For a quick definition of many of the terms used here, you may refer to the Glossary.
External references for this section: [Gou], [Rih]
A Dirichlet series is an infinite series of the form
(It's traditional to denote the function's argument, an arbitrary complex number, by s.)
Such series can easily be shown to converge in a right half-plane of the
form Re(s) > t for some t 
R, and in that
region they represent
analytic functions. For many such series of interest, the corresponding
function can be extended to a meromorphic function on the whole plane by
a process known as "analytic continuation".
Dirichlet functions are of great importance in number theory because of
the following formal property. Suppose that {a
} is
a sequence of
numbers that are defined for all integer primes p. Then we can formally
define an infinite product, called an Euler product:
and it is easy to show that if this product converges, then
where a
are defined for non-prime n such that
if 
, then
. This is well-defined precisely because n has a
unique representation as a product of primes.
Not all Dirichlet series are of this simple form. In fact, a Dirichlet
series has this form if and only if its coefficients {a}
have the property that a
=
a a.
An Euler product whose factors have the form 
is called a linear Euler product. The next simplest
kind of Euler product ("quadratic") has factors which are the reciprocals
of polynomials that are quadratic in 
, and we shall
have examples
of these as well. The relationships among the coefficients of the
corresponding Dirichlet series are rather more complicated, however.
A Dirichlet series need not have an Euler product form at all, of course.
The simplest possible example occurs when all a = 1.
This yields the famous Riemann zeta function:
In view of its definition,
it shouldn't be surprising that 
(s) has some remarkable
number-theoretic properties. Because of a number of results and conjectures
stated by Riemann regarding (s), it attracted a lot of
attention, and many interesting properties have been discovered.
For instance,
where
(x) is the number of primes 
x.
where
(n) is the Mobius function.
where
(n) is the number of divisors of n.
where
(n) is the number of different prime factors of n.
Some of the analytic properties of (s) are just as
interesting. In particular, (s) has a meromorphic
continuation to the whole complex plane and satisfies a functional equation:
There are also various integral formulas for (s),
in particular
where [x] is the largest integer
x,
and provided 0 <. Re(s) <. 1.
We mention this because of its similarity to a "Mellin transform",
which will be important a little later.
The functional equation in turn implies other properties, such as the
fact that (s) = 0 when s=-2k for positive integers k.
These are known as the "trivial zeros" of (s).
Also, (s) 
0 if Re(s)
> 1 or if Im(s) = 0 and 0 <. s <. 1. So all "non-trivial" zeros lie
in the "critical strip" 0 Re(s) 1
and are distributed
symmetrically with respect to both the real axis and the line
Re(s) = 1/2. The famous Riemann Hypothesis says that all of the non-trivial
zeros actually lie on Re(s) = 1/2.
Since this is a part of a discussion of Fermat's Last Theorem, we can't
fail to mention that (s) is also related to the
mysterious and intriguing "Bernoulli numbers" B,
which are defined by the generating function
This implies various recurrence relations, and these further imply that all B
are rational.
Specifically, it is elementary, and noted by Euler, that
B are related to the values of (s)
at even positive integers:
The B have many curious number theoretic properties.
But the most intriguing fact appears if p is an odd prime number that
does not divide the numerators of any of B
, ...,
B
. Such a p is
said to be a "regular prime". Kummer discovered that Bernoulli numbers
play a part in the theory of "cyclotomic fields" (i. e. extensions
of Q by an nth root of unity). Specifically, if p is an odd
prime, then p does not divide the "class number" of the extension
Q() if and only if p is regular (where
is a primitive pth root
of unity). Further, and most importantly,
Fermat's equation for exponent p has no nontrivial solution in integers
if p is a regular prime.
Unfortunately for early attempts to prove FLT, it hasn't yet been shown
that there are an infinite number of regular primes, even
though it is known there are an infinite number of irregular primes.
There are many more
number theoretic applications of (s), but one of the
deepest is in
the proof of the "prime number theorem", which gives an asymptotic
formula for (x), the number of primes
x. This theorem, first proved by Hadamard, states that
I. e.,
(x) is asymptotically equal to x/ln(x). Since
ln(x) grows
much more slowly than x, this says that there really are a lot of primes.
In fact, around a large number x the average gap between primes is roughly
2.3 times the number of digits in the decimal expansion of x.
The proof of this result depends on a fact which can be deduced from the
functional equation that there are actually no zeros of (s)
on the
line Re(s) = 1. It turns out that if the much stronger fact expressed in
the Riemann hypothesis is true, then much better estimates of the
distribution of primes can be given.
Throughout number theory and related fields, such as algebraic geometry and the theory of automorphic functions, many generalizations of the zeta function and Dirichlet series have been found useful. The L-functions of elliptic curves and modular forms we are about to discuss provide one important source of examples. In general, it has been fruitful to define L-functions in terms of certain representations of Galois groups, and we will be seeing some of this eventually.
External references for this section: [Sil]
We are interested in the question of, in some sense, "how many" rational
points there are on a given elliptic curve E over Q.
In the introduction to
the discussion of elliptic curves, we mentioned that global questions can
often be studied by looking at them locally, i. e. mod p for all primes p.
So we should consider the question of how many points there are on the
reduction of E at p, i. e. the corresponding curve over F.
Since there are only p elements in F, there are at
most p+1 points on any
curve over F (counting the "point at infinity").
Let A be the number
of points of the curve actually in F. Then define
a = p + 1 - A, which
represents, roughly, how many points are "missing". Note that
A, and
hence a, is defined even if E isn't an elliptic curve over
F because
its equation has repeated roots mod p ("bad reduction").
(But if E is an elliptic curve, A is the order of E(p),
the group of points of E in F.)
