Refined norm counting in quaternion algebras

Gross and Zagier related the differences between singular moduli to counting

isomorphisms between the mod ell^n reductions of the CM elliptic curves under

consideration. Later, Howard and Yang refined this by introducing the CM-degree of an isomorphism, which takes trace 1 values in a real quadratic field.

At supersingular primes, the relevant endomorphism rings are orders in quaternion algebras, and the degree corresponds to the norm. In this talk, we explore the above, and explain how one can count elements in quaternion algebras with a given CM norm.