Number Theory: An Introduction to Mathematics

5 Further Results and Conjectures 569

Hence forψ◦φ(P)=P∗=(x∗,y∗)we have

x∗=(x^2 −B)^2 / 4 y^2 ,

y∗=(x^2 −B)[(x^2 +Ax+B)^2 −(A^2 − 4 B)x^2 ]/ 8 y^3.

On the other hand, if the tangent toCA,BatPintersectsCA,Bagain at(x ̄,y ̄),then
2 P=(x ̄,− ̄y). The cubic equation

(mx+c)^2 =X^3 +AX^2 +BX

hasxas a double root andx ̄as its third root. Hencex ̄=(c/x)^2. Using the formula for
cgiven previously, we obtain

x ̄=(x^2 −B)^2 / 4 y^2 =x∗.

Furthermore, using the formula formgiven previously,

y ̄=mx ̄+c=[( 3 x^2 + 2 Ax+B)x ̄−x(x^2 −B)]/ 2 y

=(x^2 −B)[( 3 x^2 + 2 Ax+B)(x^2 −B)− 4 xy^2 )]/ 8 y^3.

Substitutingx^3 +Ax^2 +Bxfory^2 , we obtainy ̄=−y∗. Thusψ◦φ(P)= 2 P,as
claimed.
Sinceφ(E)has finite index inE′, and likewiseψ(E′) has finite index inE,itfol-
lows that 2E=ψ◦φ(E)has finite index inE. (The proof shows that the index is
at most 2α+β+^2 ,whereαis the number of distinct prime divisors ofBandβis the
number of distinct prime divisors ofA^2 − 4 B.)
By the remarks after the proof of Proposition 12, Mordell’s theorem has now been
completely proved in the case whereEcontains an element of order 2.

5 FurtherResultsandConjectures

LetCa,bbe the elliptic curve defined by the polynomial

Y^2 −(X^3 +aX+b),

wherea,b∈Zandd:= 4 a^3 + 27 b^2 =0. By Mordell’s theorem, the abelian group
E=Ea,b(Q)of all rational points ofCa,bis finitely generated. It follows from the
structure theorem for finitely generated abelian groups (Chapter III,§4) thatEis
the direct sum of a finite abelian groupEtand a ‘free’ abelian groupEf, which is the
direct sum ofr≥0 infinite cyclic subgroups. The non-negative integerris called
therankof the elliptic curve andEtitstorsion group.
The torsion group can, in principle, be determined by a finite amount of computa-
tion. A theorem of Nagell (1935) and Lutz (1937) says that ifP=(x,y)is a point of
Eof finite order, thenxandyare integers and eithery=0ory^2 dividesd. Thus there
are only finitely many possibilities to check.