For a quick definition of many of the terms used here, you may refer to the Glossary.
If you aren't familiar with the concept of a Riemann surface, you may want to consult this summary before proceding with this section.
External references for this section: [Cox], [Hus], [Maz], [Ser], [Sil])
Recall that both main steps (Theorm A and Theorem B) of the proof of Fermat's Last Theorem refer to a particular property of an elliptic curve, that of being "modular". But we said hardly anything about what that means. It turns out that there are several different ways of defining that property - and each of them has very interesting consequences.
The term "modular" comes from "modular group". The modular group is a
group 
, consisting of certain "fractional linear
transformations" of the
complex plane. A fractional linear transformation is merely the simplest
sort of rational function of the form g(z) = (az + b)/(cz + d), where the
coefficients a, b, c, d are integers and ad - bc = 1.
The group operation of such functions is composition. Simple substitution shows that elements of the modular group behave under composition just like the multiplication of 2-by-2 matrices, if one uses the matrix
to correspond to the transformation with the coefficients a, b, c, d. Although there are many matrices that could yield the same rational function, the condition ad - bc = 1 (i. e. the determinant of the matrix) makes the correspondence almost unique.
The group of 2-by-2 matrices with integral coefficients and determinant 1
is called SL
(Z). It's pretty easy to see that
the map from SL(Z) to
the modular group is a surjective group homomorphism with kernel just
{I, -I}, so that the modular group is isomorphic to
SL(Z)/{I, -I}.
(Sometimes SL(Z) is taken as the modular group
so it isn't necessary to
be fussy about speaking of equivalence classes of matrices modulo {I, -I}.)
The important thing about the modular group is that it acts as a group
of transformations on the upper half of the complex plane,
H = {z | Im(z) > 0}. That is, if T is in and z is in H,
T(z) is also in H. Like the full complex plane, H can be treated as
a Riemann surface.
One way to think of is as a group of "symmetries"
on the geometric object H.
In an analogous way, translations
can be viewed as symmetry operations on the full complex
plane. Given any two complex numbers 
and 
,
that aren't multiples
of each other by a real number, one can construct the "free abelian group"
on two generators, which are translation by
and . This group has a
"fundamental domain" with the property that any point in the whole plane is
a transformation of a point in the fundamental domain by an element
of the group. A fundamental domain for the group generated by two
translations is simply a parallelogram with vertices 0,
, , +.
The fundamental domain of this group of translations
looks suspiciously like the period parallelogram
of an elliptic function, and in fact, for any given pair of non-collinear
points , , an elliptic function can
be constructed that has the given
numbers as primitive periods. The period parallelogram is then the
fundamental domain of a transformation group with the property that for
any T in the group, f(Tz) = f(z) for any z in C, where f is any elliptic
function with the given primitive periods. In other words, f is invariant
with respect to the action of the symmetry group.
Returning to the modular group of symmetries
of H, we can define a modular function as any meromorphic
function on H which is invariant under the action of . In other words,
f( (az + b)/(cz + d) ) = f(z)for any complex numbers a, b, c, d.
This makes modular functions closely analogous to
elliptic functions, where we have just chosen a slightly different
domain of definition (H instead of C) and symmetry group . Modular and elliptic functions are both special cases
of the concept of an automorphic function, which is a meromorphic
function of 1 or more complex variables defined on a particular complex
manifold and invariant under a particular group of analytic
transformations (symmetries) of the manifold.
In practice, we want to consider a slightly more general class of
functions that are not strictly invariant under transformations in
. We say that a function f(z) on H is modular
of weight k if
f( (az + b)/(cz + d) ) = (cz + d)for all transformations inf(z)
,
some integer k 
0, and z 
H.
If the weight is 0, the function is modular in the strict sense that
it is automorphic with respect to .
Note that k must be even if f isn't identically 0, since we can take
a = d = -1 and b = c = 0 to require f(z) = (-1)f(z).
Furthermore, if f(z) is modular, it is periodic
of period 1, f(z+1) = f(z), because the transformation T(z) = z+1
is in . Therefore, f(z) has a Fourier expansion:
(We make the additional requirement that the lower limit of the sum is some finite number -m.) This is a Laurent series in q =
.
