L-functions are one of those tricksy bits of
mathematics for which there is
no particularly satisfying general definition, nor a terrifically obvious justification
for why we're interested in them. We are, though - the study of L-functions is one
of the main branches of modern
number theory. Although they look strange at first, L-functions turn out to connect with everything from homological algebra to Fermat's Last Theorem, as well as throwing up some fascinating, elegant and occasionally surprising maths of their own.
Mostly, an L-function associated with foo is something of the form
infty
--
\
\ a(n) n^{-s},
/
/
--
n=0
where
a(n) is some function of
n dependant on
foo. Hopefully,
this converges for some values of the complex variable
s to give an
analytic
function on some subset of the
complex plane. Some examples:
- The Riemann zeta function - a(n) = 1. This is the most famous
L-function, and crops up absolutely everywhere.
- The Dirichlet L-function. Suppose we are given a number N and a Dirichlet
character - a homomorphism f from (Z/NZ)^{*} into C^{*},
where ^{*} denotes the multiplicative group.
Then we can extend f to a function on the whole of N by defining
f(n) = f(n mod N) if n mod N is in (Z/NZ)^{*}
= 0 otherwise.
This allows us to define the Dirichlet L-function in the manner given above, using
a(n) = f(n).
- Shintani L-functions - these are a technical but useful generalisation
of Dirichlet L-functions, basically using a weighted sum over N^{n}
rather than just N. They aren't particularly pretty, but many other L-functions
can be written in terms of them, so results about Shintani L-functions often extend easily.
- Hecke L-functions - these are a generalisation of Dirichlet L-functions
to arbitrary algebraic number fields. Rather than summing over the integer ring of the field
(bearing in mind that N is the integer ring of Q), we instead sum over the
ideal class group - L-functions are arithmetic objects, and we tend to be interested
in ideal arithmetic rather than integer arithmetic because it's nicer - things like
prime factorisation tend to hold.
- p-adic Hecke L-functions - as the name implies, these further generalise Hecke
L-functions to arbitrary p-adic fields. The definition is pretty technical and involves
p-adic measure theory and similar unpleasantness, but the L-functions give a very interesting
and rich theory. Defining p-adic L-functions is one of the motivations for Iwasawa theory.
- L-functions associated with modular forms. If you have a modular form, it can be
written as a q-expansion (essentially a Fourier expansion):
infty
--
\
f(tau) = \ a(n)q^{n}.
/
/
--
n=0
If f is a cusp form (ie a(0) = 0) then we can associate an L-function with it: infty
--
\
L(f,s) = \ a(n)n^{-s}.
/
/
--
n=0
- The L-function of an elliptic curve. This is a little different from the previous
examples in that it is defined as a product rather than a sum. This makes sense in
view of the product formulae explained below. In this case, the L-function associated with
the elliptic curve E is
-------
| | 1
L(E,s) = | | -----
| | L_{p}(p^{-s})
| |
p prime
where L_{p}(T) is a polynomial in T determined by the properties of E when it is
reduced modulo p.
There are various
properties that L-functions tend to have, although they don't always hold
and establishing them can sometimes be very difficult indeed.
The neatest property is probably product expansions - many sorts of L-function can be written as
a product over the (rational) primes. For instance, the Riemann zeta function has the
expansion
infty
-- -------
\ | |
\ n^{-s} = | | (1-p^{-s}).
/ | |
/ | |
-- p prime
n=0
This scary looking expression is actually quite easy to justify - work from the right
hand side, and remember that each natural number can be written uniquely as a product of primes....
The other property that tends to come up a lot is the analytic continuation, and associated
functional equations. Again, take the zeta function as an example. If the real part of
s is greater than 1, then it's a standard result that the series will converge absolutely
and uniformly, and so the sum defines an analytic function on {s in C : Re(s)>1}.
If Re(s) is less than or equal to one, however, we aren't so fortunate:
zeta(0) = 1+1+1+1+... and
zeta(-1) = 1+2+3+4+... , neither of which is going to help us to find an analytic function.
However, we can find an analytic continuation - an analytic function defined on the whole
of C apart from a few poles (in this case at s = 1), which is defined by a different
expression from zeta(s), but agrees with it on those areas of the complex plane where
zeta converges. Having established his, we can talk about zeta(-1) and other things that
were previously meaningless.
In this case, proving the analytic continuation only requires a bit of fiddling with complex analysis.
In other cases, such as the case of the L-function of an elliptic curve, the establishment
of the analytic continuation is a very difficult and technical result.
Associated with the analytic continuation, we normally get some sort of functional
equation - that is, an equation relating values of the L-function at one point to its
value at another.
All this leaves one big question, though - why do we care?
L-functions aren't the sort of thing that scream 'study me'... at first glance they look
rather artificial. Why do we care about all these n^{-s}'s? Why not some other function?
I don't know anywhere near enough to give a full answer, but as far as I can tell,
it's basically because they keep turning up. It's fairly easy to associate L-functions with
things, and once you have them they just keep turning out to have connections to other things -
homology groups, K-groups, the Tate-Shafarevich Group, the prime number theorem, and all sorts
of other rather unexpected stuff. The Taniyama-Shimura conjecture (or Wiles-Taylor-Conrad-Diamond
theorem, as it now is) implies Fermat's Last Theorem, but basically just says that all
L-functions arising from elliptic curves also arise from modular forms. The ABC conjecture
is linked to L-functions. They probably come into the Langlands program somewhere, although
that's not something I know much about. L-functions seem to produce all sorts of
deep and mysterious links between interesting objects, and this makes them deep, mysterious and
interesting objects to study in their own right.
I think that the other main reason that we study them is that the theory they throw up is just plain nice -
the sort of combination of elegant and weird that got people interested in number theory in
the first place.
Disclaimer: I'm not an expert, just a humble grad student.
If you have any comments, corrections or omissions then feel free to /msg me.