ŮÚū. 1(‚ŮŚ„Ó 3)—őńŇ–∆ņÕ»Ň >>
Chapter 9
Elliptic Curves over C

The goal of this chapter is to show that an elliptic curve over the complex
numbers is the same thing as a torus. First, we show that a torus is isomor-
phic to an elliptic curve. To do this, we need to study functions on a torus,
which amounts to studying doubly periodic functions on C, especially the
Weierstrass ‚„˜-function. We then introduce the j-function and use its proper-
ties to show that every elliptic curve over C comes from a torus. Since most
of the Ô¬Āelds of characteristic 0 that we meet can be embedded in C, many
properties of elliptic curves over Ô¬Āelds of characteristic 0 can be deduced from
properties of a torus. For example, the n-torsion on a torus is easily seen to
be isomorphic to Zn ‚ä• Zn , so we can deduce that this holds for all elliptic
curves over algebraically closed Ô¬Āelds of characteristic 0 (see Corollary 9.22).

9.1 Doubly Periodic Functions
Let ŌČ1 , ŌČ2 be complex numbers that are linearly independent over R. Then

L = ZŌČ1 + ZŌČ2 = {n1 ŌČ1 + n2 ŌČ2 | n1 , n2 ‚ąą Z}

is called a lattice. The main reason we are interested in lattices is that C/L
is a torus, and we want to show that a torus gives us an elliptic curve.
The set
F = {a1 ŌČ1 + a2 ŌČ2 | 0 ‚Č¤ ai < 1, i = 1, 2}
(see Figure 9.1) is called a fundamental parallelogram for L. A diÔ¬Äer-
ent choice of basis ŌČ1 , ŌČ2 for L will of course give a diÔ¬Äerent fundamental
parallelogram. Since it will occur several times, we denote

ŌČ3 = ŌČ1 + ŌČ2 .

A function on C/L can be regarded as a function f on C such that f (z +
ŌČ) = f (z) for all z ‚ąą C and all ŌČ ‚ąą L. We are only interested in meromorphic

257

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258 CHAPTER 9 ELLIPTIC CURVES OVER C

‚„¦3
‚„¦1

‚„¦2
0
Figure 9.1
The Fundamental Parallelogram

functions, so we deÔ¬Āne a doubly periodic function to be a meromorphic
function
f :C‚Ü’C‚ą™‚ąě
such that
f (z + ŌČ) = f (z)
for all z ‚ąą C and all ŌČ ‚ąą L. Equivalently,
f (z + ŌČi ) = f (z), i = 1, 2.
The numbers ŌČ ‚ąą L are called the periods of f .
If f is a (not identically 0) meromorphic function and w ‚ąą C, then we can
write
f (z) = ar (z ‚ą’ w)r + ar+1 (z ‚ą’ w)r+1 + ¬· ¬· ¬· ,
with ar = 0. The integer r can be either positive, negative, or zero. DeÔ¬Āne
the order and the residue of f at w to be
r = ordw f
a-1 = Resw f.
Therefore, ordw f is the order of vanishing of f at w, or negative the order of
a pole. The order is 0 if and only if the function is Ô¬Ānite and nonvanishing at
w. It is not hard to see that if f is doubly periodic, then ordw+ŌČ f = ordw f
and Resw+ŌČ f = Resw f for all ŌČ ‚ąą L.
A divisor D is a formal sum of points:
D = n1 [w1 ] + n2 [w2 ] + ¬· ¬· ¬· + nk [wk ],
where ni ‚ąą Z and wi ‚ąą F . In other words, we have a symbol [w] for each
w ‚ąą F , and the divisors are linear combinations with integer coeÔ¬Écients of
these symbols. The degree of a divisor is

deg(D) = ni .

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SECTION 9.1 DOUBLY PERIODIC FUNCTIONS

DeÔ¬Āne the divisor of a function f to be
div(f ) = (ordw f )[w].
w‚ąąF

THEOREM 9.1
Let f be a doubly periodic function for the lattice L and let F be a fundamental
parallelogram for L.
1. If f has no poles, then f is constant.
Resw f = 0.
2. w‚ąąF

3. If f is not identically 0,
deg(div(f )) = ordw f = 0.
w‚ąąF

4. If f is not identically 0,
w ¬· ordw f ‚ąą L.
w‚ąąF

5. If f is not constant, then f : C ‚Ü’ C ‚ą™ ‚ąě is surjective. If n is the sum
of the orders of the poles of f in F and z0 ‚ąą C, then f (z) = z0 has n
solutions (counting multiplicities).
6. If f has only one pole in F , then this pole cannot be a simple pole.
All of the above sums over w ‚ąą F have only Ô¬Ānitely many nonzero terms.

PROOF Because f is a meromorphic function, it can have only Ô¬Ānitely
many zeros and poles in any compact set, for example, the closure of F .
Therefore, the above sums have only Ô¬Ānitely many nonzero terms.
If f has no poles, then it is bounded in the closure of F , which is a compact
set. Therefore, f is bounded in all of C. Liouville‚Ä™s theorem says that a
bounded entire function is constant. This proves (1).
Recall Cauchy‚Ä™s theorem, which says that

f (z)dz = 2ŌÄi Resw f,
‚ą‚F w‚ąąF

where ‚ą‚F is the boundary of F and the line integral is taken in the coun-
terclockwise direction. Write (assuming ŌČ1 , ŌČ2 are oriented as in Figure 9.1;
otherwise, switch them in the following)

f (z)dz =
‚ą‚F
ŌČ2 ŌČ2 +ŌČ1 ŌČ1 0
f (z)dz + f (z)dz + f (z)dz + f (z)dz.
0 ŌČ2 ŌČ1 +ŌČ2 ŌČ1

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260 CHAPTER 9 ELLIPTIC CURVES OVER C

Since f (z + ŌČ1 ) = f (z), we have
ŌČ1 0 ŌČ2
f (z)dz = ‚ą’
f (z)dz = f (z)dz.
ŌČ1 +ŌČ2 ŌČ2 0

Similarly,
ŌČ1 +ŌČ2 0
f (z)dz = ‚ą’ f (z)dz.
ŌČ2 ŌČ1
Therefore, the sum of the four integrals is 0. There is a small technicality
that we have passed over. The function f is not allowed to have any poles on
the path of integration. If it does, adjust the path with a small detour around
such points as in Figure 9.2. The integrals cancel, just as in the above. This
proves (2).

