Elliptic Curves over C

that is both analytic (as a mapping of complex manifolds) and algebraic: addition of points in E(C) corresponds to addition in C modulo the lattice L. This correspondence between lattices and elliptic curves over C is known as the Uniformization Theorem; we will spend this lecture and part of the next proving it. To make the correspondence explicit, we need to specify the map Φ from C/L and an elliptic curve E/C. This map is parameterized by elliptic functions, specifically the Weierstrass ℘-function. We will introduce general properties of elliptic functions in Section 16.1, specialize to the ℘-function in Section 16.3, and then construct the map Φ in Section 16.4. Once we have fleshed out this correspondence, we obtain a powerful method to construct elliptic curves with desired properties. The arithmetic properties of lattices over C are usually easier to understand than those of the corresponding elliptic curve. In particular, by choosing an appropriate lattice, we can construct an elliptic curve with a given endomorphism ring. In the case of elliptic curves over C, the endomorphism ring must either be Z or an order O in an imaginary quadratic field (a fact we will prove). The order O may be viewed as a lattice, and we will see that the elliptic curve corresponding to the torus C/O has endomorphism ring O. This has important implications for elliptic curves over finite fields. If we choose a suitable prime p, we can reduce an elliptic curve E/C with complex multiplication to a curve Ē/Fp with the same endomorphism ring O. The endomorphism ring determines, in particular, the trace of the Frobenius endomorphism π (up to a sign), which in turn determines the group order #E(Fp) = p + 1 − tr(π). This allows us to construct elliptic curves over finite fields that have a prescribed group order, using what is known as the CM method. As we will see, this has many practical applications, including cryptography and primality proving.