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4

Families of Elliptic Curves and Geometric

Properties of Torsion Points

In this chapter we consider families of elliptic curves by studying cubics in normal
form with coefficients depending on a parameter. The most important example is
2
the Legendre family E : y = x(x − 1)(x − λ) over k(λ), where k is a field of
characteristic unequal to 2. The points {(0, 0), (1, 0), (λ, 0)} are the three 2-torsion
points on E λ for each value of λ ∈ k −{0, 1}, and they are specified with a given
ordering.
There are families for points of order 3 and points of order 4. Then various fam-
ilies for higher-order points are considered. For example, there is the Tate normal
form of the cubic

3
2
2
E(b, c) : y + (1 − c)xy − by = x − bx ,
where with polynomial conditions on b and c the point (0, 0) has a given order n.
We close the chapter with an explicit isogeny, that is, a homomorphism of elliptic
curves given by algebraic change of coordinates. The curve in question is E[a, b]:
2
3
2
y = x + ax + bx which in characteristic unequal to 2 is nonsingular for b and
2
2
a − 4b different from 0. The isogeny is a morphism E[a, b] → E[−2a, a − 4b]
and has kernel of order 2 containing (0, 0). This isogeny when composed with its
dual is multiplication by 2, hence is called a 2-isogeny.
§1. The Legendre Family

Consider a normal cubic equation with a i (t) ∈ k[t]
2 3 2
y + a 1 (t)xy + a 3 (t)y = x + a 2 (t)x + a 4 (t)x + a 6 (t)
giving an elliptic curve E over k(t), then we can substitute in any value for t ∈ T ,the
parameter space, and obtain a normal cubic equation, and, hence, an elliptic curve E t
over k at all points T where (E t ) = 0. Each point P(t) = (x(t), y(t)) ∈ E(k(t))
can be viewed as a mapping t → P(t) ∈ E(k(t)) = E t (k) by substitution of specific
values of t, defining a map T → E. Such a map is called a cross-section.