‚„¦3
‚„¦1

‚„¦2
0
Figure 9.2

Suppose r = ordw f . Then f (z) = (z ‚ą’ w)r g(z), where g(w) is Ô¬Ānite and
nonzero. Then
r g (z)
f (z)
= + ,
z‚ą’w
f (z) g(z)
so
f
Resw = r.
f
If f is doubly periodic, then f is doubly periodic. Therefore, (2) applied to
f /f yields
f
2ŌÄi ordw f = 2ŌÄi Resw = 0.
f
w‚ąąF w‚ąąF

This proves (3).
For (4), we have
f f
w ¬· ordw f = 2ŌÄi
2ŌÄi Resw z = z dz.
f f
‚ą‚F
w‚ąąF w‚ąąF

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SECTION 9.1 DOUBLY PERIODIC FUNCTIONS

However, in this case, the function zf /f is not doubly periodic. The integral
may be written as a sum of four integrals, as in the proof of (2). The double
periodicity of f and f yield
ŌČ1 0
f (z) f (z)
z dz = (z + ŌČ1 ) dz
ŌČ1 +ŌČ2 f (z) f (z)
ŌČ2
ŌČ2 ŌČ2
f (z) f (z)
=‚ą’ dz ‚ą’ ŌČ1
z dz.
f (z) f (z)
0 0

But ŌČ2
1 f (z)
dz
2ŌÄi f (z)
0

is the winding number around 0 of the path

0 ‚Č¤ t ‚Č¤ 1.
z = f (tŌČ2 ),

Since f (0) = f (ŌČ2 ), this is a closed path. The winding number is an integer,
so
ŌČ2 ŌČ1
f (z) f (z)
z dz + z dz
f (z) ŌČ1 +ŌČ2 f (z)
0
ŌČ2
f (z)
= ‚ą’ŌČ1 dz ‚ąą 2ŌÄiZŌČ1 .
f (z)
0

Similarly,
ŌČ1 +ŌČ2 0
f (z) f (z)
dz ‚ąą 2ŌÄiZŌČ2 .
z dz + z
f (z) f (z)
ŌČ2 ŌČ1

Therefore,
w ¬· ordw f ‚ąą 2ŌÄiL.
2ŌÄi
w‚ąąF

This proves (4).
To prove (5), let z0 ‚ąą C. Then h(z) = f (z) ‚ą’ z0 is a doubly periodic
function whose poles are the same as the poles of f . By (3), the number
of zeros of h(z) in F (counting multiplicities) equals the number of poles
(counting multiplicities) of h, which is n. This proves (5).
For (6), suppose f has only a simple pole, say at w, and no others. Then
Resw f = 0 (otherwise, the pole doesn‚Ä™t exist). The sum in (2) has only one
term, and it is nonzero. This is impossible, so we conclude that either the
pole cannot be simple or there must be other poles.

REMARK 9.2 As we saw in the proof of (5), part (3) says that the number
of zeros of a doubly periodic function equals the number of poles. This is a
general fact for compact Riemann surfaces (such as a torus) and for projective
algebraic curves (see [42, Ch. 8, Prop. 1] or [49, II, Cor. 6.10]).

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262 CHAPTER 9 ELLIPTIC CURVES OVER C

If (6) were false for a function f , then f would give a bijective (by (5)) map
from the torus to the Riemann sphere (= C ‚ą™ ‚ąě). This is impossible for many
topological reasons (the torus has a hole but the sphere doesn‚Ä™t).

So far, we do not have any examples of nonconstant doubly periodic func-
tions. This situation is remedied by the Weierstrass ‚„˜-function.

THEOREM 9.3
Given a lattice L, deÔ¬Āne the Weierstrass ‚„˜-function by
1 1 1
‚ą’2
‚„˜(z) = ‚„˜(z; L) = + . (9.1)
(z ‚ą’ ŌČ)2
z2 ŌČ
ŌČ‚ąąL
ŌČ=0

Then
1. The sum deÔ¬Āning ‚„˜(z) converges absolutely and uniformly on compact
sets not containing elements of L.
2. ‚„˜(z) is meromorphic in C and has a double pole at each ŌČ ‚ąą L.
3. ‚„˜(‚ą’z) = ‚„˜(z) for all z ‚ąą C.
4. ‚„˜(z + ŌČ) = ‚„˜(z) for all ŌČ ‚ąą L.
5. The set of doubly periodic functions for L is C(‚„˜, ‚„˜ ). In other words,
every doubly periodic function is a rational function of ‚„˜ and its deriva-
tive ‚„˜ .

PROOF Let C be a compact set, and let M = Max{|z| | z ‚ąą C}. If z ‚ąą C
and |ŌČ| ‚Č• 2M , then |z ‚ą’ ŌČ| ‚Č• |ŌČ|/2 and |2ŌČ ‚ą’ z| ‚Č¤ 5|ŌČ|/2, so

z(2ŌČ ‚ą’ z)
1
1
‚ą’2=
(z ‚ą’ ŌČ)2 (z ‚ą’ ŌČ)2 ŌČ 2
ŌČ
(9.2)
10M
M (5|ŌČ|/2)
‚Č¤ = .
|ŌČ|4 /4 |ŌČ|3

The preceding calculation explains why the terms 1/ŌČ 2 are included. With-
out them, the terms in the sum would be comparable to 1/ŌČ 2 . Subtracting
this 1/ŌČ 2 makes the terms comparable to 1/ŌČ 3 . This causes the sum to con-
verge, as the following lemma shows.

LEMMA 9.4
If k > 2 then
1
|ŌČ|k
ŌČ‚ąąL
ŌČ=0

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SECTION 9.1 DOUBLY PERIODIC FUNCTIONS

converges.

PROOF Let F be a fundamental parallelogram for L and let D be the
length of the longer diagonal of F . Then |z| ‚Č¤ D for all z ‚ąą F . Let ŌČ =
m1 ŌČ1 + m2 ŌČ2 ‚ąą L with |ŌČ| ‚Č• 2D. If x1 , x2 are real numbers with mi ‚Č¤ xi <
mi + 1, then ŌČ and x1 ŌČ1 + x2 ŌČ2 diÔ¬Äer by an element of F , so
1
|m1 ŌČ1 + m2 ŌČ2 | ‚Č• |x1 ŌČ1 + x2 ŌČ2 | ‚ą’ D ‚Č• |x1 ŌČ1 + x2 ŌČ2 | ‚ą’ |m1 ŌČ1 + m2 ŌČ2 |,
2
since |m1 ŌČ1 + m2 ŌČ2 | ‚Č• 2D. Therefore,
2
|m1 ŌČ1 + m2 ŌČ2 | ‚Č• |x1 ŌČ1 + x2 ŌČ2 |.
3
Similarly,
|x1 ŌČ1 + x2 ŌČ2 | ‚Č• D.
Comparing the sum to an integral yields

(3/2)k
1
‚Č¤ (1/area of F ) dx1 dx2 .
|ŌČ|k |x1 ŌČ1 + x2 ŌČ2 |k
|ŌČ|‚Č•2D |x1 ŌČ1 +x2 ŌČ2 |‚Č•D

The change of variables deÔ¬Āned by u + iv = x1 ŌČ1 + x2 ŌČ2 changes the integral
to
‚ąě
2ŌÄ
1 1
r dr dőł < ‚ąě,
C du dv = C k
(u2 + v 2 )k/2 őł=0 r=D r
|u+iv|‚Č•D

where C = (3/2)k /(area of F ). Therefore, the sum for |ŌČ| ‚Č• 2D converges.
Since there are only Ô¬Ānitely many ŌČ with |ŌČ| < 2D, we have shown that the
sum converges.

