Klein Quartic Curve¶
The Klein quartic curve is the central geometric object in the Choptyuk problem.
Definition¶
The Klein quartic is the algebraic curve defined by:
\[
x^3 y + y^3 z + z^3 x = 0 \subset \mathbb{CP}^2
\]
This is a smooth plane curve of degree 4 with remarkable symmetry properties.
Key Properties¶
| Property | Value |
|---|---|
| Genus | \(g = 3\) |
| Degree | 4 |
| Automorphism group | \(\mathrm{PSL}(2, 7)\) of order 168 |
| Scalar curvature | \(R = -2\) (hyperbolic metric) |
| First homology | \(H_1(\Sigma, \mathbb{Z}) \cong \mathbb{Z}^6\) |
| Euler characteristic | \(\chi = -4\) |
Automorphism Group¶
The automorphism group \(\mathrm{Aut}(\Sigma) \cong \mathrm{PSL}(2, 7)\) has order 168, which is the maximum possible for a genus-3 Riemann surface (Hurwitz bound: \(|\mathrm{Aut}| \leq 84(g-1) = 168\)). This makes the Klein quartic a Hurwitz surface.
The group \(\mathrm{PSL}(2, 7)\) is the simple group of order 168, isomorphic to \(\mathrm{GL}(3, 2)\). It has a presentation:
\[
\mathrm{PSL}(2, 7) = \langle a, b \mid a^2 = b^3 = (ab)^7 = [a, b]^4 = 1 \rangle
\]
Spinor Phases¶
The spinor phases are determined by the automorphism group:
\[
\delta_A = \frac{\pi}{2}, \quad \delta_B = \frac{\pi}{3}, \quad \delta_C = \frac{\pi}{7}
\]
These correspond to the orders of the generators \(a\), \(b\), and \(ab\) in the presentation above.
Computational Access¶
from src.core.klein_curve import KleinCurve
curve = KleinCurve()
print(f"Genus: {curve.genus}") # 3
print(f"Automorphism order: {curve.aut_order}") # 168
print(f"Scalar curvature: {curve.scalar_curvature}") # -2
print(f"δ_C = π/7 = {curve.delta_C:.6f}") # 0.448799
References¶
- Klein, F. "Über die Transformationen siebenter Ordnung der elliptischen Funktionen." Math. Ann. 14, 1879.
- Elkies, N. "The Klein quartic in number theory." The Eightfold Way, MSRI, 1998.