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434 Appendix III: Elliptic Curves and Topological Modular Forms

(4) The above construction defines a functor C → (Grpoid) where for a morphism
Z → Z composition on the left induces a morphism between groupoids.

The main example of these ideas will be the category of Weierstrass equations
and their transformations which leads to the category of elliptic curves over a field
k. This is discussed in the next sections.

§4. The Category WT(R) and the Weierstrass Hopf Algebroid

We consider a category WT (R) for each commutative ring R whose objects are
Weierstrass polynomials and morphisms are triangular changes of variables over the
ring R. The relation to elliptic curves is explained in 3(2.3)–3(2.7) where a Weier-
strass polynomial is called a cubic in normal form.

(4.1) Definition. Let R be a commutative ring. The objects of the category WT (R)
are polynomials F(a)(x, y) in R[x, y] of the form
2 3 2
F(a)(x, y) = y + a 1 xy + a 3 y − x − a 2 x − a 4 x − a 6
5
where (a) = (a 1 , a 2 , a 3 , a 4 , a 6 ) ∈ R .
The morphisms φ r,s,t,v : F(a ) → F(a) are given by r, s, t ∈ R and v ∈ R ∗

where the follow relation is satisfied
2
2
3

F(a )(x , y ) = F(a)(v x + r,v y + v sx + t).

This is the triangular change of variable morphism where
2
2
3

x = v x + r and y = v y + v sx + t

so that F(a )(x , y ) = F(a)(x, y) is the previous condition. Composition is given
by substitution of these variables.

(4.2) Remark. We can describe composition of two morphisms φ r,s,t,v : F(a ) →
F(a) and φ r ,s ,t ,v : F(a ) → F(a ) explicitly. It is given by a substitution within a

substitution which has the same triangular change of variable morphism

φ r ,s ,t ,v = φ r,s,t,v φ r ,s ,t ,v : F(a ) → F(a)

To derive the rule of composition, we consider a substitution within a substitution
starting with the following two expressions
2

3

2

F(a )(x , y ) = F(a)(v x + r,v y + v sx + t) = F(a)(x, y),
and

2
2

F(a )(x , y ) = F(a)(v x + r ,v y + v s x + t ).

3