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436 Appendix III: Elliptic Curves and Topological Modular Forms

2
F(a )(x , y ) = F(a)(x, y) = F(a)(v x + r,v y + v sx + t)

3
2

2
2
2

3

2
3
= (v y + v sx + t) + a 1 (v y + v sx + t)(v x + r)
2
3
3
2
2

+ a 3 (v y + v sx + t) − (v x + r) − a 2 (v x + r) 2
2
− a 4 (v x + r) − a 6 .
6 2 5 3 6 3
= v (y ) + v (a 1 + 2s)x y + v (a 3 + a 1 r + 2t)y − v (x )
4 2 2
− v (a 2 + 3r − s − a 1 s)(x )
2
2
− v (a 4 + 2ra 2 + 3r − sa 3 − a 1 (rs + t) − 2st)x
2
2
3
− (a 6 + a 4 r + a 2 r + r − a 3 t − a 1 rt − t ).
i
Thus we can write v a = a i + δ i (r, s, t,v) where δ i (r, s, t,v) is a polynomial
i
over the integers in all a j with j < i. We have the relations where the polynomials
δ i = δ i (r, s, t, 1) are explicitly given by

a = a 1 + 2s thus δ 1 = 2s
1

a = a 2 − a 1 s + 3r − s 2 thus δ 2 =−a 1 s + 3r − s 2
2

a = a 3 + a 1 r + 2t thus δ 3 = a 1 r + 2t
3
2
a = a 4 + 2a 2 r − a 1 (rs + t) − a 3 s + 3r − 2st

4
2
thus δ 4 = 2a 2 r − a 1 (rs + t) − a 3 s + 3r − 2st
2
3

a = a 6 + a 4 r + a 2 r + r − a 3 t − a 1 rt − t 2
6
2
3
2
thus δ 6 = a 4 r + a 2 r + r − a 3 t − a 1 rt − t .
3
∗
Given a cubic F(a)(x, y) and a four tuple (r, s, t,v) ∈ T × R there exists a unique

cubic F(a )(x, y) with the morphism

φ r,s,t,v : F(a ) → F(a).
This means that the groupoid WT (R) is of the form of group object G(R) acting on
the set of WP(R) of Weierstrass polynomials over R by substitution, or as in (2.9)
we have WT (R) = WP(R)
G(R) .

(4.4) Remark. Let w : R → R be a morphism of commutative rings. There is an

associated functor WT (w) : WT (R ) → WT (R ) given by WT (w)(F(a)(x, y)) =
F(w(a))(x, y) and
WT (w)(φ r,s,t,v ) = φ w(r),w(s),w(t),w(v) .

For a second morphism v : R → R of rings we have the composition of functors
WT (v)WT (w) = WR(vw).
We have constructed a functor from commutative rings to the category of small
categories, and now we show that it is representable by a Hopf algebroid, that is, a
cocategory in the category of commutative rings.