REVIEW SHEET FOR LECTURE 11/17
MING-LUN HSIEH
1. Elliptic curves over complex numbers
Let ω1, ω2 be two complex numbers. Suppose that ω1 and ω2 are linearly independent over R. In other words, C = Rω1+ Rω2. Let L = Zω1+ Zω2 be a Z-lattice in C. It is clear that the quotient group C/L ≃ R/Z×R/Z is a torus. We are going to study complex functions on C/L and relate C/L to elliptic curves over C. Let [·] : C → C/L, z 7→ [z] be the natural quotient map. Recall that a function F : C/L → C can regarded as a function f : C → C such that f (z + l) = f (z) for all l ∈ L and F ([z]) = f(z). Such kind of functions are called doubly periodic functions with periods ω1 and ω2. The following lemma can be deduced from the residue formula in complex analysis.
Lemma 1.1. Let f : C→ C be a meromorphic and doubly periodica function. Then
∑
[a]∈C/L
orda(f ) = 0.
Here orda(f )∈ Z is the order of f(z) at z = a. Moreover,
∑
[a]∈C/L
orda(f )· [a] = 0 ∈ C/L.
We define ℘(z) : C→ C by the series
℘(z) := 1 z2 +
∑′
ω∈L
1
(z− ω)2 − 1 ω2. Here ∑′
means the sum over nonzero elements in L. To check ℘(z) is convergent, for each positive integer n, we let
S(n) :={ω ∈ L | n ≤ |ω| < n + 1} .
Let c(n) = #S(n) be the cardinality of S(n). We have the following lemma:
Lemma 1.2. There exists a constant C > 0 such that c(n)≤ C · n for all n.
In particular, for λ > 2, the sum
∑′ ω∈L
1
|ω|λ converges .
Date: December 1, 2011.
1
2 M.L. HSIEH
We write
℘(z) = 1
z2 + (−z2)·
∑′
ω∈L
1
ω2(z− ω)2 + (2z)·
∑′
ω∈L
1 ω(z− ω)2.
Let R > 0 be a positive number. For each z ∈ C such that R ≤ |z| ≤ 2R and z ̸∈ L, we can find a constant d(R) such that
|℘(z)| ≤ d(R) + 4R2·
∑′
ω∈L
1
|ω|4 + 4R·
∑′
ω∈L
1
|ω|3.
Thus the series ℘(z) converges absolutely and uniformly on every compact sets of the form {z | R ≤ |z| ≤ 2R}. By the theory of complex analysis, this shows that ℘(z) defines a holo- morphic function on C−L. In addition, the derivative ℘′(z) can be obtained by differentiating
℘(z) term by term. We therefore have
℘′(z) =∑
w∈L
−2 (z− ω)3
It is easy to show ℘(z) (resp. ℘′(z)) has a pole of order two (resp. three) at each point in L.
The function ℘(z) and ℘′(z) are called the Weirstrass ℘-functions.
Proposition 1.3. ℘(z) and ℘′(z) are doubly periodic meromorphic functions. In addition,
℘(z) is an even function while ℘′(z) is an odd function.
When |z| is sufficiently small, we have an expansion
℘(z) = 1 z2 +
∑∞ n=1
cnzn, where
cn=
∑′ ω∈L
n + 1 ωn+2. We compute
℘′(z)2 = 4
z6 − 8c2
z2 − 16c4+ o(z),
℘(z)3 = 1
z6 +3c2
z2 + 3c4+ o(z),
℘(z) = 1
z2 + o(z).
Then we find immediately that the doubly periodic function
φ(z) := ℘′(z)2 − 4℘(z)3+ 20c2℘(z) + 28c4
has no pole at z ∈ L and φ(0) = 0. In other words, φ(z) is a doubly periodic entire function on C such that φ(0) = 0. By Liouville’s theorem, φ(z) = 0. We thus proved the following:
Theorem 1.4. For every z∈ C, (x, y) = (℘(z), ℘′(z)) satisfies the equation y2 = 4x3− g4(L)x− g6(L),
where
g4(L) = 60·
∑′
ω∈L
1
ω4, g6(L) = 140·
∑′
ω∈L
1 ω6.
REVIEW SHEET FOR LECTURE 11/17 3
Proposition 1.5. The polynomial fL(x) := 4x3− g2(L)x− g6(L) has three distinct roots e1 = ℘(ω1
2 ), e2 = ℘(ω2
2 ) and e3 = ℘(ω1+ ω2 2 ).
Theorem 1.6. Let E : y2 = fL(x) be the elliptic curve defined over C. Then we have an isomorphism as abelian groups:
Φ : C/L→E(C)∼ [z]7→
{
[℘(z), ℘′(z), 1] · · · [z] ̸= [0], [0, 1, 0] · · · [z] = [0].
2. Exercises Exercise 2.1. Show that
℘′′(z) = 6(℘(z))2− 1
2· g4(L)
by comparing the Laurent expansion around z = 0 between both sides.
Exercise 2.2. Let e1, e2 and e3 be as in Prop. 1.5. Show that
℘(z +ω1
2 ) = e1+ (e1− e2)(e1− e3)
℘(z)− e1
and that
℘(ω1
4 ) = e1± {(e1− e2)(e1− e3)}12 .
Exercise 2.3. Use Lemma 1.1 to show that the map Φ in Theorem 1.6 is surjective.
Department of Mathematics, National Taiwan University, Taipei, Taiwan E-mail address: [email protected]
