases in which such an order is and where is the genus. We prove that if is odd (respectively even and ) then there exists a unique Riemann surface of genus with a group of automorphisms of order (respectively ). A similar conclusion can be derived in terms of orientably-regular hypermaps. In addition, we determine the full automorphism group of such Riemann surfaces and provide decompositions of their Jacobians.
Paper Structure (10 sections, 9 theorems, 153 equations)

This paper contains 10 sections, 9 theorems, 153 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Statement of the results
  4. Orientably-regular hypermaps
  5. Proof of Theorem \ref{['unicidad']}
  6. Proof of Theorem \ref{['aut']}
  7. Proof of Theorem \ref{['jaco']}
  8. Proof of Theorem \ref{['gpar']}
  9. Proof of Theorem \ref{['final']}
  10. Proof of Theorem \ref{['gimpar']}

Key Result

Theorem 3.1

Let $X$ be a compact Riemann surface of genus $g \geqslant 3.$ Assume $g$ odd and different from 21. Then $X$ is endowed with a group of automorphisms of order $3g$ if and only if Assume $g$ even. Then $X$ is endowed with a group of automorphisms of order $3g+3$ if and only if

Theorems & Definitions (15)

  • Theorem 3.1
  • Theorem 3.2
  • Corollary 3.3
  • Theorem 3.4
  • Proposition 3.5
  • Theorem 3.6
  • Theorem 3.7
  • Theorem 3.8
  • Theorem 4.1
  • Remark 7.1
  • ...and 5 more