From: nikl+sm000121@pchelwig1.mathematik.tu-muenchen.de (Gerhard Niklasch) Newsgroups: sci.math Subject: Re: Dirichlet Series and Fermat's Last Theorem Date: 19 Jun 1998 15:39:40 GMT In article <17F79DC5DS86.RVANRAAM@bcsc02.gov.bc.ca>, RVANRAAM@bcsc02.gov.bc.ca (Ray Van Raamsdonk) writes: |> Someone was trying to explain to me once the relationship of |> Dirichlet series to Fermat's theorem in that they are used |> for the theory of elliptic functions and also for the study |> of modular forms. Can anyone enlighten me on this topic a bit |> more? Very rough sketch in very broad strokes -- maybe an expert can fill in more detail: Thanks to previous work by Hellegouarch and Frey, it was known by 1986 that any counterexample to FLT would give rise to an `Elliptic Curve defined over Q' with a rather strange combination of properties (and Wiles' proof ended up showing that no EC/Q could possibly have this strange combination of properties). Sweeping some technicalities under the rug, you can think of an EC/Q as being given by an equation with integer coefficients a,b Y^2 = X^3 + a X + b together with its sets of solutions in any field in which the cubic polynomial on the righthand side has three distinct roots (the integers a,b are naturally mapped to field elements by adding up 1s or -1s the right number of times), and together with all the structure that can naturally be put on these solution sets (actually that of an abelian group on which any field automorphisms will act). The finite fields Z/pZ as p ranges over primes, as well as their finite extensions with p^r elements (one for each prime p and each integer r > 0, up to isomorphism) are among the admissible environments for solution sets, provided that a finite number of primes is excluded. For each good prime, one defines an integer a_p as a_p = p + 1 - (number of solutions (X,Y) of the equation with X,Y in Z/pZ) and a quadratic polynomial P_p(T) = 1 - a_p T + p T^2 , and for the bad primes one invents something similar (actually simpler) which will play the same role. (At this stage one may stop to note that deep theorems relate these a_p and P_p(T) to other things, like the action of automorphisms on solution sets, so they are not as artificial as the above introduction makes them appear.) One builds the `L series of the curve' as a Dirichlet series out of an Euler product L(E,s) = product_{p prime} P_p(p^(-s))^(-1) = product_{p good prime} (1 - a_p p^(-s) + p^(1-2s))^(-1) * product_{p bad prime} (the appropriate thing) (with finitely many factors for bad primes), and proves that this converges absolutely, and uniformly on compact sets, in the half-plane Re(s) > 3/2; conjecturally, it represents an analytic function defined on the entire complex plane, and satisfies a functional equation relating the values at s and 2-s (and the central value L(E,1), or rather the leading nonzero coefficient of the Taylor expansion of L around that point, should conjecturally also carry arithmetic information, but this is a different topic). Formally expanding the factors into series in powers of p^(-s) and multiplying out the product gives you the explicit Dirichlet L-series sum_{n>0} a_n / n^s , where a quick calculation shows that for n = p (a good prime), a_n coincides with the previous a_p. (The expansion is actually justified in the domain of absolute convergence, so this series represents the same L-function for Re(s) > 3/2.) And now we boldly write down a power series with the same coefficients: F(q) = sum_{n>0} a_n q^n (a step which can also be regarded as applying a suitable integral transform to the L-series); this converges for |q| < 1, and by substituting q = exp(2i\pi\tau) we obtain a function f(\tau) = F(q) defined on the upper half plane Im(\tau) > 0, and 1-periodic by construction: f(\tau + 1) = f(\tau). The Taniyama-Shimura-Weil conjecture may be phrased as stating that f has even more symmetries than mere periodicity -- that it is, in technical language, a modular (cusp) form of weight 2 and level N, where N (the `conductor' of the curve) can be computed from the behaviour of the original equation defining our EC at the bad primes (and N will be a product of powers of the bad primes, each appearing at least to the first power). This way to put it, however, shows just the tip of the modular iceberg, and hides most of what is going on -- the existence of a nonconstant rational map from the `modular curve X_0(N)' to our EC, and the intricate effects which modularity has on all those automorphism actions. ECs which satisfy this conjecture are called `modular elliptic curves', and Wiles and Taylor proved that a great many elliptic curves defined over Q are indeed modular -- sufficiently many to include the Frey curves. However, by work of Serre, Ribet and others, completed around 1986-87, it had been known that the Frey curves arising from a putative FLT counterexample could not possibly be modular (if they were, they would, after some extra level-reducing steps made possible by the special structure of the Frey equations, give rise to a nonzero cusp form of weight 2 and level 2, and it is quite elementary to show that such a beast does not exist). And now my virtual rug is all shuffled and crumpled up from the huge amount of things I've been sweeping under it, so I'll pass on the torch here. ;^) I'll stop just long enough to give not-the-most-quoted reference: Dale Husemöller, `Elliptic Curves', Springer GTM 111 (1987). (Finding the most-quoted one is left as an exercise to the reader. :) Enjoy, Gerhard -- * Gerhard Niklasch * spam totally unwelcome * http://hasse.mathematik.tu-muenchen.de/~nikl/ ******* all browsers welcome * This .signature now fits into 3 lines and 77 columns * newsreaders welcome