Lemma 9.4 and Equation 9.2 imply that the sum of the terms in Equa-
tion 9.1 with |ŌČ| ‚Č• 2M converges absolutely and uniformly for z ‚ąą C. Since
only Ô¬Ānitely many terms have been omitted, we obtain (1). Since a uniform
limit of analytic functions is analytic, ‚„˜(z) is analytic for z ‚ąą L. If z ‚ąą L,
then the sum of the terms for ŌČ = z is analytic near z, so the term 1/(z ‚ą’ ŌČ)2
causes ‚„˜ to have a double pole at z. This proves (2).
To prove (3), note that ŌČ ‚ąą L if and only if ‚ą’ŌČ ‚ąą L. Therefore, in the sum
for ‚„˜(‚ą’z), we can take the sum over ‚ą’ŌČ ‚ąą L. The terms of this sum are of
the form
1 1 1
1
‚ą’ ‚ą’ 2.
=
(z ‚ą’ ŌČ)
2 2 2
(‚ą’z + ŌČ) (‚ą’ŌČ) ŌČ
Therefore the sum for ‚„˜(‚ą’z) equals the sum for ‚„˜(z).
The proof of (4) would be easy if we could ignore the terms 1/ŌČ 2 , since
changing z to z + ŌČ would simply shift the summands. However, these terms

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264 CHAPTER 9 ELLIPTIC CURVES OVER C

are needed for convergence. With some care, one could justify rearranging the
sum, but it is easier to do the following. DiÔ¬Äerentiating ‚„˜(z) term by term
yields
1
‚„˜ (z) = ‚ą’2 .
(z ‚ą’ ŌČ)3
ŌČ‚ąąL

Note that ŌČ = 0 is included in the sum. This sum converges absolutely (by
comparison with the case k = 3 in Lemma 9.4) when z ‚ąą L, and changing z
to z + ŌČ shifts the terms in the sum. Therefore,

‚„˜ (z + ŌČ) = ‚„˜ (z).

This implies that there is a constant cŌČ such that

‚„˜(z + ŌČ) ‚ą’ ‚„˜(z) = cŌČ ,

for all z ‚ąą L. Setting z = ŌČ/2 yields

cŌČ = ‚„˜(‚ą’ŌČ/2) ‚ą’ ‚„˜(ŌČ/2) = 0,

by (3). Therefore ‚„˜(z + ŌČ) = ‚„˜(z). This proves (4).
Let f (z) be any doubly periodic function. Then

f (z) + f (‚ą’z) f (z) ‚ą’ f (‚ą’z)
f (z) = +
2 2
expresses f (z) as the sum of an even function and an odd function. Therefore,
it suÔ¬Éces to prove (5) for even functions and for odd functions. Since ‚„˜(‚ą’z) =
‚„˜(z), it follows that ‚„˜ (‚ą’z) = ‚ą’‚„˜ (z), so ‚„˜ (z) is an odd function. If f (z)
is odd, then f (z)/‚„˜ (z) is even. Therefore, it suÔ¬Éces to show that an even
doubly periodic function is a rational function of ‚„˜(z).
Let f (z) be an even doubly periodic function. We may assume that f is
not identically zero; otherwise, we‚Ä™re done. By changing f , if necessary, to
af + b
cf + d

for suitable a, b, c, d with ad ‚ą’ bc = 0, we may arrange that f (z) does not have
a zero or a pole whenever 2z ‚ąą L (this means that we want f (0) = 0, ‚ąě and
f (ŌČi /2) = 0 for i = 1, 2, 3). If we prove (af + b)/(cf + d) is a rational function
of ‚„˜, then we can solve for f and obtain the result for f .
Since f (z) is even and doubly periodic, f (ŌČ3 ‚ą’ z) = f (z), so

ordw f = ordŌČ3 ‚ą’w f.

We can therefore put the Ô¬Ānitely many elements in F where f (z) = 0 or
where f (z) has a pole into pairs (w, ŌČ3 ‚ą’ w). Since we have arranged that
w = ŌČ3 /2, the two elements of each pair are distinct. There is a slight

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SECTION 9.1 DOUBLY PERIODIC FUNCTIONS

problem if w lies on a side of F . Suppose w = xŌČ1 with 0 < x < 1. Then
ŌČ3 ‚ą’ w = (1 ‚ą’ x)ŌČ1 + ŌČ2 ‚ąą F . In this case, we translate by ŌČ2 to get
(1 ‚ą’ x)ŌČ1 ‚ąą F . Since w = ŌČ1 /2, we have x = 1/2, hence xŌČ1 = (1 ‚ą’ x)ŌČ1 , and
again the two elements of the pair are distinct. The case w = xŌČ2 is handled
similarly.
For a Ô¬Āxed w, the function ‚„˜(z) ‚ą’ ‚„˜(w) has zeros at z = w and z = ŌČ3 ‚ą’ w.
By Theorem 9.1(5), these are the only two zeros in F and they are simple
zeros. Therefore, the function
ordw f
(‚„˜(z) ‚ą’ ‚„˜(w))
h(z) =
(w, ŌČ3 ‚ą’w)

(the product is over pairs (w, ŌČ3 ‚ą’ w)) has a zero of order ordw f at w and
at ŌČ3 ‚ą’ w when ordw f > 0 and has a pole of the same order as f when
ordw f = 0 by Theorem 9.1, the poles at z ‚ąą L of the
ordw f < 0. Since
factors in the product cancel. Therefore, f (z)/h(z) has no zeros or poles in F .
By Theorem 9.1(1), f (z)/h(z) is constant. Since h(z) is a rational function
of ‚„˜(z), so is f (z). This completes the proof of Theorem 9.3.

In order to construct functions with prescribed properties, it is convenient
to introduce the Weierstrass ŌÉ-function. It is not doubly periodic, but it
satisÔ¬Āes a simple transformation law for translation by elements of L.

