Page 92 -

§2. Normal Forms for Cubic Curves 69

Now for the second derivation of the normal form of the cubic equation for an
elliptic curve we use Hartshorne [1977], Chapter 4 as background reference. The
reader unfamiliar with this general curve theory can skip to the next section.
For the remainder of this section a curve C is a complete, nonsingular curve over
an algebraically closed field k and 0 is a point of C(k).

(2.6) Riemann–Roch for Curves of Genus 1. Let O C be the structure sheaf on C
of germs of regular functions. Let O C (m · 0) be the sheaf of germs of functions
having at most an mth order pole at 0. For the vector space of sections
(O C (m · 0))
the Riemann–Roch theorem gives the following formula when C is of genus 1:

m for m > 0,
dim k
(O C (m · 0)) =
1 for m = 0.

We can choose a basis for these spaces
(O C (m · 0)) for small m as follows

where we make use of the inclusions
(O C (m · 0)) ⊂
(O C (m · 0)) for m ≤ m :

(O C (1 · 0)) =
(O C (0 · 0)) = k · 1,

(O C (2 · 0)) = k · 1 ⊕ k · x,

(O C (3 · 0)) = k · 1 ⊕ k · x ⊕ k · y,
2

(O C (4 · 0)) = k · 1 ⊕ k · x ⊕ k · y ⊕ k · x ,
2

(O C (5 · 0)) = k · 1 ⊕ k · x ⊕ k · y ⊕ k · x ⊕ k · xy.
Here x has a pole of order 2 and y of order 3 at 0. In
(O C (6 · 0)) there are seven
natural basis elements,
2 3 2
1, x, y, x , xy, x , and y ,
but the space is six dimensional. Hence there is a linear relation which is the defining
equation of the image under

(1: x : y) : C → P 2 (k).

Further, given a local uniformizing parameter z at 0 generating the maximal idea
of the local ring O C of C at 0, we can specify that the formal analytic expansion of
x and y be of the form

1 1
x = +· · · and y =− + ··· .
z 2 z 3
Note that x is unique up to a constant and y up to a linear combination of x and a
2
3
constant. With this normalization the linear relation satisfied by 1, x, y, x , xy, x ,
3
and y is exactly the normal form of the cubic.
(2.7) Remark. The origin of admissible changes of variables can be seen from the
point of view of changing z to a new local uniformizing parameter uz =¯z by mul-
tiplying z by a nonzero element of the ground field. Then x and y are changed to