Bianchi cusps and ideal classes

Statement

For $K=\mathbb Q(\sqrt{-d})$ and $\Gamma=\operatorname{PSL}_2(\mathcal O_K)$ , the cusps of the Bianchi orbifold admit bijections

\{\text{cusps}\}\ \cong\ \Gamma\backslash\mathbb P^1(K) \ \xrightarrow{\ \sim\ }\operatorname{Cl}(K), \qquad [a:b]\longmapsto[a\mathcal O_K+b\mathcal O_K].

Here $(a,b)\ne(0,0)$ ; homogeneous coordinates are taken up to simultaneous nonzero scaling. The right side is the ideal class group. In particular the number of cusps equals the class number $h_K$ .

Why the displayed map is well defined

Scaling $(a,b)$ multiplies its fractional ideal by a principal factor, leaving its class unchanged. A determinant-one integral matrix replaces $a,b$ by integral linear combinations; its integral inverse proves that the generated ideal is unchanged.

This checks well-definedness, not the whole bijection: identifying all cusp orbits and proving surjectivity and injectivity requires the ideal-class theorem.

The standard cusp

The boundary point $\infty=[1:0]$ maps to the principal ideal class. Thus class number one means precisely that every cusp is equivalent to this standard cusp.

References

T. Church, B. Farb, and A. Putman, Integrality in the Steinberg module and the top-dimensional cohomology of SL_n O_K, Example 5.6, p. 31 of the linked version. Author-hosted paper