A tame example: y² = x³ - x #
Let W be the Weierstrass curve y² = x³ - x (TauCeti.CurveX3SubX.curve) over a field K of
characteristic other than two, for instance ℚ, and let K(W) = K(x, y) be its function field, a
quadratic extension of K(x). Its discriminant is 64, so W is an elliptic curve. This file
computes the ramification of K(W) / K(x) and checks it against the Hurwitz genus formula. Every
ramified place has ramification index 2, invertible in K, so the ramification is tame, in
contrast to y² + y = x³ over 𝔽₂, whose place at infinity is wild
(TauCeti.ArtinSchreier.isWild_infinity).
The points of order dividing two are the point at infinity O and the three points
T₀ = (0, 0), T₁ = (1, 0) and T₋₁ = (-1, 0) on the line y = 0, all rational over K.
Their four places are the places of K(W) ramified over K(x): each has ramification index 2
and different exponent 1, every other place is unramified, and
Diff(K(W) / K(x)) = O + T₀ + T₁ + T₋₁.
The Hurwitz genus formula then reads 2 · 1 - 2 = 2 · (0 - 2) + 4: the genus 1 of K(W) on the
left is computed from the Riemann–Roch spaces L(n · O), while the right side uses only the
degree [K(W) : K(x)] = 2, the genus 0 of K(x) and the four ramification indices.
Main results #
TauCeti.CurveX3SubX.different_eq:Diff(K(W) / K(x)) = O + T₀ + T₁ + T₋₁.TauCeti.CurveX3SubX.one_lt_ramificationIdx_iff: the ramified places are the places of the four points of order dividing two.TauCeti.CurveX3SubX.ramificationIdx_pointZeroand its siblings: those places have ramification index2.TauCeti.CurveX3SubX.hurwitz_genus_formula: the numerical Hurwitz genus identity2 · 1 - 2 = 2 · (0 - 2) + 4, withe - 1summed over the four ramified places.
References #
- H. Stichtenoth, Algebraic Function Fields and Codes, 2nd ed., GTM 254, Springer, 2009, Corollary 3.4.14, Theorem 3.5.1, Corollary 3.5.6 and Proposition 6.1.3.
The different of y² = x³ - x over K(x) is the sum of the places of the four points
of order dividing two: Diff(K(W) / K(x)) = O + T₀ + T₁ + T₋₁.
The ramified places of y² = x³ - x over K(x) are the places of the four points of
order dividing two: the place at infinity and the places of (0, 0), (1, 0) and (-1, 0).
The place of (0, 0) has ramification index 2 over K(x).
The place of (1, 0) has ramification index 2 over K(x).
The place of (-1, 0) has ramification index 2 over K(x).
The Hurwitz genus formula for y² = x³ - x over K(x), as a numerical identity: with
e - 1 summed over the four ramified places,
2 g - 2 = [K(W) : K(x)] (2 g(K(x)) - 2) + ∑ (e - 1),
which reads 2 · 1 - 2 = 2 · (0 - 2) + 4. The genus g = 1 on the left is computed from the
Riemann–Roch spaces L(n · O) (WeierstrassCurve.Affine.genus_functionField), independently of
the ramification data on the right.