18.783 Elliptic Curves Spring 2013 Lecture #16 04/09/2013
Elliptic curves over C
We now consider elliptic curves over the complex numbers. Our main tool will be the correspondence between elliptic curves over C and tori C/L defined by lattices L in C (see Definition 16.1). We will proceed to show the following: 1. Every lattice L can be used to define an elliptic curve E/C.
2. Every elliptic curve E/C arises from a lattice L.
3. If E/C is the elliptic curve corresponding to the lattice L, then there is an isomorphism
Φ C/L −→ E/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 Uniformiza- tion 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 Weier- strass ℘-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 usu- ally 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 endomor- phism 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 E/¯ Fp with the same endomorphism ring O. The endomorphism ring determines, in particular, the trace of the Frobenius endomorphism π (uptoasign),whichinturn 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.
16.1 Elliptic functions We begin with the definition of a lattice in the complex plane.
Definition 16.1. A lattice L =[ω1,ω2] is an additive subgroup = ω1ZZ+ω2 of C generated by complex numbers ω1 and ω2 that are independent over R.
1 Andrew V. Sutherland Example 16.2. Let τ be the root of a monic quadratic equation x2 + bx + c with integer coefficients and negative discriminant. Then the lattice [1,τ] is the additive group of an imaginary quadratic order O = Z[τ]. Conversely, if O is an imaginary quadratic order Z[τ], then the additive group of O is the lattice [1,τ]. If we take the quotient of the complex plane C modulo a lattice L,wegetatorusC/L.
Definition 16.3. A fundamental parallelogram for L =[ω1,ω2] is a set of the form
Fα = {α + t1ω1 + t2 ω 2 : α ∈ C, 0 ≤ t1,t2 < 1}.
We can identify the points in a fundamental parallelogram with the points of C/L.
ω1
ω2
Figure 1: A lattice [ω1,ω2] with a fundamental parallelogram shaded.
In order to define the correspondence between complex tori and elliptic curves over C, we need to define the notion of an elliptic function on C. As complex analysis is not a prerequisite for this course, we first define the terminology we need, which can be found in any standard textbook on the subject, including [1, 3].1 A function f(z)definedonsomeopenneighborhoodofapointz0 ∈ C is said to be holomorphic at z0 if the complex derivative