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42 1. Elementary Properties of the Chord-Tangent Group Law on a Cubic Curve

3
0 = x + (a − λ)x + (b − β)
= (x − x 1 )(x − x 2 )(x − x 3 )

and x 1 + x 2 + x 3 = 0. Hence the set of points of A defined over k, denoted A(k),
3
has the structure of a group where (0, b) = 0and −(x, y) =−(x, x + ax + b) =
3
(−x,(−x) −ax +b). Again three points add to zero if and only if they lie on a line.
(5.1) Remark. This example does not give rise to a new group with the group struc-
2
3
ture depending on the coefficients a and b as in the case of y = x + ax + b,or,
3
more generally, a cubic in normal form. In fact the function f (t) = (t, t + at + b)
is an isomorphism f : k → A(k) of the additive group of the line k onto A(k).
The above example can be used to study the group law on E ns (k), the set of
2
3
2
3
nonsingular points of y = x , i.e., all (x, y) = (0, 0) with y = x . Making the
change of variable
u 1 x 1
x = , y = , or u = , v = ,
v v y y
2 3 3 2 3
the equation y = x is transformed into (u/v) = 1/v or v = u . A line ax +by +
c = 0 is transformed into the line with equation au + b + cv = 0, and the point 0 at
infinity in the x, y-plane is transformed to (0, 0) in the u, v-plane. Thus we have:
2 3
(5.2) Proposition. The function g(t) = (1/t , 1/t ) is an isomorphism g : k →
E ns (k) of the additive groupk onto the groupof nonsingular points of the cuspidal
2
3
cubic curve y = x .
(5.3) Remark. The origin (0, 0) cannot be included in the set with the chord-tangent
3
2
group law on E(k) for E defined by y = x since any line y = λx through (0, 0)
3
2
intersects the cubic at only one other point (x, y) = (λ ,λ ). The chord-tangent
group law is defined with all lines not passing through (0, 0).
2
2
3
2
Consider the cubic y = x (x + 1) = x + x with a double point (or node) at
the origin (0, 0). Using the substitution
y + x 1
u = and v = ,
y − x y − x
and the calculations
2x
u − 1 =
y − x
and

3
3
3
3
2
2
(y − x) (u − 1) = 8x = 8(y − x ) = 8uv(y − x) ,
3
we obtain the equation (u − 1) = 8uv, which is similar to the cubic in (5.1). Lines
in the x, y-plane are transformed into lines in the u, v-plane with some exceptions,