Congruence-cover characterization of the braid group of an elliptic fibration

Determine whether, for a genus-one elliptic fibration over the sphere, the group of fiber-preserving braid mapping classes is equal to the intersection over all positive integers N of the groups of braid mapping classes that lift to the level-N congruence branched covers.

Background

The paper associates to a genus-one elliptic fibration π: M → S a family of branched covers π[N]: Σπ[N] → S arising from the reduction of the mapping-class-group monodromy modulo N. For a disk base, the authors establish relationships between the braid group Br(π) of fiber-preserving mapping classes and the liftable groups Br(π[N]) associated to these congruence covers, including a detailed computation for level 2.

The authors note that repeated applications of the abelian-lifting lemma suggest that Br(π) can be recovered as the intersection of the groups Br(π[2^a]) when the base is a disk, using the residual solvability of SL₂(Z). They then ask whether the analogous equality holds for a genus-one elliptic fibration over the closed sphere and for the full family of congruence levels N. They explain that this is equivalent to a local-global principle for a conjugacy problem for parabolic SL₂(Z)-representations of the punctured sphere.