The branch places of a hyperelliptic function field #
Let F / k be a function field with exact constants, 2 ≠ 0 in k, and x ∈ F transcendental
with [F : k(x)] = 2 and F / k(x) separable. Completing the square, F = k(x)(y) with
y² = u ∈ k(x), so the ramification of F / k(x) is that of a radical extension of prime
exponent 2: every place of F has different exponent 0 or 1 over k(x), and exponent 1
exactly at the ramified places. The Hurwitz genus formula over k(x), of genus zero, then says
that the different has degree 2g + 2; over an algebraically closed k every place is rational,
so F / k(x) has exactly 2g + 2 ramified places, each the only place over its restriction, and
their restrictions, the branch places of k(x), are 2g + 2 rational places.
When moreover g ≥ 2, every automorphism of F / k preserves k(x) and permutes the branch
places through its restriction to k(x): this is the invariant finite set of rational places of
k(x) on which Aut(F / k) acts, the input to the finiteness of Aut(F / k).
Main results #
TauCeti.differentExponent_adjoin_le_oneandTauCeti.differentExponent_adjoin_eq_one_iff_one_lt_ramificationIdx: the different exponents ofF / k(x)are0or1, and1exactly at the ramified places.TauCeti.degree_different_adjoin:deg Diff(F / k(x)) = 2g + 2.TauCeti.card_support_different_adjoin: over an algebraically closedk, exactly2g + 2places ofFramify overk(x).TauCeti.eq_of_restrict_eq_of_mem_support_different: restriction tok(x)is injective on the ramified places.TauCeti.card_branchPlaces: the branch places ofk(x)number2g + 2.TauCeti.smul_mem_branchPlaces: forg ≥ 2the restriction of an automorphism ofF / kpermutes the branch places.
References #
- H. Stichtenoth, Algebraic Function Fields and Codes, 2nd ed., GTM 254, Springer, 2009, Proposition 3.7.3, Corollary 3.4.14 and Section VI.2.
A quadratic extension of k(x) is radical away from characteristic two: some y with
y² = u for a nonzero u ∈ k(x) generates F over k(x).
The different exponents of F / k(x) are at most one.
The different exponent is one exactly at the ramified places of F / k(x).
A place of F ramifies over k(x) exactly when it lies in the support of the different.
The different of F / k(x) has degree 2g + 2: the Hurwitz genus formula over k(x),
of genus zero, reads 2g - 2 = 2 · (-2) + deg Diff.
Exactly 2g + 2 places of F ramify over k(x) when k is algebraically closed: every
ramified place has different exponent 1 and degree 1, so the degree 2g + 2 of the different
counts them.
The branch places of k(x): the restrictions to k(x) of the places of F that ramify
over k(x).
Equations
- TauCeti.branchPlaces hx = Finset.image (fun (Q : TauCeti.Place k F) => TauCeti.Place.restrict k (↥k⟮x⟯) Q) (TauCeti.Divisor.different k F ⋯).support
Instances For
Restriction is injective on the ramified places: a place Q of F that ramifies over
k(x) is the only place of F over its restriction, because e(Q ∣ k(x)) = 2 = [F : k(x)]
exhausts the fibre.
There are 2g + 2 branch places when k is algebraically closed.
The automorphisms of F / k permute the branch places through their restriction to
k(x), for g ≥ 2: restriction of places is equivariant, and ramification is preserved.