The cuspidal cohomology conjecture for prime ideals in imaginary quadratic fields

Let $K$ be an imaginary quadratic field, let ${\mathcal O}_K$ be its ring of integers, and let $\Gamma_0({\mathfrak p})$ denote the corresponding congruence subgroup for a prime ideal ${\mathfrak p}$ of residue degree one. Cuspidal cohomology conjecture. For every $K$, there are infinitely many such prime ideals ${\mathfrak p}$ for which

$$H^1_{cusp}(\Gamma_0({\mathfrak p}),\mathbb C) \not= 0.$$

The conjecture concerns the apparent uniform distribution of the prime ideals for which cuspidal cohomology does not vanish; the surrounding data suggest that vanishing occurs for roughly $90\%$ of the relevant primes. It is also related to the existence of abelian varieties over imaginary quadratic fields with special endomorphism rings and restricted ramification.