On the Stratification Conjecture for CM Torsion Subgroups

Work of Bourdon–Clark–Pollack shows that the set of degrees d∈Z+ for which the classification of torsion subgroups of CM elliptic curves over all number fields of degree d is the same as the classification of CM torsion of elliptic curves over Q has positive asymptotic density. Based on this, I conjectured that for every d0∈Z+, the set of d∈Z+ such that the classification of CM torsion in degree d as the same as in degree d0 has positive density. This was proven for all odd d0 by Bourdon–Pollack in 2017. Here we give the first results on even d0: the conjecture holds for d0=2 and for d0=2p0 for a set of primes p0 of relative density one. However, we will also explain why the conjecture seems likely to be false for d0=2p0 where p0 lies in an infinite set of prime numbers, including p0=3.