Mandelbrot Set and Fermat’s Last Theorem
113
This mapping is, in effect, a parameterization of the elliptic curve by points in a
“fundamental parallelogram” in the complex plane.
The topological space that results from identifying opposite sides of a period
parallelogram is called a complex torus. The fundamental periods that define the
parallelogram generate a lattice in C consisting of all sums of integral multiples of
ω 1 and ω 2 . If L denotes the lattice, then L = Z ω 1 ⊕ Z ω 2 . The complex torus can
then be described as C/L. What this all means, therefore, is that a (complex) torus is
the “natural” domain of definition of the function or any doubly periodic complex
function.
The Fermat equation is the prime example. In general, an elliptic curve has the
form y
2
= Ax
2
+ Bx
2
+ C x + D, but for considering arithmetical questions, it is
natural to restrict our attention to the case where A, B, C, D are all rational. This
assumption will usually be in effect when we are considering properties of elliptic
curves involving arithmetical questions (as opposed to their more general analytic
properties). If all coefficients are rational, the elliptic curve is said to be defined over
Q. The all-important Taniyama-Shimura conjecture concerns only elliptic curves
defined over Q.
The fact that any elliptic curve (not necessarily defined over Q) has an abelian
group structure means that we can learn a lot about it by studying various of its
subgroups. For considering arithmetical (i.e., number theoretic) questions, we restrict
our attention to curves defined over Q. In that case, there are several interesting
subgroups we can consider.
The first is the group of all points on the curve E which have an order that divides
m for some particular integer m. That is, m “times” such a point is the identity
element. Such points are called “m-division points”, and the subgroup they make up
is denoted E[m]. The reason for the name is that any point in E[m] generates a cyclic
subroup of E (and E[m]) whose order divides m. If the order is actually m, then the
points in the cyclic group generated by the point divide E into m segments.
It isn’t necessarily the case that the coordinates of a point in E[m] have integral or
rational coordinates. However, the coordinates will be algebraic numbers (i.e., roots
of an algebraic equation with coefficients in Q). It’s relatively easy to show that as
an abstract group E[m] is just the direct sum of two cyclic groups of order m i.e.,
Z /m Z ⊕ Z /m Z, so its order is m. We shall see later that its real interest lies in the
fact that we can construct representations of other groups of transformations that act
on E[m]. Such representations will consist of 2 × 2 matrices with integral entries
i.e., elements of G L 2 (Z ).
Another interesting subgroup of E is the set of all points whose coordinates are
rational. Such points are said to be rational points. If the curve is defined over Q, then
it is a simple fact that the set of all rational points (if there are any) is a subgroup.
Another interesting subgroup of E is the set of all points whose coordinates are
rational. Such points are said to be rational points. If the curve is defined over Q, then
it is a simple fact that the set of all rational points (if there are any) is a subgroup.
The definition of L(E, s) will be made based on details about a series of other
groups connected with E. These arise by considering E as an elliptic curve over the
finite fields F P . This is the same as taking the original equation and reducing the
113
This mapping is, in effect, a parameterization of the elliptic curve by points in a
“fundamental parallelogram” in the complex plane.
The topological space that results from identifying opposite sides of a period
parallelogram is called a complex torus. The fundamental periods that define the
parallelogram generate a lattice in C consisting of all sums of integral multiples of
ω 1 and ω 2 . If L denotes the lattice, then L = Z ω 1 ⊕ Z ω 2 . The complex torus can
then be described as C/L. What this all means, therefore, is that a (complex) torus is
the “natural” domain of definition of the function or any doubly periodic complex
function.
The Fermat equation is the prime example. In general, an elliptic curve has the
form y
2
= Ax
2
+ Bx
2
+ C x + D, but for considering arithmetical questions, it is
natural to restrict our attention to the case where A, B, C, D are all rational. This
assumption will usually be in effect when we are considering properties of elliptic
curves involving arithmetical questions (as opposed to their more general analytic
properties). If all coefficients are rational, the elliptic curve is said to be defined over
Q. The all-important Taniyama-Shimura conjecture concerns only elliptic curves
defined over Q.
The fact that any elliptic curve (not necessarily defined over Q) has an abelian
group structure means that we can learn a lot about it by studying various of its
subgroups. For considering arithmetical (i.e., number theoretic) questions, we restrict
our attention to curves defined over Q. In that case, there are several interesting
subgroups we can consider.
The first is the group of all points on the curve E which have an order that divides
m for some particular integer m. That is, m “times” such a point is the identity
element. Such points are called “m-division points”, and the subgroup they make up
is denoted E[m]. The reason for the name is that any point in E[m] generates a cyclic
subroup of E (and E[m]) whose order divides m. If the order is actually m, then the
points in the cyclic group generated by the point divide E into m segments.
It isn’t necessarily the case that the coordinates of a point in E[m] have integral or
rational coordinates. However, the coordinates will be algebraic numbers (i.e., roots
of an algebraic equation with coefficients in Q). It’s relatively easy to show that as
an abstract group E[m] is just the direct sum of two cyclic groups of order m i.e.,
Z /m Z ⊕ Z /m Z, so its order is m. We shall see later that its real interest lies in the
fact that we can construct representations of other groups of transformations that act
on E[m]. Such representations will consist of 2 × 2 matrices with integral entries
i.e., elements of G L 2 (Z ).
Another interesting subgroup of E is the set of all points whose coordinates are
rational. Such points are said to be rational points. If the curve is defined over Q, then
it is a simple fact that the set of all rational points (if there are any) is a subgroup.
Another interesting subgroup of E is the set of all points whose coordinates are
rational. Such points are said to be rational points. If the curve is defined over Q, then
it is a simple fact that the set of all rational points (if there are any) is a subgroup.
The definition of L(E, s) will be made based on details about a series of other
groups connected with E. These arise by considering E as an elliptic curve over the
finite fields F P . This is the same as taking the original equation and reducing the