Identifying the conformal equivalence class of a 2-torus subgroup of the cubical 3-torus

Question

Let K, L, M be integers with gcd(K,L,M) = 1. They determine a connected Lie subgroup G = G(K,L,M) of the cubical 3-torus (ℝ/ℤ)³ via

G = {(x,y,z) ∊ (ℝ/ℤ)³ | Kx + Ly + Mz = 0}

(where 0 denotes the identity element of ℝ/ℤ).

G is a 2-torus and inherits a Riemannian metric of everywhere zero Gaussian curvature from (ℝ/ℤ)³, which belongs to a unique conformal equivalence class of Riemann surfaces of genus 1 (or at most two such classes, depending on the choice of orientation of G).

As is well-known, every Riemannian flat 2-torus is conformally equivalent to the quotient ℂ/L for some lattice L = ⟨1, 𝛕⟩ for some 𝛕 in the set

X = {z ∊ ℂ | Im(z) > 0 and |z| ≥ 1 and |Re(𝛕)| ≤ 1/2}.

Such a 𝛕 is determined uniquely unless i ≠ 𝛕 ∊ ∂X, in which case the boundary points 𝛕 and conj(-𝛕) correspond to the same class. Now suppose we have chosen an orientation for G.

In terms of (K,L,M), what is the parameter 𝛕 corresponding to the Riemannian flat torus G ? Equivalently, what is the j-invariant of G ?

Answer 1

This may be easier than I thought. Assume WLOG that |K| ≥ |L| ≥ |M| with gcd(K,L,M) = 1 as above.

Then the subgroup H ⊂ ℤ³ of integer translations of ℝ³ that preserves the 2-plane P, defined by

P = {(x,y,z) ∊ ℝ³ | Kx + Ly + Mz = 0}

(where 0 denotes 0) is generated by v = (-L,K,0) and w = (0,-M,L):

H = ⟨v, w⟩ ⊂ P

as a subgroup of P, which itself is a subgroup of ℝ³. We can assume the counterclockwise angle in P from v to w is less then pi radians, and this condition will determine the orientation on P.

But the 2-torus in question is the quotient P / H. Therefore the parallelogram generated by v = (-L,K,0) and w = (0,-M,L) is a fundamental domain in P for this group action.

Therefore the parameter 𝜏 is determined a) by cos(angle(1,𝜏)) = cos(angle(v,w)), where

cos(angle(v,w) = -KM / (√(K^2 + L^2) √(L^2 + M^2))

and b) by the condition that |𝜏| = ||v|| / ||w||, where

||v|| / ||w||  =  √(K^2+L^2) / √(L^2+M^2).

The above assumes that either division by zero is a valid operation, or else that the case (K,L,M) = (1,0,0) is excluded a priori as trivial.