Lecture on Shimura Curves 6: Special Points and Canonical Models

As mentioned several times in class, the arithmetic of quaternionic Shimura varieties is strongly controlled by the behavior of the class of CM points (just as in the case of modular curves). Just as for elliptic curves, if o is an order in an imaginary quadratic field K, one has a notion of a QM-surface with o-CM. Namely, let ι : O ↪→ A be a QM structure on an abelian surface (we are still working over C). We then have the notion of the QM-endomorphism ring and endomorphism algebra of A: EndQM(A) is equal to the set of O-equivariant endomorphisms of A: i.e., the set of endomorphisms α of A such that for all x ∈ O, ι(x)α = αι(x). In other words, EndQM(A) is the centralizer of O in End(A). By our classification of endomorphism algebras of abelian surfaces, there are essentially two possibilities: either Z or an order in an imaginary quadratic field.