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86 4. Families of Elliptic Curves and Geometric Properties of Torsion Points

(1.1) Remark. The group E(k(t)) of points of E over k(t) is the group of rational
cross-sections of the algebraic family E → T of elliptic curves E t over k.One such
cross-section of the family is always the zero cross-section.

(1.2) Definition. For a field k of characteristic = 2, the Legendre family of elliptic
2
curves is E λ : y = x(x − 1)(x − λ).
The curve E λ is nonsingular for λ = 0, 1, so that over k −{0, 1}, it is a family of
nonsingular elliptic curves. There are four basic cross-sections:

0(λ) = 0, e 1 (λ) = (0, 0), e 2 (λ) = (1, 0), e 3 (λ) = (λ, 0).

The values in E λ (k) of these four cross-sections give the group 2 E λ (k) of 2-
division points in E λ , and the four cross-sections as points in E(k(λ)) give the
group 2 E(k(λ)) of 2-division in E. The Legendre family E λ of cubics over the
2
2
λ-line has two singular fibres which are nodel cubics E 0 : y = x (x − 1) and
2
2
E 1 : y = x(x − 1) .At λ = 0 the cross-sections e 1 and e 3 take the same value
equal to the double point (0, 0) and at λ = 1 the cross-sections e 2 and e 3 take the
same value equal to the double point (1, 0).
There are six possible orderings for the 2-division points on an elliptic curve E,
or equivalently, six possible bases (e 1 , e 2 ) for the subgroup 2 E. Now we consider
the effect of some of these permutations on the Legendre form where the basis is
implicitly given from the form of the cubic.

EXAMPLE 1. The effect of the transposition of (0, 0) and (1, 0) in this three-element
set in 2 E −{0} can be viewed as a mapping E λ → E λ , such that (0, 0) corresponds

to (1, 0), (1, 0) to (0, 0),and (λ, 0) to (λ , 0) on E λ . In this case λ = 1 − λ,and

the mapping is given by (x, y) → (1 − x, iy) in terms of a change of variable, since
2
(iy) = (1 − x)(1 − x − 1)(1 − x − λ), becomes
2
y = x(x − 1) (x − (1 − λ)) .
EXAMPLE 2. The effect of the transposition of (1, 0) and (λ, 0) can be viewed as
a mapping E λ → E λ such that (0, 0) corresponds to (0, 0), (λ, 0) to (1, 0),and

(1, 0) to (λ , 0) on E λ . In this case λ = 1/λ, and the mapping is given by (x, y) →

2
(λx,λ 3/2 y) in terms of a change of variable, since (λ 3/2 y) = (λx)(λx −1)(λx −λ),
becomes
1

2
y = x(x − 1) x − .
λ
These changes of variable generate the group G of order 6 acting on P 1 , which
is nonabelian and hence is isomorphic to the symmetric group on three letters. The
changes of variable are defined over the field k(λ 1/2 , i), and this field is an extension
field of k(λ), the field of definition of the Legendre curve.