570 XIII Connections with Number Theory
A deep theorem of Mazur (1977) says that the torsion group must be one of the
following:
(i) a cyclic group of ordern( 1 ≤n≤10 orn= 12 ),
(ii) the direct sum of a cyclic group of order 2 and a cyclic group of order
2 n( 1 ≤n≤ 4 ).
It was already known that each of these possibilities occurs. It is easy to check if the
torsion group is of type (i) or type (ii), since in the latter case there are three elements
of order 2, whereas in the former case there is at most one. Mazur’s result shows that
an element has infinite order, if it does not have order≤12.
It is conjectured that there exist elliptic curves overQwith arbitrarily large rank.
(Examples are known of elliptic curves with rank≥22.) At present no infallible algo-
rithm is known for determining the rank of an elliptic curve, let alone a basis for the
torsion-free groupEf. However, Manin (1971) devised a conditional algorithm, based
on the strong conjecture of Birch and Swinnerton-Dyer which will be mentioned later.
This conjecture is still unproved, but is supported by much numerical evidence.
An important way of obtaining arithmeticinformation about an elliptic curve is by
reduction modulo a primep. We regard the coefficients not as integers, but as integers
modp, and we look not forQ-points, but forFp-points. Since the normal formCa,b
was obtained by assuming that the field had characteristic= 2 ,3, we now adopt a more
general normal form.
LetW =W(a 1 ,...,a 6 )be the projective completion of the affine cubic curve
defined by the polynomial
Y^2 +a 1 XY+a 3 Y−(X^3 +a 2 X^2 +a 4 X+a 6 ),whereaj∈Q(j= 1 , 2 , 3 , 4 , 6 ). It may be shown thatW is non-singular if and only
if thediscriminant∆=0, where
∆=−b^22 b 8 − 8 b 43 − 27 b^26 + 9 b 2 b 4 b 6and
b 2 =a^21 + 4 a 2 ,
b 4 =a 1 a 3 + 2 a 4 ,
b 6 =a^23 + 4 a 6 ,
b 8 =a^21 a 6 −a 1 a 3 a 4 + 4 a 2 a 6 +a 2 a^23 −a^24.(We retain the name ‘discriminant’, although∆=− 16 dforW =Ca,b.) The defini-
tion of addition onWhas the same geometrical interpretation as onCa,b, although the
corresponding algebraic formulas aredifferent. They are written out in§7.
For anyu,r,s,t∈Qwithu=0, the invertible linear change of variables
X=u^2 X′+r, Y=u^3 Y′+su^2 X′+treplacesWby a curveW′of the same form with discriminant∆′=u−^12 ∆. By means
of such a transformation we may assume that the coefficientsajare integers and that∆,