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

TL;DR

This paper proves Bernstein-Schwarzman's conjecture for a specific 3D crystallographic reflection group related to Klein's simple group, by computing the algebra of invariant theta functions which is not freely generated.

Contribution

It establishes the conjecture for the Klein group case in three dimensions and computes the invariant theta functions algebra, overcoming previous obstacles.

Findings

The quotient is a 3D weighted projective space with weights 1, 2, 4, 7.

The invariant algebra of theta functions is not free polynomial.

The conjecture holds for the Klein group in dimension 3.

Abstract

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