If f(z) is analytic for all zH then we say that
it is a modular form, which may have non-zero weight, and is an
important special case. This condition means that f(z) has no
"poles" (singularities) in H. In particular, f(z) is analytic at
, so there are no terms in the Fourier
series with negative indices, and we define f() =
c
. An even more special case is if
f() = c = 0, and then
we say that f(z) is a cusp form.
The condition of analyticity on a modular form is very restrictive.
It turns out that the only modular forms of weight 0 are constants,
although there are certainly non-trivial modular functions of weight 0
(but they have singularities). In fact, a modular form that isn't
trivial must have a weight that is even and 4. A
cusp form must have a weight 12.
We stress the terminology of "symmetry", because it is a very apt
term for certain transformations of a geometric object. In elementary
geometry, a symmetry is some operation on the object which leaves it
"unchanged". In other words, it has to do with the concept of "sameness"
in spite of difference. Given any geometric object, two distinct points
and on the object can be
considered the "same", or "equivalent", if there is
some element T of a transformation group, i. e. a symmetry operation,
such that T() = . The fact that the
set of transformations form a
group means that this relationship is reflexive (there's an identity
element), symmetric (there's a group inverse), and transitive (because
of the group operation). So the relationship has the defining
characteristics of what is called an equivalence relation.
Any time there is an equivalence relation on a set, the set can be partitioned into disjoint subsets of equivalence classes. A single equivalence class is sometimes called an "orbit", since it consists of all images of a given point under some element of the group. For instance, the set of all rotations of the plane about the origin is a group, and the orbit of any particular point in the plane is a circle whose radius is the distance of the point from the origin.
If the set which is acted upon has a topology, then the set of orbits also has a topology which is called the "quotient topology", and the resulting topological space is called the "quotient space". Since a Riemann surface has a topology, a group of analytic transformations acting upon it defines a quotient space, which is also a Riemann surface. In this way, whenever we have a class of functions on a Riemann surface that are invariant under the operation of a symmetry group, we can regard the functions as actually defined on the quotient space.
In particular, for the modular group , one can
consider the quotient space H/ of the upper half
plane. This consists of the space of orbits of points lying in the
fundamental domain of . The fundamental domain D of
consists of the set of points in the strip {z |
|Re(z)| 
1/2, |z| 1 } lying above
the circle |z| = 1. Then every point in the upper half plane is T(z) for
some T in and z in D.
Every element of H/ is an orbit, and there is one
and only one point in the fundamental domain that lies in the orbit.
This means there is a 1:1 correspondence of points in H/ (orbits) and points in D. In fact, H/ and D are topologically equivalent, and essentially the
same as Riemann surfaces. So the automorphic functions on H with respect
to , i. e. the modular functions, are essentially
the meromorphic functions on D considered as a Riemann surface by its
isomorphism with H/.
We have stressed this idea of symmetry because of the way it relates the analytic and geometric properties of an object like a Riemann surface to the algebraic properties of a group. From long experience with symmetries of simple plane geometric figures we have a lot of intuitive "knowledge" of how to think in terms of symmetries. We know, for instance, that most geometric objects have only certain specific symmetries that go a long way to actually defining the object. For example, the possession of a finite cyclic group of order 5 as a symmetry group, but no larger group, pretty much characterizes a regular pentagon among all convex polygons. Symmetry is one of the fundamental concepts of mathematics.
In the case of a Riemann surface viewed as a geometric object, there are other constructs that say a lot about the object, and in particular, the space of meromorphic functions defined on the surface. If the surface happens to be a quotient space with respect to a symmetry group on another surface, then the space of all its meromorphic functions corresponds to a very special class of functions on the "larger" surface: the automorphic functions.
Thus the elliptic functions are essentially the automorphic functions
on the extended complex plane corresponding to the group of translations by two
non-collinear values. If we look at a smaller group, consisting just of
translations by one quantity w, we get a larger space of automorphic
functions that also includes all rational functions of the
exponentials 
. (Since elliptic functions actually have
two distinct periods, they are also in this space.)
