Abstract: Take two CM elliptic curves over a number field, each equipped with a level structure preserved by all endomorphisms. At which primes $p$ does there exist an isomorphism between the elliptic curves mod $p$ that preserves the level structures? This question was answered for $\Gamma_0(N)$ level structure by Gross, Kohnen, and Zagier in 1987. A key step in their proof is showing that an isomorphism mod $p$ determines embeddings of the (quadratic) CM orders into a common (quaternion) Eichler order. In this talk, we will discuss what changes in the argument when we replace $\Gamma_0(N)$ with the normalizer of a non-split Cartan subgroup. This is joint work with Elie Studnia and Jan Vonk.