Abelian p-groups and the Halting problem

We investigate which effectively presented abelian p  -groups are isomorphic relative to the halting problem. The standard approach to this and similar questions uses the notion of Δ20-categoricity (to be defined). We partially reduce the description of Δ20-categorical p-groups of Ulm type 1 to the analogous problem for equivalence structures. Using this reduction, we solve a problem left open in [5]. For the sake of the reduction mentioned above, we introduce a new notion of effective Δ20-categoricity that lies strictly in-between plain Δ20-categoricity and relative Δ20-categoricity (to be defined). We then reduce the problem of classifying effective Δ20-categoricity to a question stated in terms of Σ20-sets. Among other results, we show that for c.e. Turing degrees bounding such sets is equal to being complete.