1.2. Elliptic functions 9
The Weierstrass function is basic in the sense that it generates the sub-
field C (p) of all even elliptic functions in E(A). Indeed if / G E(A) is even,
then by comparing singularities (the zeros and the poles) we find that
f(U) = c n (p(«) -
pwrf{w)
w (mod
A)
where c is a constant. Here the product takes only one singular point of /
out of each pair ±w(mod A).
Differentiating p(u) we obtain the elliptic function
(1.24) p\u) = - 2
J2(u
~
u
) ~
3
UJEA
which is odd,
pf{u)
= p'{—u). Therefore the field of all elliptic functions
for a lattice A is generated by p and p',
(1.25) £(A) = C ( p , p ' ) .
The functions p(tx), p'{u) are linearly independent over the constant field
C, but they are algebraically related through the Weierstrass equation
(1.26) p\uf = 4 (p(u) - ei) (p(u) - e2) (p(u) - e3)
with ei, e2, es given by (1.22). To derive this equation recall that p(u) e\
has exactly one double zero at u =
CJI/2,
which is also a simple zero of p'{u).
The same holds true for the other factors. Therefore the ratio
f(u) = p\uf (p(u) -
ex)"1
(p(u) - ea)-
1
(p(u) -
ez)~l
is an elliptic function whose only possible pole (mod A) is at the origin. But
u = 0 is a regular point of f(u) since
pf(u)
~
2u~3
and p(u) ~
u~2
as
u —* 0. Therefore /(u) is constant, in fact /(u) = 4 as claimed.
Prom (1.20) one derives easily that
oo
(1.27) p(u) =
u~2
+ ^ (m +
l)Gm+2Um
m=l
where
(1.28) Gk =
Y^"-k-
ueA
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