558 XIII Connections with Number Theory
Birational equivalence may be defined in the following way. Arational transfor-
mationof the projective plane with pointsX=(X 1 ,X 2 ,X 3 )is a mapX→Y=φφφ(X),
where
φφφ(X)=(φ 1 (X),φ 2 (X),φ 3 (X))andφ 1 ,φ 2 ,φ 3 are homogeneous polynomials without common factor of the same
degreem, say. (In the corresponding affine plane the coordinates are transformed by
rationalfunctions.) The transformation isbirationalif there exists an inverse map
Y→X=ψψψ(Y),where
ψψψ(Y)=(ψ 1 (Y),ψ 2 (Y),ψ 3 (Y))andψ 1 ,ψ 2 ,ψ 3 are homogeneous polynomials without common factor of the same
degreen,say,suchthat
ψψψ[φφφ(X)]=ω(X)X,φφφ[ψψψ(Y)]=θ(Y)Yfor some scalar polynomialsω(X),θ(Y). Two irreducible projective plane curvesC
andDover the fieldK, defined respectively by the homogeneous polynomialsF(X)
andG(Y)(not necessarily of the same degree), arebirationally equivalentif there
exists a birational transformationY=φ(X)with inverseX=ψ(Y)such thatG[φ(X)]
is divisible byF(X)andF[ψ(Y)] is divisible byG[(Y)].
It is clear that birational equivalence is indeed an equivalence relation, and that
irreducible projective curves which are projectively equivalent are also birationally
equivalent. Birational transformations are often used to simplify the singular points
of a curve. Indeed the theorem onresolution of singularitiessays that any irreducible
curve is birationally equivalent to a non-singular curve, although it may be a curve in
a higher-dimensional space rather than in the plane. The algebraic geometry of curves
may be regarded as the study of those properties which are invariant under birational
equivalence.
It was shown by Poincar ́e (1901) that any non-singular curve ofgenus1defined
over the fieldQof rational numbers and with at least one rational point is birationally
equivalent overQto a cubic curve. Such a curve is now said to be anelliptic curve
(for the somewhat inadequate reason that it may be parametrized by elliptic functions
over the field of complex numbers.) However, for our purposes it is sufficient to define
an elliptic curve to be a non-singular cubic curve of the formW, over a fieldKof
arbitrary characteristic, or of the formCa,b, over a fieldKof characteristic= 2 ,3.
4 Mordell’sTheorem
We showed in the previous section that, for any fieldKof characteristic= 2 ,3, the
K-points of the elliptic curveCa,bdefined by the polynomial
Y^2 −X^3 −aX−b,wherea,b∈Kandd:= 4 a^3 + 27 b^2 =0, form an abelian group,E(K)say. We
now restrict our attention to the case whenK=Qis the field of rational numbers, and