Page 39 -

16 Introduction to Rational Points on Plane Curves

direct sum with the group of order 2. Since Tors E(Q) embeds into E(R) as a finite
subgroup, we have from this that Tors E(Q) is either finite cyclic or the direct sum
of a finite cyclic group with the group of order 2.
The question of a uniform bound on Tors E(Q) as E varies over all curves E
defined over Q was studied from the point of view of modular curves by G. Shimura,
A. Ogg, and others, see Chapter 11, §3. In 1976 Barry Mazur proved the following
deep result which had been conjectured by Ogg.
(5.3) Theorem (Mazur). For an elliptic curve E defined over Q the group Tors E(Q)
of torsion points is isomorphic to either
Z/mZ for m = 1, 2, 3,... , 10, 12
or
Z/mZ ⊕ Z/2Z for m = 2, 4, 6, or 8.
In particular there is no element of order 11, 13, or 14 in the group of rational
points on an elliptic curve over Q. There are examples which show that all above
cases can occur.
This leaves the question of the rank g. There are examples of curves known with
rank up to at least 24. It is unknown whether or not the rank is bounded as E varies
over curves defined over Q. Such a bound is generally considered to be unlikely.
With our present understanding of elliptic curves the rank g is very mysterious and
difficult to calculate in a particular case. See also Rubin and Silverberg [2002].
(5.4) Remark. Let E be an elliptic curve defined over Q by the equation y 2 =
3
x + ax + b. In fact, after a change of variable every elliptic curve over the rational
numbers has this form. There is no known effective way to determine the rank of
E from these two coefficients, a and b. In fact, there is no known effective way of
determining whether or not E(Q) is finite. Of course E(Q) is finite if and only if the
rank g = 0.

This is one of the basic problems in arithmetic algebraic geometry or diophantine
geometry. In 16 we will associate an L-function L E (s) to E. Conjecturally it has an
analytic continuation to the complex plane. This L-function was first introduced by
Hasse and was studied further by A. Weil.

(5.5) Birch, Swinnerton–Dyer Conjecture. The rank g of an elliptic curve E de-
fined over the rational numbers is equal to the order of the zero of L E (s) at s = 1.
Birch and Swinnerton–Dyer gathered a vast amount of supporting evidence for
this conjecture. Coates and Wiles in 1977 made the first real progress on this con-
jecture for curves with complex multiplication and recently R. Greenberg has shown
that the converse to some of their statements also holds. This subject has exploded
in the last twenty years and we will not treat any of these developments. The reader
should consult the book by K. Rubin, Euler Systems, Annals of Math Studies.The
final part of the book is devoted to an elementary elaboration of this conjecture.
A refinement of their conjecture explains the number lim s→1 (s − 1) −g L E (s).
The final part of the book is devoted to an elementary elaboration of this conjecture.