PROPOSITION 9.5
Let
z (z/w)+ 1 (z/w)2
1‚ą’ e .
ŌÉ(z) = ŌÉ(z; L) = z 2
ŌČ
ŌČ‚ąąL
ŌČ=0

Then

1. ŌÉ(z) is analytic for all z ‚ąą C

2. ŌÉ(z) has simple zeros at each ŌČ ‚ąą L and has no other zeros
d2
log ŌÉ(z) = ‚ą’‚„˜(z)
3. dz 2

4. given ŌČ ‚ąą L, there exist a = aŌČ and b = bŌČ such that

ŌÉ(z + ŌČ) = eaz+b ŌÉ(z)

for all z ‚ąą C.

PROOF The exponential factor is included to make the product converge.
A short calculation yields the power series expansion
2
1
(1 ‚ą’ u)eu+ 2 u = 1 + c3 u3 + c4 u4 + ¬· ¬· ¬· .

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266 CHAPTER 9 ELLIPTIC CURVES OVER C

Therefore, there is a constant C such that
2
1
|(1 ‚ą’ u)eu+ 2 u ‚ą’ 1| ‚Č¤ C|u|3

for u near 0. In particular, this inequality holds when u = z/ŌČ for |ŌČ| suÔ¬É-
|an | converges,
ciently large and z in a compact set. Recall that if a sum
then the product (1+an ) converges. Moreover, if (1+an ) = 0 for all n, then
|z/ŌČ|3 converges by Lemma 9.4 with k = 3,
the product is nonzero. Since
the product deÔ¬Āning ŌÉ(z) converges uniformly on compact sets. Therefore,
ŌÉ(z) is analytic. This proves (1). Part (2) follows since the product of the
factors, omitting one ŌČ, is nonzero at z = ŌČ.
To prove (3), diÔ¬Äerentiate the logarithm of the product for ŌÉ(z) to obtain
1 1 1 z
d
log ŌÉ(z) = + + +2 .
z‚ą’ŌČ ŌČ ŌČ
dz z ŌČ‚ąąL
ŌČ=0

Taking one more derivative yields the sum for ‚ą’‚„˜(z). This proves (3).
Let ŌČ ‚ąą L. Since
d2 ŌÉ(z + ŌČ)
log = 0,
2
dz ŌÉ(z)
there are constants a = aŌČ and b = bŌČ such that
ŌÉ(z + ŌČ)
= az + b.
log
ŌÉ(z)
Exponentiating yields (4). We can restrict z in the above to lie in a small re-
gion in order to avoid potential complications with branches of the logarithm.
Then (4) holds in this small region, and therefore for all z ‚ąą C, by uniqueness
of analytic continuation.

We can now state exactly when a divisor is a divisor of a function. The
following is a special case of what is known as the Abel-Jacobi theorem,
which states when a divisor on a Riemann surface, or on an algebraic curve,
is the divisor of a function.

THEOREM 9.6
ni [wi ] be a divisor. Then D is the divisor of a function if and
Let D =
ni wi ‚ąą L.
only if deg(D) = 0 and

PROOF Parts (3) and (4) of Theorem 9.1 are precisely the statements
ni wi ‚ąą L.
that if D is the divisor of a function then deg(D) = 0 and
ni wi = ‚ąą L. Let
Conversely, suppose deg(D) = 0 and
ŌÉ(z)
ŌÉ(z ‚ą’ wi )ni .
f (z) =
ŌÉ(z ‚ą’ ) i

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SECTION 9.2 TORI ARE ELLIPTIC CURVES

If ŌČ ‚ąą L, then

f (z + ŌČ)
= eaŌČ z+bŌČ ‚ą’aŌČ (z‚ą’ )‚ą’bŌČ ni (aŌČ (z‚ą’wi )+bŌČ )
e = 1,
f (z)

since ni = 0 and ni wi = . Therefore, f (z) is doubly periodic. The
divisor of f is easily seen to be D, so D is the divisor of a function.

Doubly periodic functions can be regarded as functions on the torus C/L,
and divisors can be regarded as divisors for C/L. If we let C(L)√— denote the
doubly periodic functions that do not vanish identically and let Div0 (C/L)
denote the divisors of degree 0, then much of the preceding discussion can be
expressed by the exactness of the sequence

div sum
0 ‚ą’‚Ü’ C√— ‚ą’‚Ü’ C(L)√— ‚ą’‚Ü’ Div0 (C/L) ‚ą’‚Ü’ C/L ‚ą’‚Ü’ 0. (9.3)

The ‚Äúsum‚ÄĚ function adds up the complex numbers representing the points in
the divisor mod L. The exactness at C(L)√— expresses the fact that a function
with no zeros and no poles, hence whose divisor is 0, is a constant. The
exactness at Div0 (C/L) is Theorem 9.6. The surjectivity of the sum function
is easy. If w ‚ąą C, then sum([w] ‚ą’ ) = w mod L.

9.2 Tori are Elliptic Curves
The goal of this section is to show that a complex torus C/L is naturally
isomorphic to the complex points on an elliptic curve.
Let L be a lattice, as in the previous section. For integers k ‚Č• 3, deÔ¬Āne the
Eisenstein series

ŌČ ‚ą’k .
Gk = Gk (L) = (9.4)
ŌČ‚ąąL
ŌČ=0

By Lemma 9.4, the sum converges. When k is odd, the terms for ŌČ and ‚ą’ŌČ
cancel, so Gk = 0.

PROPOSITION 9.7
For 0 < |z| < Min 0=ŌČ‚ąąL (|ŌČ|),
‚ąě
1
(2j + 1)G2j+2 z 2j .
‚„˜(z) = 2 +
z j=1

¬© 2008 by Taylor & Francis Group, LLC
268 CHAPTER 9 ELLIPTIC CURVES OVER C

When |z| < |ŌČ|,
PROOF

1 1 1
‚ą’ 2 = ŌČ ‚ą’2 ‚ą’1
(z ‚ą’ ŌČ)2 (1 ‚ą’ (z/ŌČ))2
ŌČ
‚ąě
zn
‚ą’2
=ŌČ (n + 1) .
ŌČn
n=1

Therefore,
‚ąě
zn
1
‚„˜(z) = 2 + (n + 1) n+2 .
z ŌČ
n=1
ŌČ=0

Summing over ŌČ Ô¬Ārst, then over n, yields the result.

THEOREM 9.8
Let ‚„˜(z) be the Weierstrass ‚„˜-function for a lattice L. Then

‚„˜ (z)2 = 4‚„˜(z)3 ‚ą’ 60G4 ‚„˜(z) ‚ą’ 140G6 .

