Effective finiteness conjecture for good Atkin–Lehner quotients

Fix a totally real number field $F$, an almost indefinite quaternion algebra $B$ over $F$, and an ideal ${\mathfrak a}$. Let $Q_0({\mathfrak N})_{\mathfrak a}$ denote the associated Atkin–Lehner quotient for an ideal ${\mathfrak N}$, and call it good when it satisfies the automorphic criterion implying vanishing of the modified diagonal cycle.

Effective finiteness conjecture. The set of ideals ${\mathfrak N}$ for which $Q_0({\mathfrak N})_{\mathfrak a}$ is good is an effectively computable finite set.

This is a more explicit fixed-data version of the finiteness program for good Shimura curves. It is stated as a conjecture in the discussion of Atkin–Lehner quotients and is intended to support the general finiteness results.