We’ve done a lot of work so far just to try to define the terms in the Taniyama-Shimura conjecture, but today we should finally make it. Our last piece of information is to write down what the L-function of a modular form is. Since I don’t want to build a whole bunch of theory needed to define the special class of modular forms we’ll be considering, I’ll just say that we actually need to restrict our definition of “modular form” to “normalized cuspidal Hecke eigenform”. I’ll point out exactly why we need this, but it doesn’t change anything in the conjecture except that every elliptic curve actually corresponds to an even nicer type of modular form.

Let be a weight cusp form with -expansion . Since this is an analytic function on the disk, we have the tools and theorems of complex analysis at our disposal. We can perform something called the Mellin transform. It is just a standard integral transform given by the formula .

After some computation you find that this transformed function is a product of really nice functions. We get , where is the usual Gamma function. Now if you actually went through and worked this out you would find out that has a really nice form in terms of the Fourier coefficients. The so-called L-series associated to the Mellin transform is given by

.

If your eyes glazed over for the Mellin transform talk, then just think of the L-function of the modular form as taking all of its Fourier coefficients and throwing them in the numerator of this series to make a new function. A quick remark is that if all the are (this won’t happen) we recover the Riemann zeta function. Thus you could think of the L-function we get as some sort of generalization of the zeta function. If you’ve been through some elementary number theory you have probably even seen a proof that where the product is over all primes called an Euler product. Now in general if I hand you a sequence of integers that has some reasonable growth condition, then will be a nice convergent series, probably with an analytic continuation to the plane. The tricky part is to figure out what types of sequences allow this Euler product decomposition.

This is where we have to use that was of this special form. In the theory of modular forms there is something called Atkin-Lehner theory which tells us that the for a cusp form of this special type actually satisfy some nice relations such as when . These relations are precisely the ones needed to conclude that there is a nice Euler product expansion and it is given by

We say that a variety is modular if coincides with up to finitely many primes for some . We’ve been ignoring the technicalities of dealing with the primes of bad reduction and the primes that divide the level (a surprisingly hard problem to determine when these are the same set!), but now we see that for the definition of a variety being modular this doesn’t even matter. There are other subtleties in defining all of this for when the variety does not have -dimensional middle cohomology, but again for our immediate purposes you can trust that people have made the suitable adjustments.

Now we see the truly shocking results of Taniyama-Shimura. We take this incredibly symmetric analytic object (so symmetric it is surprising any exist at all) and we take this completely algebraic variety defined over and the conjecture claims that we can always find one of these symmetric things that match up with this action on the cohomology. Wiles and Taylor are often credited with proving it in 1994, but it wasn’t actually proved until 2003 by Breuil, Conrad, Diamond, and Taylor. This was the elliptic curve case.

Just last year Gouvea and Yui proved that all rigid Calabi-Yau threefolds are modular. It is a conjecture that all Calabi-Yau varieties over should be modular, so this includes K3 surfaces. It might seem weird that K3 surfaces haven’t been proven but the threefold case has been. This just has to do with those technicalities of what to do if the middle cohomology is bigger than 2-dimensional, which it always is. There you have it. The famous Taniyama-Shimura conjecture which led to a proof of Fermat’s Last Theorem.