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Family: genus-zero-curve — Genus 0 curve #23

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Family: genus-zero-curve — Genus 0 curve #23

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22 changes: 22 additions & 0 deletions diophantine_classifier/data/families/genus-zero-curve.yaml
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# Family: Genus 0 curve
# Schema: see data/families/README.md; prose documentation in docs/FAMILIES.md.
# References are keys into data/references.bib, each with a 'why' annotation.
slug: genus-zero-curve
name: Genus 0 curve
priority: 1
status: algorithmic
class: curve
form: C(x, y) = 0 irreducible of genus 0
parents:
- general-curve
matcher: true
methods:
- conic reduction
- parametrization
- Pell-type analysis of integral points
software:
sage: Conic; Curve.parametrization
magma: Parametrization
references:
- key: PoulakisVoskos2000
why: practical algorithms for genus-zero equations
9 changes: 9 additions & 0 deletions docs/FAMILIES.md
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Expand Up @@ -210,6 +210,15 @@ PARI `hyperellratpoints`, Magma `Chabauty`, `PointSearch`.
**References.** Siegel 1929; Faltings 1983; Bombieri–Gubler, *Heights in Diophantine
Geometry*; Stoll, "Rational points on curves" (survey, 2011).

### `genus-zero-curve` — Genus 0 curves — P1, algorithmic
**Form.** C(x, y) = 0 irreducible of genus 0.
**Status.** Rational points: none or a ℙ¹-parametrization (conic step:
Hasse principle + Cremona–Rusin/Simon); integral points on the affine model:
reduce along the parametrization to Pell-like/divisor conditions (finite iff ≥ 3
points at infinity, Siegel; effective — Alvanos–Poulakis give complete algorithms).
**Software.** Sage: `Conic`, `parametrization`; Magma: `Conic`, `Parametrization`.
**References.** Hilbert–Hurwitz 1890; Poulakis 2002; Alvanos–Poulakis 2011.

### `genus-one-curve` — Genus 1 curves (non-Weierstrass models) — P1, algorithmic*
**Form.** C(x, y) = 0 irreducible of genus 1 (any plane model).
**Status.** With a rational point: birational to an elliptic curve (Nagell/Riemann–Roch
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1 change: 1 addition & 0 deletions tests/test_classify.py
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("y^3 = x^4 + 2", "", "superelliptic"),
("x^3*y + y^3*z + z^3*x = 0", "", "general-curve"), # Klein quartic
("x^2*y^2 = x^3 + 1", "", "genus-one-curve"),
("y^2 = x^3", "", "genus-zero-curve"), # cuspidal
# Fermat-type
("x^2 + y^4 = z^3", "", "generalized-fermat"),
("2*x^3 + 3*y^3 = 5*z^3", "", "generalized-fermat"),
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