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§3. The Weierstrass ℘-Function 173

Proof. Every elliptic function f (z) is the sum of an even and an odd elliptic function

f (z) + f (−z) f (z) − f (−z)
f (z) = + .
2 2

Since ℘ (z) times an odd elliptic function is an even elliptic function, it suffices to
show that C(℘(z)) is the field of even elliptic functions.
If f (z) is an even elliptic function, then ord 0 f (z) = 2m is even and f (z) =
℘(z) −m g(z), where g(z) is an even elliptic function with no zeros or poles on the
associated lattice L.If a is a zero of ℘(z) − c,thensois ω − a for ω ∈ L,and if a is
a zero or pole of g(z),thensois ω − a.If2a ∈ L, then the zero (or pole) is of order

at least 2 since g (−z) =−g (z) and so g (a) = g (−a) =−g (a). Thus

(℘(z) − ℘ (a i ))
i
g(z) = c · ,
(℘(z) − ℘ (b i ))
i
where {a i ,ω − a i } are the zeros of g(z) and {b i ,ω − b i } are the poles of g(z) in a
fixed fundamental domain. This proves the theorem.
Let M L denote the field of elliptic meromorphic functions for the lattice L. This
can be viewed as the field of meromorphic functions on the complex torus C/L = T
as in (2.7).
(3.4) Definition. Adivisor D on C/L is a finite formal integral linear combination

D = n u (u) of points u in C/L, and its degree deg(D) = n u is an integer.
u u
The group Div(C/L) of all divisors on C/L is the group of all divisors. A principal

divisor D is one of the form ( f ) = ord u ( f )(u), where f is a nonzero meromor-
u
phic function in M L .WedenotebyDiv 0 (C/L) and Div p (C/L), respectively, the
subgroups of divisors of degree zero and principal divisors.
We have an exact sequence
deg
0 → Div 0 (C/L) → Div(C/L) → Z → 0,
and Div p (C/L) is a subgroup of Div 0 (C/L) by (2.5).

(3.5) Theorem (Abel–Jacobi). The function f ( n u (u)) = n u · ufrom
u u
Div(C/L) to C/L induces a function also denoted f :Div 0 (C/L)/Div p (C/L) →
C/L which is an isomorphism of groups.
Proof. Since u = f ((u)−(0)) for any u in C/L, the group homomorphism is surjec-
tive. The subgroup Div p (C/L) is carried to zero by f from (2.5). The construction

following the proof of Theorem (3.3) above shows that n u · u = 0in C/L is suf-
u
ficient for the existence of an elliptic function f with ( f ) = u n u (u). This proves
the theorem.
Another proof that Div p (C/L) is thekernelof f results by using the σ-function
introduced in Exercises 3, 4, and 5 at the end of this section.