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§4. Tate’s Normal Form for a Cubic with a Torsion Point 93

In particular, (0, 0) is not a point of order 2. By (2.2) the point (0, 0) has order 3 if
and only if a 2 = 0and a 3 = 0.
Assume that (0, 0) is not of order 2 or 3, that is , both a 2 and a 3 are each unequal
2
3
to 0. By changing x to u x and y to u y, we can make a 3 = a 2 =−b, and we
obtain the following form of the equation of the curve which now depends on two
parameters. Observe that all of the above discussion is carried out in a given field k,
and the changes of variable did not require field extensions.
(4.1) Definition. The Tate normal form of an elliptic curve E with point P = (0, 0)
is
2 3 2
E = E(b, c) : y + (1 − c)xy − by = x − bx .
Further a calculation, using the relations in 3(3.3), yields the following formula
for the discriminant = (b, c) of E(b, c):
2 4
3 3
4 3
(b, c) = (1 − c) b − (1 − c) b − 8(1 − c) b
4 4 5
+ 36(1 − c)b − 27b + 16b .
(4.2) Remark. The Tate normal form gives a description in terms of equations for
the set of pairs (E, P) consisting of an elliptic curve E together with a point P on E
where P,2P, and 3P are all unequal to zero. These pairs correspond to pairs (b, c)
with both b = 0and (b, c) = 0. The curve in the family corresponding to (b, c)
is E(b, c) and the point is P = (0, 0). In the two-parameter Tate family E(b, c)
there will be many cases where curves in different fibres E(b, c) are isomorphic, for
example, E(b, 1) and E(b, −1) are isomorphic curves.
If we require, further, that nP = 0 for some integer n > 3, then there is a
polynomial equation f n (b, c) = 0 over Z which b and c must satisfy. The result is
that the relations

T n : f n (b, c) = 0, b = 0, (b, c) = 0
define an open algebraic curve with a family E(b, c) of elliptic curves over it together
with a given n division point P, i.e., a cross-section P in the family of order n.We
will determine f n explicitly in several cases, see (4.6).

(4.3) Remarks. Since this family contains, up to isomorphism, all elliptic curves E
with torsion point P of order n, the curve T n maps onto the open curve Y 1 (n) which
is the parameter space for isomorphism classes of pairs (E, P) of elliptic curves E
together with a point P of order n. The curve Y 1 (n) has a completion X 1 (n) which
is nonsingular where the completing points, called cusps, correspond to degenerate
elliptic curves. We return to this later.
(4.4) Remark. There is an elliptic curve E over a field k with torsion point P of
order n over the field k if and only if the open algebraic curve T n has k rational points,
or in other words, the set T n (k) is nonempty. This is equivalent to the statement
that Y 1 (n)(k) is nonempty. This is also equivalent to X 1 (n)(k) having noncuspidal k
rational points. For further discussion of Y 1 (n) and X 1 (n) see Chapter 11, §2and §3.