Number Theory Seminar

Friday 9/19/03 12:00-1:00 PM in Hylan 1106A:
Ken McMurdy- "p-adic Properties of CM j-invariants" (Part I)

ABSTRACT:

Every elliptic curve E at least has endomorphisms by Z (just coming from the group law). When End(E) is bigger than Z, the curve is said to have complex multiplication, or CM for short. For curves defined over the complex numbers, this can only mean that End(E) is isomorphic to an order in some imaginary quadratic field. Turning the problem around, we can fix an order R in some imaginary quadratic field K and then ask which elliptic curves have R as their endomorphism ring. It turns out that the j-invariants of these curves are conjugate algebraic integers and can be readily calculated by working out the class group of K (and using Mathematica). In the first installment of this talk, I will review the general theory of CM curves over the complex numbers, and work out a few of the examples that have motivated some conjectures regarding p-adic properties of these CM j-invariants.