Canonizing certain Borel equivalences for Silver forcing

Following the recent research of J. Zapletal we study Borel equivalence relations with connection to the Silver forcing. We prove that ℓ p relations and generally equivalence relations given by F σ P-ideals can be canonized to the full equivalence relation or to a subset of E 0 . Next we classify a certain subclass of subequivalences of E 0 on the Silver forcing and find perfectly many mutually different subequivalences of E 0 with respect to the Silver forcing.