The rational subfield of index two of a hyperelliptic function field

Let F / k be a function field with exact constant field k and genus g, and let x ∈ F be transcendental with [F : k(x)] = 2. The divisor (g - 1) · (x)_∞ has degree 2g - 2 and its Riemann–Roch space contains 1, x, …, x^{g-1}, so it is a canonical divisor, and its Riemann–Roch space is exactly the polynomials in x of degree less than g. From this follows Stichtenoth's Proposition 6.2.4(a): every z ∈ F with [F : k(z)] ≤ g lies in k(x). Indeed, with B = (z)_∞ the Riemann–Roch theorem gives ℓ((g - 1)(x)_∞ - B) ≥ 1, so some nonzero u has u · L(B) ⊆ L((g - 1)(x)_∞) ⊆ k(x), and z = (z u) / u.

In particular, for a hyperelliptic function field, g ≥ 2, the rational subfield of index two is unique: any z with [F : k(z)] = 2 has k(z) = k(x).

Main results

• TauCeti.divisorClass_sub_one_zsmul_poles_eq_canonicalClass: (g - 1) · (x)_∞ is canonical.
• TauCeti.mem_adjoin_of_mem_riemannRochSpace_sub_one_zsmul_poles: L((g - 1) · (x)_∞) ⊆ k(x).
• TauCeti.mem_adjoin_of_finrank_adjoin_le_genus: [F : k(z)] ≤ g forces z ∈ k(x).
• TauCeti.adjoin_eq_adjoin_of_finrank_adjoin_eq_two: the rational subfield of index two of a hyperelliptic function field is unique.

References

• H. Stichtenoth, Algebraic Function Fields and Codes, 2nd ed., GTM 254, Springer, 2009, Proposition 6.2.4.

theorem TauCeti.divisorClass_sub_one_zsmul_poles_eq_canonicalClass {k : Type u_1} {F : Type u_2} [Field k] [Field F] [Algebra k F] (hF : IsFunctionField k F) (hex : IsIntegrallyClosedIn k F) {x : F} (hx : Transcendental k x) (hdeg : Module.finrank (↥k⟮x⟯) F = 2) : (Place.orderSystem hF).divisorClass ((↑(genus k F) - 1) • Divisor.poles hF (Units.mk0 x ⋯)) = canonicalClass hF hex

(g - 1) · (x)_∞ is a canonical divisor when [F : k(x)] = 2: it has degree 2g - 2, and its Riemann–Roch space contains the g independent functions 1, x, …, x^{g-1}.

theorem TauCeti.riemannRochSpace_sub_one_zsmul_poles_eq_span {k : Type u_1} {F : Type u_2} [Field k] [Field F] [Algebra k F] (hF : IsFunctionField k F) (hex : IsIntegrallyClosedIn k F) {x : F} (hx : Transcendental k x) (hdeg : Module.finrank (↥k⟮x⟯) F = 2) : riemannRochSpace ((↑(genus k F) - 1) • Divisor.poles hF (Units.mk0 x ⋯)) = Submodule.span k (Set.range fun (i : Fin (genus k F)) => x ^ ↑i)

L((g - 1) · (x)_∞) is spanned by 1, x, …, x^{g-1} when [F : k(x)] = 2: the g powers are independent, and ℓ((g - 1) · (x)_∞) = g since the divisor is canonical.

theorem TauCeti.mem_adjoin_of_mem_riemannRochSpace_sub_one_zsmul_poles {k : Type u_1} {F : Type u_2} [Field k] [Field F] [Algebra k F] (hF : IsFunctionField k F) (hex : IsIntegrallyClosedIn k F) {x : F} (hx : Transcendental k x) (hdeg : Module.finrank (↥k⟮x⟯) F = 2) {f : F} (hf : f ∈ riemannRochSpace ((↑(genus k F) - 1) • Divisor.poles hF (Units.mk0 x ⋯))) : f ∈ k⟮x⟯

L((g - 1) · (x)_∞) ⊆ k(x) when [F : k(x)] = 2.

theorem TauCeti.mem_adjoin_of_finrank_adjoin_le_genus {k : Type u_1} {F : Type u_2} [Field k] [Field F] [Algebra k F] (hF : IsFunctionField k F) (hex : IsIntegrallyClosedIn k F) {x : F} (hx : Transcendental k x) (hdeg : Module.finrank (↥k⟮x⟯) F = 2) {z : F} (hz : Module.finrank (↥k⟮z⟯) F ≤ genus k F) : z ∈ k⟮x⟯

Stichtenoth, Proposition 6.2.4(a): if [F : k(x)] = 2 and z ∈ F has [F : k(z)] ≤ g, then z ∈ k(x). For a constant z this is exactness of k; otherwise, with B = (z)_∞, the Riemann–Roch theorem at B with the canonical divisor (g - 1) · (x)_∞ gives a nonzero u with u · L(B) ⊆ L((g - 1) · (x)_∞) ⊆ k(x), and z = (z u) / u.

theorem TauCeti.adjoin_eq_adjoin_of_finrank_adjoin_eq_two {k : Type u_1} {F : Type u_2} [Field k] [Field F] [Algebra k F] (hF : IsFunctionField k F) (hex : IsIntegrallyClosedIn k F) (hg : 2 ≤ genus k F) {x z : F} (hx : Transcendental k x) (hdeg : Module.finrank (↥k⟮x⟯) F = 2) (hdegz : Module.finrank (↥k⟮z⟯) F = 2) : k⟮z⟯ = k⟮x⟯

The rational subfield of index two of a hyperelliptic function field is unique (Stichtenoth, Proposition 6.2.4): if g ≥ 2 and x, z ∈ F are transcendental with [F : k(x)] = [F : k(z)] = 2, then k(z) = k(x).