Introduction

Kleinian functions, also known as the generalised Wierstrass \(\wp_{ij}\) functions, are important tools for studying hyperelliptic curves. The main objective of this notebook is to present code that can compute Kleinian functions for a hyperelliptic curve of genus 2. While the primary focus is on the computation aspect, related concepts and background information are provided for context.

This notebook is mainly inspired by the work of Julia Bernatska [1] and the code attached to an article posted on WolframCommunity [2]. It also draws from the works of [3], [4], [4], and others referenced in later sections. Additionally, it references the SageMath Riemann surfaces documentation SageMath Riemann surfaces documentation [6]. All additional references are listed in the Literature section at the end of the notebook.

• 1. Preliminaries
• 2. Genus two with real brunch points case
• 3. Genus two with real coefficients

Bibliography

[1]

Julia Bernatska. Computation of \$\wp\$-functions on plane algebraic curves. July 2024. URL: https://arxiv.org/abs/2407.05632v3 (visited on 2024-10-17).

[2]

Uniformization of a genus 4 hyperelliptic curve with arbitrary complex branch points - Online Technical Discussion Groups—Wolfram Community. URL: https://community.wolfram.com/groups/-/m/t/3243472 (visited on 2024-10-17).

[3]

V.Z. Enolski, E. Hackmann, V. Kagramanova, J. Kunz, and C. Lämmerzahl. Inversion of hyperelliptic integrals of arbitrary genus with application to particle motion in general relativity. Journal of Geometry and Physics, 61(5):899–921, May 2011. URL: https://linkinghub.elsevier.com/retrieve/pii/S0393044011000027 (visited on 2024-10-28), doi:10.1016/j.geomphys.2011.01.001.

[4]

Julia Bernatska. Abelian function fields on Jacobian varieties. December 2024. arXiv:2412.05455 [math]. URL: http://arxiv.org/abs/2412.05455 (visited on 2025-01-06), doi:10.48550/arXiv.2412.05455.

[5]

Julia Bernatska. An analogue of Vélu's formulae in genus two. December 2024. arXiv:2412.10284 [math]. URL: http://arxiv.org/abs/2412.10284 (visited on 2025-01-06), doi:10.48550/arXiv.2412.10284.

[6]

Riemann matrices and endomorphism rings of algebraic Riemann surfaces - Plane and Space Curves. URL: https://doc.sagemath.org/html/en/reference/curves/sage/schemes/riemann_surfaces/riemann_surface.html#sage.schemes.riemann_surfaces.riemann_surface.RiemannSurface (visited on 2024-10-17).