The modular group is a rather less intuitive group
of symmetries of the
upper half plane than the group of translations of the plane, but it
plays essentially the same role. It is an algebraic object that encodes
geometric information about the half plane H. Note that while H admits
a symmetry of translation by a real number, it does not admit a translation
by any non-real number. However, it does admit the transformation
z -> -1/z, which is inversion in the unit circle.
This latter transformation has finite order 2. It turms out
that the modular group has a presentation with generators T(z) = z + 1
and S(z) = -1/z and relations S
=
(ST)
= 1.
Modular functions are, then, the automorphic functions on the upper half
plane under the action of the modular group. They correspond to
the space of all meromorphic functions on the quotient space of the
upper half plane under the action of .
Associating a group with a geometric object provides a very powerful way of studying the object, since the algebraic structure of the group has a close relation to geometric properties of the object. For instance, a regular hexagon has a cyclic group of order 6 as a symmetry group. But a cyclic group of order 6 is a "direct sum" of cyclic groups of orders 2 and 3, i. e. it is "generated" by elements of orders 2 and 3. Correspondingly, a regular hexagon has 2-fold and 3-fold rotational symmetry as well as 6-fold symmetry.
The modular group is infinite, so it has quite a bit of structure. Since it is defined to consist of matrices with integral entries, it is natural to consider arithmetic properties of entries of members of the group. It turns out that there are a number of interesting subgroups defined by congruence conditions.
There is, first of all, the principal congruence subgroup of level
N, where N is a positive integer. This is denoted by (N). It is defined by the congruence conditions that
a=d=1 (mod N), and c=b=0 (mod N). This just means that members of (N) are congruent to the identity matrix mod N. So it's
not surprizing that this is a subgroup (i. e. it is closed under the
group multiplication and inverse operations). If N is 1, (1) is , since any element of is congruent to the identity mod 1.
(N) is in fact a "normal" subgroup, since it is the
kernel of the map of reduction mod N. So the "quotient group" /(N) can be defined. Moreover, the
index of (N) in , which is the
order of /(N), is finite and
equal to
if N > 2 (and 6 if N=2).
Other subgroups of finite index in are called
congruence subgroups if they contain (N) for
some N. If ' is such a subgroup, ' is said to have level N if N is the least integer with
' 
(N).
(Note that if M is a
multiple of N the congruence conditions for (M) are
stronger than for (N), so (N) (M).)
By relaxing the congruence conditions on (N) a
little, we can get larger groups of the same level N. For instance, if
we require only a=d=1 (mod N) and c=0 (mod N), we get 
(N), and only c=0 (mod N) we get 
(N) (i. e., upper triangular matrices, mod N). Note
that (N) 
(N) (N).
In the theory of elliptic curves, we will often have to deal with
subgroups of rather than the full modular group.
We will be working with functions that are automorphic only with
respect to such subgroups, which is a weaker condition than full
modularity, since fewer transformations are involved. In such cases,
we shall continue to say things like f(z) is a modular function or a
modular form "with respect to the subgroup".
References: [Hus], [Sil]
We saw above that if is the (full) modular group,
then H/ is a Riemann surface that is isomorphic to
the fundamental domain D of . So it seems plausible
that if ' is a subgroup, we should be able to
consider H/' as a Riemann surface.
If we have a subgroup of the modular group, we can construct a Riemann
surface that is related to the subgroup in the same way that the
quotient space H/ is related to . For any subgoup '
,
the fundamental domain D' of '
contains the fundamental domain D of . (It's larger
because ' is smaller than , so
there must be more points in the fundamental domain to allow any point
of H to be a transform of a point in D' by an element of '.) Since the quotient spaces H/'
and H/ are isomorphic as Riemann surfaces to the
fundamental domains D' and D respectively, H/' is
in some sense larger than H/.
Just as the complex plane can be "compactified" by adding a "point at
infinity" to give the "Riemann sphere", the space H/ can be compactified. The result is denoted X(). The same can be done for H/' if
' is any subgroup of of
finite index, and the result is X(').
