Long–Maclachlan–Reid conjectures on rational homology 3-spheres

For a prime ideal $\mathfrak{p}$, let $amma_0(\mathfrak{p})$ denote the corresponding Bianchi congruence subgroup, and let $H^2_{cusp}(amma_0(\mathfrak{a}),\mathbb Q)$ be its second cuspidal cohomology. The ring $\mathbb Z[i]$ is the ring of Gaussian integers, and $\mathbf Nmathfrak{q}$ denotes the norm of an ideal.

Long–Maclachlan–Reid conjectures.

1. There exist infinitely many pairs of prime ideals $\mathfrak{p}_1,\mathfrak{p}_2$ in $\mathbb Z[i]$ such that

$$H^2_{cusp}(amma_0(\mathfrak{p}_1\mathfrak{p}_2),\mathbb Q)=0.$$

2. If $\mathfrak{p}=(1+i)$, then there are infinitely many prime ideals $\mathfrak{q}subsetmathbb Z[i]$ with $\mathbf Nmathfrak{q}\equiv 1\pmod {12}$ such that

$$H^2_{cusp}(amma_0(\mathfrak{p}\mathfrak{q}),\mathbb Q)=0.$$

These conjectures concern the existence of infinitely many commensurability classes of arithmetic rational homology 3-spheres. The source attributes them to Long, Maclachlan, and Reid; no resolution is given here.