PROOF From Proposition 9.7,

‚„˜(z) = z ‚ą’2 + 3G4 z 2 + 5G6 z 4 + ¬· ¬· ¬·
‚„˜ (z) = ‚ą’2z ‚ą’3 + 6G4 z + 20G6 z 3 + ¬· ¬· ¬· .

Cubing and squaring these two relations yields

‚„˜(z)3 = z ‚ą’6 + 9G4 z ‚ą’2 + 15G6 + ¬· ¬· ¬·
‚„˜ (z)2 = 4z ‚ą’6 ‚ą’ 24G4 z ‚ą’2 ‚ą’ 80G6 + ¬· ¬· ¬· .

Therefore,

f (z) = ‚„˜ (z)2 ‚ą’ 4‚„˜(z)3 + 60G4 ‚„˜(z) + 140G6 = c1 z + c2 z 2 + ¬· ¬· ¬·

is a power series with no constant term and with no negative powers of z.
But the only possible poles of f (z) are at the poles of ‚„˜(z) and ‚„˜ (z), namely,
the elements of L. Since f (z) is doubly periodic and, as we have just shown,
has no pole at 0, f (z) has no poles. By Theorem 9.1, f (z) is constant. Since
the power series for f (z) has no constant term, f (0) = 0. Therefore, f (z) is
identically 0.

It is customary to set

g2 = 60G4
g3 = 140G6 .

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SECTION 9.2 TORI ARE ELLIPTIC CURVES

The theorem then says that

‚„˜ (z)2 = 4‚„˜(z)3 ‚ą’ g2 ‚„˜(z) ‚ą’ g3 . (9.5)

Therefore, the points (‚„˜(z), ‚„˜ (z)) lie on the curve

y 2 = 4x3 ‚ą’ g2 x ‚ą’ g3 .

It is traditional to leave the 4 as the coeÔ¬Écient of x3 , rather than performing a
change of variables to make the coeÔ¬Écient of x3 equal to 1. The discriminant
of the cubic polynomial is 16(g2 ‚ą’ 27g3 ).
3 2

PROPOSITION 9.9
‚ąÜ = g2 ‚ą’ 27g3 = 0.
3 2

PROOF Since ‚„˜ (z) is doubly periodic, ‚„˜ (ŌČi /2) = ‚„˜ (‚ą’ŌČi /2). Since
‚„˜ (‚ą’z) = ‚ą’‚„˜ (z), it follows that

‚„˜ (ŌČi /2) = 0, i = 1, 2, 3. (9.6)

Therefore, each ‚„˜(ŌČi /2) is a root of 4x3 ‚ą’ g2 x ‚ą’ g3 , by (9.5). If we can show
that these roots are distinct, then the cubic polynomial has three distinct
roots, which means that its discriminant is nonzero. Let

hi (z) = ‚„˜(z) ‚ą’ ‚„˜(ŌČi /2).

Then hi (ŌČi /2) = 0 = hi (ŌČi /2), so hi vanishes to order at least 2 at ŌČi /2. Since
hi (z) has only one pole in F , namely the double pole at z = 0, Theorem 9.1(5)
implies that ŌČi /2 is the only zero of hi (z). In particular,

hi (ŌČj /2) = 0, when j = i.

Therefore, the values ‚„˜(ŌČi /2) are distinct.

The proposition implies that

E : y 2 = 4x3 ‚ą’ g2 x ‚ą’ g3

is the equation of an elliptic curve, so we have a map from z ‚ąą C to the
points with complex coordinates (‚„˜(z), ‚„˜ (z)) on an elliptic curve. Since ‚„˜(z)
and ‚„˜ (z) depend only on z mod L (that is, if we change z by an element of
L, the values of the functions do not change), we have a function from C/L
to E(C). The group C/L is a group, with the group law being addition of
complex numbers mod L. In concrete terms, we can regard elements of C/L
as elements of F . When we add two points, we move the result back into F by
subtracting a suitable element of L. For example, (.7ŌČ1 + .8ŌČ2 ) + (.4ŌČ1 + .9ŌČ2 )
yields .1ŌČ1 + .7ŌČ2 .

¬© 2008 by Taylor & Francis Group, LLC
270 CHAPTER 9 ELLIPTIC CURVES OVER C

THEOREM 9.10
Let L be a lattice and let E be the elliptic curve y 2 = 4x3 ‚ą’ g2 x ‚ą’ g3 . The
map

ő¦ : C/L ‚ą’‚Ü’ E(C)
z ‚ą’‚Ü’ (‚„˜(z), ‚„˜ (z))
0 ‚ą’‚Ü’ ‚ąě

is an isomorphism of groups.

PROOF The surjectivity is easy. Let (x, y) ‚ąą E(C). Since the function
‚„˜(z) ‚ą’ x has a double pole, Theorem 9.1 implies that it has zeros, so there
exists z ‚ąą C such that ‚„˜(z) = x. Theorem 9.8 implies that

‚„˜ (z)2 = y 2 ,

so ‚„˜ (z) = ¬±y. If ‚„˜ (z) = y, we‚Ä™re done. If ‚„˜ (z) = ‚ą’y, then ‚„˜ (‚ą’z) = y and
‚„˜(‚ą’z) = x, so ‚ą’z ‚Ü’ (x, y).
Suppose ‚„˜(z1 ) = ‚„˜(z2 ) and ‚„˜ (z1 ) = ‚„˜ (z2 ), and z1 ‚Č° z2 mod L. The only
poles of ‚„˜(z) are for z ‚ąą L. Therefore, if z1 is a pole of ‚„˜, then z1 ‚ąą L and
z2 ‚ąą L, so z1 ‚Č° z2 mod L. Now assume z1 is not a pole of ‚„˜, so z1 is not in
L. The function
h(z) = ‚„˜(z) ‚ą’ ‚„˜(z1 )
has a double pole at z = 0 and no other poles in F . By Theorem 9.1, it has
exactly two zeros. Suppose z1 = ŌČi /2 for some i. From Equation 9.6, we
know that ‚„˜ (ŌČi /2) = 0, so z1 is a double root of h(z), and hence is the only
root. Therefore z2 = z1 . Finally, suppose z1 is not of the form ŌČi /2. Since
h(‚ą’z1 ) = h(z1 ) = 0, and since z1 ‚Č° ‚ą’z1 mod L, the two zeros of h are z1
and ‚ą’z1 mod L. Therefore, z2 ‚Č° ‚ą’z1 mod L. But

y = ‚„˜ (z2 ) = ‚„˜ (‚ą’z1 ) = ‚ą’‚„˜ (z1 ) = ‚ą’y.