Furthermore, there is a natural many-to-1 mapping H/' -> H/, since every orbit in
H/' is contained in an orbit in H/. Technically this map is what's called a covering,
since each point of H/ has an open neighborhood U
whose pre-image is a disjoint union of open sets which are homeomorphic
to U. Intuitively, this means that H/' is (locally)
like multiple copies of H/. The covering can be
done for the compactified spaces X(') and X() also.
So far, what we have seen is that for subgroups '
of finite index in , the spaces X(') are (compact) abstract Riemann surfaces, essentially
the quotient spaces. But much more is actually true - for certain ', X(') is in fact an algebraic
curve, that is, a locus of points (x, y) in C where x and y are related by a polynomial equation
f(x, y) = 0. (Technically, X(') is what is termed a
complete algebraic curve.) When ' is a congruence
subgroup (N), the corresponding curve X(N) is
called a modular curve. If ' is (N), the corresponding curve is written X(N).
Quite a lot of technical effort is required to verify all the necessary
details to prove that these Riemann surfaces are actually algebraic
curves. In general, one can explicitly construct a map j:H/'->X(') and this map has specific,
significant properties. In particular, this can be done when ' is (N), (N), or (N).
For example, if ' is the full modular group , X() is the Riemann sphere (i. e.
the 1-dimensional complex projective line). And the map j:H/ -> X() is given by J(z), which was
studied in the classical theory of modular functions and is called the
"fundamental modular function". There is a simple explicit formula for
J(z).
If ' is (N) so that X(') is X(N), then for some N it
turns out that something rather surprising can happen. Namely, we are
able to find an elliptic curve E over Q and a surjective map f:
X(N) -> E. This is called a "parameterization of
the elliptic curve by modular functions". (We'll explain the terminology
below.) N will be the "conductor" of E, which is (roughly) the product
of primes where E has "bad reduction".
It is here that the importance of the modular curves lies, because when
an elliptic curve is parameterized by modular functions in this sense,
there is a modular form (of weight 2) which has an L-function (suitably
defined, as we will do later) that is the same as the L-function of E
(again suitably defined). If f(z) is this modular form, it turns out
that f(z)dz is a differential 1-form, invariant under the action of , which is the "pull-back" using the map X(N) -> E of the "fundamental" differential 1-form on E.
Furthermore, the L-function of the elliptic curve is especially nice in that it has an analytic continuation to the whole plane and satisfies a functional equation. There is a conjecture known as the Hasse-Weil conjecture which says this is true for the L-function of any elliptic curve over Q. The Hasse-Weil conjecture is in turn part of a larger research program named after Langlands. We will go into much more more detail on L-functions later.
The property of an elliptic curve of being parameterized by modular functions is one way of defining a modular elliptic curve, and the Taniyama-Shimura conjecture asserts that every elliptic curve is modular. Before Wiles' recent results, only elliptic curves with the property known as "complex multiplication" had been shown to be parameterised by modular functions (by Shimura in 1971).
There's only one thing left to do here: to explain why we call a map f:
X(N) -> E a "parametrization of E by modular
functions". But this is simple. Since E is an elliptic curve, it
consists of points (x, y) in C where x
and y are related by a polynomial equation, specifically 
. So we get two functions f
, f: X(N) -> C such that 
.
Now, except at a finite number of points, a function on X(N) can be "lifted" to a function on H which is
invariant under the action of (N) - i. e. a
function that is modular with respect to (N). So
we have an explicit parameterization of the curve E by modular functions
(for a certain subgroup of of finite index).
Why does (N) play the leading role here instead of
other congruence subroups of level N such as (N)
and (N)? It is because the fact we have a covering
X(N) -> E is the "best" we can do for a particular
N. There are also coverings X(N) -> X(N) -> X(N) because (N) (N) (N). So there are coverings of E by X(N) and X(N) also (just by composition), and therefore
parameterizations of E by functions modular with respect to (N) and (N). But since those are
subgroups of (N), the same functions that are
modular for (N) are for the others as well.
Back to Fermat's Last Theorem Home Page
Copyright © 1996 by Charles Daney, All Rights Reserved
Last updated: March 12, 1996