8 1. The Classical Modular Forms
given parallelogram. Applying (1.16) to the elliptic function f(u)/f(u), we
get
(1.17) Y, rnf(w)=0.
w (mod
A)
Similarly integrating
uff(u)/f(u)
along the boundary dP we get
(1.18) ] P mf(w)w = 0 (mod A).
w (mod
A)
Define the order of / to be the sum of orders of zeros (mod A) or the
negative of the sum of orders of poles (mod A), i.e.,
(1.19) Tf— V^ max(ra/(iu),0) = Y^ min(ra/(w),0).
w (mod
A)
w (mod
A)
It follows from (1.16) that there is no elliptic function of order 1. Prom
(1.17) applied to f(u) c it follows that any complex number c is a value
of f(u) for exactly r/ points (mod A) counted with multiplicity.
The simplest elliptic function which is not constant has one pole of order
two with zero residue at the origin. Such a function can be constructed by
averaging over the lattice; for example, one obtains
(1.20) p(u) =
u-2
+ £ ' ((u -
u)'2
- or
2
)
where / indicates that UJ = 0 is skipped in the summation. Notice that p(u)
is even, i.e., p(u) p(—u). Since p(u) has order 2 we infer that
(1.21) p(u) = p(w) == u = ±w (mod A).
Moreover p(u) p(w) has a double zero at u w exactly when w = —w
(mod A). There are three such points (mod A), namely u;i/2, 0J2/2 and
(uui + 0J2) /2 (the point w = 0 is not because it is a pole of p). The numbers
are all distinct because of (1.21); thus the associated discriminant
(1.23) A - 16 (ei -
e2)2
(e2 -
e3)2
(e3 -
ex)2
is not zero. The function p(u) is called the Weierstrass function.
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