Moduli space SL_2(ℤ)\ℂ for complex elliptic curves

Visualization of the moduli space interpretation SL_2(ℤ)\ℂ as a moduli space for complex elliptic curves.

The complex upper halfplane with SL_2(ℤ) has a moduli space interpretation as parametrizing complex elliptic curves.

What the project does

Given τ, the project visualizes the elliptic curve ℂ/Λ_τ where Λ_τ = ℤ + ℤ·τ, and generates the following plots:

1. Fundamental domain (moduli) plot

• Draws the standard SL_2(ℤ) fundamental domain 𝔽 ⊂ ℍ.
• Reduces the input τ to the representative τ_FD ∈ 𝔽 (via an explicit SL_2(ℤ) transformation) and marks τ_FD on the plot.

1. Lattice plot in the complex plane

• Plots lattice points of Λ_τ in ℂ and the fundamental parallelogram.
• Marks the spanning vectors ω1 = 1 (red) and ω2 = τ (blue).

1. Torus plot for ℂ/Λ_τ

• Displays a standard embedded torus as a topological model of the quotient ℂ/Λ_τ.
• Highlights the two generator cycles corresponding to ω1 and ω2 in red/blue.

1. 3D ℘-surface visualization

• Computes Weierstrass ℘(z) and ℘′(z) using explicit truncated lattice sums on a grid in a fundamental parallelogram (avoiding poles near lattice points).
• Plots the 3D objectz ↦ (Re(℘(z)), Im(℘(z)), Re(℘′(z))) ∈ ℝ^3with clear axis labels and title.

1. Automatic on-screen animation (30 frames)

• Runs automatically when you execute main_demo.m.
• For θ running from 0 to 2π (30 frames), displays the 3D plot ofz ↦ (Re(℘(z)), Im(℘(z)), cos(θ)·Re(℘′(z)) + sin(θ)·Im(℘′(z))) ∈ ℝ^3
• Shows each frame for ~0.5 seconds and stops after the final frame.

1. Real 2D Weierstrass curve plot (only if meaningful)

• Computes invariants g2(τ), g3(τ) from truncated Eisenstein sums.
• If the associated real model is numerically sensible (e.g. j(τ) is approximately real), plots the real affine curvey^2 = 4x^3 − g2 x − g3restricted correctly to the real locus (no spurious x-axis segments).
• The plotting guarantees the correct geometry: for each fixed y there are at most 3 real x-values (as expected from a cubic), and the default example τ = i has the correct qualitative shape.

Input format

• Main input: a single complex number tau with imag(tau) > 0 (a point in the upper half-plane).
• Set tau by editing the first lines of main_demo.m, e.g. tau = 1i for the square lattice.
• Optional speed/accuracy parameters (grid sizes, lattice truncation, tolerances, animation settings) are also set in main_demo.m and have safe defaults.

How to run

1. Run startup.m (adds the project folders to the MATLAB path).
2. Run main_demo.m.
3. Edit tau in main_demo.m and rerun to explore different elliptic curves.