This means that ‚„˜ (z1 ) = y = 0. But ‚„˜ (z) has only a triple pole, so has only
three zeros in F . From Equation 9.6, we know that these zeros occur at ŌČi /2.
This is a contradiction, since z = ŌČi /2. Therefore, z1 ‚Č° z2 mod L, so ő¦ is
injective.
Finally, we need to show that ő¦ is a group homomorphism. Let z1 , z2 ‚ąą C
and let
ő¦(zi ) = Pi = (xi , yi ).
Assume that both P1 , P2 are Ô¬Ānite and that the line through P1 , P2 intersects
E in three distinct Ô¬Ānite points (this means that P1 = ¬±P2 , that 2P1 +P2 = ‚ąě,
and that P1 + 2P2 = ‚ąě). For a Ô¬Āxed z1 , this excludes Ô¬Ānitely many values of
z2 . There are two reasons for these exclusions. The Ô¬Ārst is that the addition
law on E has a diÔ¬Äerent formula when the points are equal. The second is

¬© 2008 by Taylor & Francis Group, LLC
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SECTION 9.2 TORI ARE ELLIPTIC CURVES

that we do not need to worry about the connection between double roots
in the algebraic calculations and double roots of the corresponding analytic
functions.
Let y = ax + b be the line through P1 , P2 . Let P3 = (x3 , y3 ) be the third
point of intersection of this line with E and let (x3 , y3 ) = P3 = ő¦(z3 ) with
z3 ‚ąą C. The formulas for the group law on E show that
2
y2 ‚ą’ y 1
1
‚ą’ x1 ‚ą’ x2
x3 =
x2 ‚ą’ x1
4
2
‚„˜ (z2 ) ‚ą’ ‚„˜ (z1 )
1
‚ą’ ‚„˜(z1 ) ‚ą’ ‚„˜(z2 ).
=
‚„˜(z2 ) ‚ą’ ‚„˜(z1 )
4

The function
(z) = ‚„˜ (z) ‚ą’ a‚„˜(z) ‚ą’ b
has zeros at z = z1 , z2 , z3 . Since (z) has a triple pole at 0, and no other
poles, it has three zeros in F . Therefore,

div( ) = [z1 ] + [z2 ] + [z3 ] ‚ą’ 3.

By Theorem 9.1(4), z1 + z2 + z3 ‚ąą L. Therefore,

‚„˜(z1 + z2 ) = ‚„˜(‚ą’z3 ) = ‚„˜(z3 ) = x3 .

We obtain
2
‚„˜ (z2 ) ‚ą’ ‚„˜ (z1 )
1
‚ą’ ‚„˜(z1 ) ‚ą’ ‚„˜(z2 ).
‚„˜(z1 + z2 ) = (9.7)
‚„˜(z2 ) ‚ą’ ‚„˜(z1 )
4

By continuity, this formula, which we proved with certain values of the zi
excluded, now holds for all zi for which it is deÔ¬Āned.
We now need to consider the y-coordinate. This means that we need to
compute ‚„˜ (z1 + z2 ). We sketch the method (the interested and careful reader
may check the details). DiÔ¬Äerentiating (9.7) with respect to z2 yields an
expression for ‚„˜ (z1 + z2 ) in terms of x1 , x2 , y1 , y2 , and ‚„˜ (z2 ). We need to
express ‚„˜ in terms of ‚„˜ and ‚„˜ . DiÔ¬Äerentiating (9.5) yields

2‚„˜ ‚„˜ = (12‚„˜2 ‚ą’ g2 )‚„˜ .

Dividing by ‚„˜ (z) (this is all right if ‚„˜ (z) = 0; the other cases are Ô¬Ālled in by
continuity) yields

2‚„˜ (z2 ) = 12‚„˜(z2 )2 ‚ą’ g2 . (9.8)

Substituting this into the expression obtained for ‚„˜ (z1 + z2 ) yields an expres-
sion for ‚„˜ (z1 + z2 ) in terms of ‚„˜(z1 ), ‚„˜ (z1 ), ‚„˜(z2 ), ‚„˜ (z2 ). Some algebraic

¬© 2008 by Taylor & Francis Group, LLC
272 CHAPTER 9 ELLIPTIC CURVES OVER C

manipulation shows that this equals the value for ‚ą’y3 obtained from the ad-
dition law for (x1 , y1 ) + (x2 , y2 ) = (x3 , ‚ą’y3 ). Therefore,

(‚„˜(z1 ), ‚„˜ (z1 )) + (‚„˜(z2 ), ‚„˜ (z2 )) = (‚„˜(z1 + z2 ), ‚„˜ (z1 + z2 )).

This is exactly the statement that

ő¦(z1 ) + ő¦(z2 ) = ő¦(z1 + z2 ). (9.9)

It remains to check (9.9) in the cases where (9.7) is not deÔ¬Āned. The
cases where ‚„˜(zi ) = ‚ąě and where z1 ‚Č° ‚ą’z2 mod L are easily checked. The
remaining case is when z1 = z2 . Let z2 ‚Ü’ z1 in (9.7), use l‚Ä™HňÜpital‚Ä™s rule, and
o
use (9.8) to obtain
2
1 ‚„˜ (z1 )
‚ą’ 2‚„˜(z1 )
‚„˜(2z1 ) =
4 ‚„˜ (z1 )
2
6‚„˜(z1 )2 ‚ą’ 1 g2
1
‚ą’ 2‚„˜(z1 )
2
= (9.10)
4 ‚„˜ (z1 )
2
6x2 ‚ą’ 1 g2
1 1
‚ą’ 2x1 .
2
=
4 y1
This is the formula for the coordinate x3 that is obtained from the addition
law on E. DiÔ¬Äerentiating with respect to z1 yields the correct formula for the
y-coordinate, as above. Therefore,

ő¦(z1 ) + ő¦(z1 ) = ő¦(2z1 ).

This completes the proof of the theorem.

The theorem shows that the natural group law on the torus C/L matches
the group law on the elliptic curve, which perhaps looks a little unnatural.
Also, the classical formulas (9.7) and (9.10) for the Weierstrass ‚„˜-function,
which look rather complicated, are now seen to be expressing the group law
for E.

9.3 Elliptic Curves over C
In the preceding section, we showed that a torus yields an elliptic curve. In
the present section, we‚Ä™ll show the converse, namely, that every elliptic curve
over C comes from a torus.
Let L = ZŌČ1 + ZŌČ2 be a lattice and let

Ō„ = ŌČ1 /ŌČ2 .

¬© 2008 by Taylor & Francis Group, LLC
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SECTION 9.3 ELLIPTIC CURVES OVER C

Since ŌČ1 and ŌČ2 are linearly independent over R, the number Ō„ cannot be
real. By switching ŌČ1 and ŌČ2 if necessary, we may assume that the imaginary
part of Ō„ is positive:
(Ō„ ) > 0.

