Genus one for elliptic function fields

Shows g(K) = 1 for a fraction field K of an elliptic coordinate ring by counting the Weierstrass monomials in L(n·∞). The zero canonical divisor is then obtained by moving an arbitrary canonical divisor with its unique nonzero Riemann–Roch section.

theorem WeierstrassCurve.Affine.Chart.divOmega_invariant_differential {k : Type u_1} [Field k] (W : Affine k) [WeierstrassCurve.IsElliptic W] (K : Type u_2) [Field K] [Algebra W.CoordinateRing K] [IsFractionRing W.CoordinateRing K] [Algebra (Polynomial k) K] [IsScalarTower (Polynomial k) W.CoordinateRing K] [Algebra (RatFunc k) K] [IsScalarTower (Polynomial k) (RatFunc k) K] [_root_.FunctionField k K] [Algebra.IsSeparable (RatFunc k) K] [Algebra k K] [IsScalarTower k (Polynomial k) K] [FunctionField.IsFullConstantField k K] : FunctionField.Chart.IsCanonical k K 0

The divisor of the invariant differential is zero (canonical class is trivial).

theorem WeierstrassCurve.Affine.Chart.genus_eq_one {k : Type u_1} [Field k] (W : Affine k) [WeierstrassCurve.IsElliptic W] (K : Type u_2) [Field K] [Algebra W.CoordinateRing K] [IsFractionRing W.CoordinateRing K] [Algebra (Polynomial k) K] [IsScalarTower (Polynomial k) W.CoordinateRing K] [Algebra (RatFunc k) K] [IsScalarTower (Polynomial k) (RatFunc k) K] [_root_.FunctionField k K] [Algebra.IsSeparable (RatFunc k) K] [Algebra k K] [IsScalarTower k (Polynomial k) K] [FunctionField.IsFullConstantField k K] : FunctionField.Chart.genus k K = 1

M7b: the genus of an elliptic function field is 1.