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§3. The Weierstrass ℘-Function 171

(2.7) Remark. An elliptic function f (z) associated with L is a function on the com-
plex plane C, but the periodicity condition is equivalent to f defining a function f ∗
∗
on the quotient torus T = C/L where for the projection p : C → C/Lf p = f .In
this way we view M L as the field of meromorphic functions on the torus T = C/L
with quotient complex structure under the bijection which assigns to f the quotient
∗
function f .
(2.8) Remark. If λ is a complex number, satisfying λL ⊂ L , then λ defines a

morphism λ : T = C/L → T = C/L , the function which assigns to f ∈ M L the

elliptic function f (λz) ∈ M L is an embedding of M L as a subfield of the field M L

of elliptic functions on T = C/L. When λ is an isomorphism, then this map M L →

M L is an isomorphism of fields. We will see later that T and T are isomorphic if and

only if M L and M are isomorphic fields. For the above embedding i λ : M L → M L
L
the degree of the field extension is given by the following formula
& '

[M L : M L ] = deg(λ) = L : λL .

§3. The Weierstrass ℘-Function
In order to prove the convergence of certain infinite expressions defining elliptic
functions, we will need the following lemma.
(3.1) Lemma. For a lattice L and a real number s > 2, the following infinite sum
converges absolutely

1
.
ω s
ω∈L−0
Proof. Let L = Zω 1 +Zω 2 where ω 2 /ω 1 is nonreal. Since L is discrete, there exists
c > 0 such that for all integers n 1 , n 2 we have |n 1 ω 1 + n 2 ω 2 |≥ c(|n 1 |+|n 2 |).
Considering the 4n pairs (n 1 , n 2 ) with |n 1 |+|n 2 |= n, we obtain

1 4 1
≤ < +∞
s s s−1
|ω| c n
ω =0 1≤n
for s − 1 > 1.
We wish to construct an elliptic function f (z) with the origin as the only pole and
z −2 as singular part. Since f (z)− f (−z) is an elliptic function with no singularities,
we see that f (z) will be an even function. Up to a constant, it will have a Laurent
2
4
development of the form z −2 + a 2 z + a 4 z +· · · + a 2k z 2k + ··· . Also it is zero at
ω 1 /2 for L = Zω 1 + Zω 2 .
(3.2) Definition. The Weierstrass ℘-function and ζ-function associated with a lat-
tice L are given by the infinite sums with and without notation for L