In other words, we assume Ō„ lies in the upper half plane

H = {x + iy ‚ąą C | y > 0}.

The lattice
LŌ„ = ZŌ„ + Z

is homothetic to L. This means that there exists a nonzero complex number
ő» such that L = ő»LŌ„ . In our case, ő» = ŌČ2 .
For integers k ‚Č• 3, deÔ¬Āne

1
Gk (Ō„ ) = Gk (LŌ„ ) = . (9.11)
(mŌ„ + n)k
(m,n)=(0,0)

We have
k
Gk (Ō„ ) = ŌČ2 Gk (L),

where Gk (L) is the Eisenstein series deÔ¬Āned for L = ZŌČ1 + ZŌČ2 by (9.4). Let

q = e2ŌÄiŌ„ .

It will be useful to express certain functions as sums of powers of q. If Ō„ =
x + iy with y > 0, then |q| = e‚ą’2ŌÄy < 1. This implies that the expressions we
obtain will converge.

PROPOSITION 9.11
‚ąě
Let ő∂(x) = n=1 n‚ą’x and let

ŌÉ (n) = d
d|n

be the sum of the th powers of the positive divisors of n. If k ‚Č• 2 is an
integer, then
‚ąě
(2ŌÄi)2k
ŌÉ2k‚ą’1 (n)q n
G2k (Ō„ ) = 2ő∂(2k) + 2
(2k ‚ą’ 1)! n=1
‚ąě
(2ŌÄi)2k j 2k‚ą’1 q j
= 2ő∂(2k) + 2 .
(2k ‚ą’ 1)! 1 ‚ą’ qj
j=1

¬© 2008 by Taylor & Francis Group, LLC
274 CHAPTER 9 ELLIPTIC CURVES OVER C

PROOF We have
eŌÄiŌ„ + e‚ą’ŌÄiŌ„
cos ŌÄŌ„
ŌÄ = ŌÄi ŌÄiŌ„
e ‚ą’ e‚ą’ŌÄiŌ„
sin ŌÄŌ„
q+1 2ŌÄi
= ŌÄi = ŌÄi +
q‚ą’1 q‚ą’1
‚ąě
= ŌÄi ‚ą’ 2ŌÄi qj . (9.12)
j=0

Recall the product expansion
‚ąě
Ō„ Ō„
1‚ą’ 1+
sin ŌÄŌ„ = ŌÄŌ„
n n
n=1

(see ). Taking the logarithmic derivative yields
‚ąě
1 1 1
cos ŌÄŌ„
=+ + . (9.13)
ŌÄ
Ō„ ‚ą’n Ō„ +n
sin ŌÄŌ„ Ō„ n=1

DiÔ¬Äerentiating (9.12) and (9.13) 2k ‚ą’ 1 times with respect to Ō„ yields
‚ąě ‚ąě
1
‚ą’ (2k ‚ą’ 1)!
2k 2k‚ą’1 j 2k‚ą’1
(2ŌÄi) j q = (‚ą’1) .
(Ō„ + n)2k
n=‚ą’‚ąě
j=1

Consider (9.11) with 2k in place of k. Since 2k is even, the terms for (m, n)
and (‚ą’m, ‚ą’n) are equal, so we only need to sum for m = 0, n > 0 and for
m > 0, n ‚ąą Z, then double the answer. We obtain
‚ąě ‚ąě ‚ąě
1 1
G2k (Ō„ ) = 2 +2
2k (mŌ„ + n)2k
n m=1 n=‚ą’‚ąě
n=1
‚ąě ‚ąě
(2ŌÄi)2k j 2k‚ą’1 mj
= 2ő∂(2k) + 2 q
(2k ‚ą’ 1)!
m=1 j=1
‚ąě ‚ąě
(2ŌÄi)2k
j 2k‚ą’1 q mj .
= 2ő∂(2k) + 2
(2k ‚ą’ 1)! m=1 j=1

Let n = mj in the last expression. Then, for a given n, the sum over j can
be regarded as the sum over the positive divisors of n. This yields the Ô¬Ārst
expression in the statement of the proposition. The expansion m‚Č•1 q mj =
q j /(1 ‚ą’ q j ) yields the second expression.

Recall that we deÔ¬Āned g2 = g2 (L) = 60G4 (L) and g3 = g3 (L) = 140G6 (L)
for arbitrary lattices L. Restricting to LŌ„ , we deÔ¬Āne

g2 (Ō„ ) = g2 (LŌ„ ), g3 (Ō„ ) = g3 (LŌ„ ).

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SECTION 9.3 ELLIPTIC CURVES OVER C

Using the facts that

ŌÄ4 ŌÄ6
ő∂(4) = and ő∂(6) = ,
90 945
we obtain
‚éõ ‚éě
‚ąě
4ŌÄ 4 4ŌÄ 4 ‚éĚ j 3 qj ‚é†
(1 + 240q + ¬· ¬· ¬· ) =
g2 (Ō„ ) = 1 + 240
1 ‚ą’ qj
3 3 j=1
‚éõ ‚éě
‚ąě
8ŌÄ 6 8ŌÄ 6 ‚éĚ j 5 qj ‚é†
(1 ‚ą’ 504q + ¬· ¬· ¬· ) = 1 ‚ą’ 504
g3 (Ō„ ) = .
1 ‚ą’ qj
27 27 j=1

Since ‚ąÜ = g2 ‚ą’ 27g3 , a straightforward calculation shows that
3 2

‚ąÜ(Ō„ ) = (2ŌÄ)12 (q + ¬· ¬· ¬· ).

DeÔ¬Āne
3
g2
j(Ō„ ) = 1728 .
‚ąÜ
+ ¬· ¬· ¬· . Including a few more terms in the above calculations
1
Then j(Ō„ ) = q
yields
1
+ 744 + 196884q + 21493760q 2 + ¬· ¬· ¬· .
j(Ō„ ) =
q
For computational purposes, this series converges slowly since the coeÔ¬Écients
are large. It is usually better to use the following.

PROPOSITION 9.12

3
j 3 qj
‚ąě
1 + 240 j=1 1‚ą’q j
j(Ō„ ) = 1728 2.
3
‚ąě ‚ąě
j 3 qj j 5 qj
‚ą’ 1 ‚ą’ 504
1 + 240 j=1 1‚ą’q j j=1 1‚ą’q j

PROOF Substitute the above expressions for g2 , g3 into the deÔ¬Ānition of
the j-function. The powers of ŌÄ and other constants cancel to yield the present
expression.

