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Bernstein--Schwarzman conjectured that the quotient of a complex affine spaceby an irreducible complex crystallographic group generated by reflections is aweighted projective space. The conjecture was proved by Schwarzman andTokunaga-Yoshida in dimension 2 for almost all such groups, and for allcrystallographic reflection groups of Coxeter type by Looijenga,Bernstein-Schwarzman and Kac-Peterson in any dimension. We prove that theconjecture is true for the crystallographic reflection group in dimension 3 forwhich the associated collineation group is Klein's simple group of order 168.In this case the quotient is the 3-dimensional weighted projective space withweights 1, 2, 4, 7. The main ingredient in the proof is the computation of thealgebra of invariant theta-functions. Unlike the Coxeter case, the invariantalgebra is not free polynomial, and this was the major stumbling block.

Action of the automorphism group on the Jacobian of Klein's quartic curve II: Invariant theta-functions