Number Theory: An Introduction to Mathematics

3 Elliptic Functions 511

we obtain

2 x 1 x 2 (x 1 x 2 ′+x′ 1 x 2 )′−(x 1 x′ 2 +x 1 ′x 2 )^2 − 2 x 1 x 2 x 1 ′x 2 ′

=x^21 { 2 x 2 fλ(x 2 )−gλ(x 2 )}+x 22 { 2 x 1 fλ(x 1 )−gλ(x 1 )}.

But ifgλ(x)=αx^3 +βx^2 +γxandfλ(x)=g′λ(x)/2, then

2 xfλ(x)−gλ(x)=x^2 ( 2 αx+β).

Hence

2 x 1 x 2 (x 1 x′ 2 +x 1 ′x 2 )′−(x 1 x 2 ′+x 1 ′x 2 )^2 = 2 x 12 x 22 {α(x 1 +x 2 )+β}+ 2 x 1 x 2 x′ 1 x 2 ′.

On the other hand,

(x 1 ′−x 2 ′)(x 1 x 2 ′+x′ 1 x 2 )=x 2 gλ(x 1 )−x 1 gλ(x 2 )+(x 1 −x 2 )x′ 1 x′ 2

=x 1 x 2 (x 1 −x 2 ){α(x 1 +x 2 )+β}+(x 1 −x 2 )x′ 1 x′ 2.

Comparing these two relations, we obtain

{ 2 x 1 x 2 (x 1 x′ 2 +x 1 ′x 2 )′−(x 1 x 2 ′+x′ 1 x 2 )^2 }(x 1 −x 2 )= 2 x 1 x 2 (x′ 1 −x 2 ′)(x 1 x 2 ′+x′ 1 x 2 ).

If we divide by 2x 1 x 2 (x 1 −x 2 )(x 1 x 2 ′+x 1 ′x 2 ), this takes the form

(x 1 x 2 ′+x′ 1 x 2 )′

x 1 x 2 ′+x′ 1 x 2

−

x 1 x 2 ′+x′ 1 x 2

2 x 1 x 2

=

x 1 ′−x 2 ′

x 1 −x 2

,

which can be integrated to give

(x 1 x 2 ′+x 1 ′x 2 )^2 =C(τ)x 1 x 2 (x 1 −x 2 )^2 ,

where the constantC(τ)depends onτ. Equivalently,

[S(u)S′(v)−S′(u)S(v)]^2 =C((u+v)/ 2 )S(u)S(v)[S(u)−S(v)]^2.

To evaluate the constant, we divide throughout byS(v)and letv →0. By (9), this
yieldsC(u/ 2 )=γ/S(u).Sinceγ=4 (for Riemann’s normal form), we obtain finally

S(u+v)= 4 S(u)S(v)[S(u)−S(v)]^2 /[S(u)S′(v)−S′(u)S(v)]^2. (10)

ThusS(u+v)is a rational function ofS(u),S(v),S′(u),S′(v). Moreover, since
(S′)^2 =gλ(S), there exists a polynomialp(x,y,z), not identically zero and with coef-
ficients independent ofuandv, such thatp[S(u+v),S(u),S(v)]=0. In other words,
the functionS(u)has analgebraic addition theorem.
The relation (10) can also be written in the form

S(u+v)=[S(u)S′(v)+S′(u)S(v)]^2 / 4 S(u)S(v)[1−λS(u)S(v)]^2 , (11)