It can be shown (see [70, p. 249]) that
‚ąě
(1 ‚ą’ q k )24 .
12
‚ąÜ = (2ŌÄ) q
k=1

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276 CHAPTER 9 ELLIPTIC CURVES OVER C

This yields the expression
3
j 3 qj
‚ąě
1 + 240
1 ‚ą’ qj
j=1
j= ,
‚ąě
q k=1 (1 ‚ą’ q k )24
which also works very well for computing j.
More generally, if L is a lattice, deÔ¬Āne
g2 (L)3
j(L) = 1728 .
g2 (L)3 ‚ą’ 27g3 (L)2
If ő» ‚ąą C√— , then the deÔ¬Ānitions of G4 and G6 easily imply that
g2 (ő»L) = ő»‚ą’4 g2 (L) g3 (ő»L) = ő»‚ą’6 g3 (L).
and (9.14)
Therefore
j(L) = j(ő»L).
‚ą’1
Letting L = ZŌČ1 + ZŌČ2 and ő» = ŌČ2 , we obtain
j(ZŌČ1 + ZŌČ2 ) = j(Ō„ ),
where Ō„ = ŌČ1 /ŌČ2 .
Recall that
ab
a, b, c, d ‚ąą Z, ad ‚ą’ bc = 1
SL2 (Z) =
cd
acts on the upper half plane H by
aŌ„ + b
ab
Ō„=
cd cŌ„ + d
for all Ō„ ‚ąą H.

PROPOSITION 9.13
ab
Let Ō„ ‚ąą H and let ‚ąą SL2 (Z). Then
cd
aŌ„ + b
j = j(Ō„ ).
cŌ„ + d

PROOF We Ô¬Ārst compute what happens with Gk :
aŌ„ + b 1
Gk =
(m aŌ„+d + n)k
+b
cŌ„ + d cŌ„
(m,n)=(0,0)
1
= (cŌ„ + d)k
(m(aŌ„ + b) + n(cŌ„ + d))k
(m,n)=(0,0)
1
= (cŌ„ + d)k .
((ma + nc)Ō„ + (mb + nd))k
(m,n)=(0,0)

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SECTION 9.3 ELLIPTIC CURVES OVER C

ab
Since has determinant 1, we have
cd
‚ą’1
d ‚ą’b
ab
= .
‚ą’c a
cd

Let
ab
(m , n ) = (m, n) = (ma + nc, mb + nd).
cd
Then
d ‚ą’b
(m, n) = (m , n ) ,
‚ą’c a
so there is a one-to-one correspondence between pairs of integers (m, n) and
pairs of integers (m , n ). Therefore,

aŌ„ + b 1
= (cŌ„ + d)k
Gk
(m Ō„ + n )k
cŌ„ + d
(m ,n )=(0,0)

= (cŌ„ + d)k Gk (Ō„ ).

Since g2 and g3 are multiples of G4 and G6 , we have
aŌ„ + b aŌ„ + b
= (cŌ„ + d)4 g2 (Ō„ ), = (cŌ„ + d)6 g3 (Ō„ ).
g2 g3
cŌ„ + d cŌ„ + d
Therefore, when we substitute these expressions into the deÔ¬Ānition of j, all
the factors (cŌ„ + d) cancel.

Let F be the subset of z ‚ąą H such that
1 1 ŌÄ ŌÄ
|z| ‚Č• 1, ‚ą’ ‚Č¤ z = eiőł for
(z) < , <őł< .
2 2 3 2
Figure 9.3 is a picture of F . Since we will need to refer to it several times, we
let
ŌĀ = e2ŌÄi/3 .

PROPOSITION 9.14
Given Ō„ ‚ąą H, there exists

ab
‚ąą SL2 (Z)
cd

such that
aŌ„ + b
= z ‚ąą F.
cŌ„ + d
Moreover, z ‚ąą F is uniquely determined by Ō„ .

¬© 2008 by Taylor & Francis Group, LLC
278 CHAPTER 9 ELLIPTIC CURVES OVER C

i
ő°

Figure 9.3
The Fundamental Domain for SL2 (Z)

The proposition says that F is a fundamental domain for the action of
SL2 (Z) on H. For a proof of the proposition, see  or .

COROLLARY 9.15
Let L ‚ä‚ C be a lattice. There exists a basis {ŌČ1 , ŌČ2 } of L with ŌČ1 /ŌČ2 ‚ąą F.
In other words,
L = (ő»)(ZŌ„ + Z)
for some ő» ‚ąą C√— and some uniquely determined Ō„ ‚ąą F.

PROOF Let {ő±, ő≤} be a basis for L and let Ō„0 = ő±/ő≤. By changing the
sign of ő± if necessary, we may assume that Ō„0 ‚ąą H. Let

ab
‚ąą SL2 (Z)
cd

be such that
aŌ„0 + b
= Ō„ ‚ąą F.
cŌ„0 + d
Let
ŌČ1 = aő± + bő≤, ŌČ2 = cő± + dő≤.
Since the matrix is in SL2 (Z),

L = Ző± + Ző≤ = ZŌČ1 + ZŌČ2 = ŌČ2 (ZŌ„ + Z).

This proves the corollary.

¬© 2008 by Taylor & Francis Group, LLC
279
SECTION 9.3 ELLIPTIC CURVES OVER C

If z ‚ąą C, recall that ordz f is the order of f at z. That is,

f (Ō„ ) = (Ō„ ‚ą’ z)ordz (f ) g(Ō„ ),

with g(z) = 0, ‚ąě. We can also deÔ¬Āne the order of f at i‚ąě. Suppose

f (Ō„ ) = an q n + an+1 q n+1 + ¬· ¬· ¬· , (9.15)

with n ‚ąą Z and an = 0, and assume that this series converges for all q close
to 0 (with q = 0 when n < 0). Then

ordi‚ąě (f ) = n.

Note that q ‚Ü’ 0 as Ō„ ‚Ü’ i‚ąě, so ordi‚ąě (f ) expresses whether f vanishes (n > 0)
or blows up (n < 0) as Ō„ ‚Ü’ i‚ąě.

PROPOSITION 9.16
Let f be a function meromorphic in H such that f is not identically zero and
such that
aŌ„ + b ab
‚ąą SL2 (Z).
= f (Ō„ ) for all
f
cd
cŌ„ + d
Then
1 1
ordi‚ąě (f ) + ordŌĀ (f ) + ordi (f ) + ordz (f ) = 0.
3 2
z=i,ŌĀ,i‚ąě

REMARK 9.17 The function f can be regarded as a function on the
 ŮÚū. 1(‚ŮŚ„Ó 3)—őńŇ–∆ņÕ»Ň >>