Meromorphic descent to the coarse Fuchsian quotient #
Let Γ ≤ PSL(2, ℝ) act properly discontinuously on the upper half-plane, so that the coarse orbit
quotient Γ \ ℍ is a Riemann surface and the orbit projection π : ℍ → Γ \ ℍ is holomorphic.
This file proves the meromorphic counterpart of the holomorphic descent criterion
Subgroup.mdifferentiable_iff_comp_quotientMk, together with the order formula at elliptic
points.
A function F on Γ \ ℍ is meromorphic at the orbit of z exactly when its pullback F ∘ π is
meromorphic at z (Subgroup.meromorphicAt_comp_quotientMk_iff). One direction is pullback along
the holomorphic map π. For the other, in the chart at the orbit of z the projection is
u ↦ u ^ m in a disc coordinate u centred at z, where m is the order of the stabilizer of
z, so the chart representative of F is the descent of a meromorphic function through the power
map, which is meromorphic by TauCeti.meromorphicAt_descendPow.
The orders satisfy
ord_z (F ∘ π) = m * ord_{π z} F
(Subgroup.meromorphicOrderAt_comp_quotientMk), the local multiplicity of π at z being m
(Subgroup.localMultiplicity_quotientMk). In particular an invariant function meromorphic on the
upper half-plane descends uniquely to a meromorphic function on Γ \ ℍ
(Subgroup.existsUnique_meromorphicAt_quotientMk), and its order at a point of stabilizer order
m is m times the order of the descended function at the image orbit. Orders upstairs may be
computed from f ∘ ofComplex by
TauCeti.UpperHalfPlane.meromorphicOrderAt_eq_meromorphicOrderAt_comp_ofComplex.
Main declarations #
Subgroup.meromorphicAt_comp_quotientMk_iff: the pullback criterion for meromorphy.Subgroup.meromorphicOrderAt_comp_quotientMk: pulling back multiplies the order by the order of the stabilizer.Subgroup.existsUnique_meromorphicAt_quotientMk: invariant meromorphic functions descend uniquely.
References #
- Hershel Farkas and Irwin Kra, Riemann Surfaces, Graduate Texts in Mathematics 71, Springer, second edition, 1992, Chapter I §§4–5.
- Rick Miranda, Algebraic Curves and Riemann Surfaces, Graduate Studies in Mathematics 5, American Mathematical Society, 1995, Chapter III §§3–4.
Meromorphic descent at an orbit. A function on the coarse quotient is meromorphic at the
orbit of z as soon as its pullback to the upper half-plane is meromorphic at z, including at
elliptic points.
The pullback criterion for meromorphy. A function on the coarse quotient is meromorphic at
the orbit of z exactly when its pullback to the upper half-plane is meromorphic at z.
The order formula for descent. Pulling a function on the coarse quotient back to the upper
half-plane multiplies its order at the orbit of z by the order m of the stabilizer of z:
ord_z (F ∘ π) = m * ord_{π z} F. Applied to the descent of an invariant meromorphic function
(Subgroup.existsUnique_meromorphicAt_quotientMk), this computes the order of the function
upstairs from that of its descent. Both sides are the junk value 0 when F is not meromorphic
at the orbit of z.
Meromorphic descent. Every invariant function on the upper half-plane that is meromorphic
at every point descends uniquely to a function on the coarse quotient that is meromorphic at every
point, with no freeness assumption. Its orders are related to those upstairs by
Subgroup.meromorphicOrderAt_comp_quotientMk.