For each p we define
if E has good reduction at p, and
otherwise. Finally, the L-function of E is defined as
It can be shown that the Euler product converges for Re(s) > 3/2, and furthermore that the Dirichlet series for L(E,s) satisfies
where a
is as defined above when n is prime.
This function is known as the Hasse-Weil L-function. As usual, we expect much more to be true about it. Specifically, the Hasse-Weil conjecture states that L(E,s) has a meromorphic extension to the whole complex plane. Furthermore, there should be a functional equation like that of the Riemann zeta function. If
where
(s) is the usual gamma function,
and N is the regulator
of E (which is, roughly, the product of the primes where there is bad
reduction), then
where w =
1 depends on the curve E.
This conjecture could be proven directly for elliptic curves with the property known as "complex multiplication". Also, the conjecture was known if E is modular in the sense described earlier, i. e. if E has a parameterization by modular functions (which was known to be true in case E has complex multiplication). Now that the Taniyama-Shimura conjecture is known for "semistable" curves E, we know that such E have a parameterization by modular functions, so the Hasse-Weil conjecture holds for them also.
The conjecture of Birch and Swinnerton-Dyer goes even farther and says that L(E,s) has a zero at s=1 of order equal to the rank of E (i. e. the number of infinite cyclic group factors in the group of rational points on E).
Clearly, the Hasse-Weil L-function of E has some pretty impressive properties. And it is but a special case in the "Langlands program", which conjectures that members of a much broader class of Dirichlet series are meromorphic and have a functional equation. In a little more detail, it is possible to define a Hasse-Weil L-function for "projective varieties" over a number field. A "projective variety" is basically a higher dimensional analog of an algebraic curve, while a number field is a finite extension of Q. A priori, such L-functions are defined only in a certain right half plane. The Langlands program involves studying these L-functions by looking at related L-functions of automorphic representations of "reductive algebraic groups". We will see the special case of this for elliptic curves when we look at Galois representations. Langlands' program is central to contemporary algebraic number theory.
External references for this section: [Hus]
And we aren't done with the Hasse-Weil L-functions yet. They turn out to be the same as the L-functions associated with modular forms by a completely different definition.
If f(z) is a modular form, f(z + 1) = f(z), so there is a Fourier expansion,
where z
H, and a
C. In other words, there is a
related function f* of q=
with f(z) = f*(q), and
is the ordinary Laurent expansion around q = 0. Recall that f is said to be holomorphic at
if f* is holomorphic at 0, in which case
Given a Fourier expansion like that, there is a standard way to construct a related Dirichlet series, using an integral transform called the Mellin transform. This is defined by
If we let f
(t) = f(it) - f(),
then we can define the L-function of f as
The classical Gamma function is defined as
i. e.
(s) = M(1/e
, s).
From this it is easily shown by a change of variables that
The function L(f,s) is called the L-function of the modular form f. If f
has weight k, so that f(-1/z) = z
f(z), then from the
foregoing, it is easy to see that the related function defined by
satisfies the functional equation
and has poles at most at s=0 and s=2.
We want to relate all this to elliptic curves. There is one technical
snag, however, in that as we have seen, the subgroup 
(N)
rather than the
full modular group (where N is the conductor of E)
is the largest symmetry
group for which we have a useful theory as far as elliptic curves are
concerned. If a function f is modular only
with respect to (N), then we don't have the
relation f(-1/z) = zf(z)
since the inversion z -> -1/z isn't in (N), so we
can't derive the desired functional equation.
Fortunately, it can still be shown that if f is a modular form of
weight k with respect to (N), if L(f,s) is defined
as before, and if we define
then L*(f,s) extends to a meromorphic function on C and it has the functional equation
where w =
1 depends on f. In particular, for forms of
weight 2,
L*(f, 2-s) = -w L*(f,s), which just happens to be the functional equation
satisfied by the L-function of an elliptic curve.
We can see where this is leading. We have already defined the L-function L(E,s) of an elliptic curve E over Q having Dirichlet series
so we would expect that the function
should be very interesting - perhaps some sort of modular function. In fact, it ought to be a modular function of weight 2 for
(N),
so that L(f,s) = L(E,s) has the proper functional equation. Although we
can easily write down the series, all the hard work lies in showing
convergence and that the resulting function is modular with weight 2.
Actually, f(z) can be defined directly from L(E,s) without explicit mention of the series coefficients by means of an "inverse Mellin transform":
using the "Mellin inversion formula". The validity of this inversion depends on specific properties of the functions involved, but it's basically similar in nature to the inversion formula for Fourier integrals.
In fact, it is a rather difficult result established by Langlands and
Deligne in 1972 that if E is an elliptic curve with a parameterization
by modular functions as defined previously (i. e. there is a surjection
of X
(N) on E over Q), then indeed f(z) is a
modular cusp form
(i. e. it vanishes at 0) for (N) of weight 2
and L(E,s) = L(f,s).
Since this result demonstrates the functional equation for L(E,s), it
verifies the Hasse-Weil conjecture when E is a modular curve (i. e.
has parameterization by modular functions).
But the Taniyama-Shimura conjecture says that all elliptic curves over Q are modular, so if it's true, the Hasse-Weil conjecture is also true. Wiles' proof that Taniyama-Shimura holds for the semistable case therefore definitely proves Hasse-Weil in this case as well.
The cusp form f(z) can be identified even more precisely when E is a
modular curve. Specifically, differential geometry says that a
surface like E has a "canonical differential", i. e. a differential
1-form. Given the map, X(N)->E, this 1-form pulls
back to a 1-form
on X(N), which is (up to a constant multiple) f(z)dz.
Since f(z) has
weight 2, the differential f(z)dz is invariant under transformations of
(N), because
Copyright © 1996 by Charles Daney, All Rights Reserved
Last updated: March 12, 1996
