From: RVANRAAM@BCSC02.GOV.BC.CA To: rusin@washington.math.niu.edu (Dave Rusin) Subject: Re: Elliptic curve and modular:Fermat Date: Thu, 24 Aug 95 23:05:13 Pdt In article <41h0d5$4ag@watson.math.niu.edu> on 24 Aug 1995 04:49:41 GMT you write >In article <17400109S86.RVANRAAM@bcsc02.gov.bc.ca>, Ray Van Raamsdonk > wrote >>I am trying to understand bits and pieces of papers relating >>to Fermat's theorem and Wiles proof. I was wondering if anyone >>would care to discuss what an elliptic curve being modular means >>and what the intuitive or geometric idea and significance of this is > >OK, I'll chance this Here is another response I also got SUBJECT: Modularity Received: from cleo.bc.edu by BCSC02.GOV.BC.CA (IBM VM SMTP V2R2) with Tcp Thu, 24 Aug 95 19:34:20 Pdt Received: from hermes.bc.edu (hermes.bc.edu [136.167.2.13]) by cleo.bc.edu (8.6.12/8.6.12) with SMTP id WAA43524 for ; Thu, 24 Aug 1995 22:33:18 -0400 Message-ID: Subject: Modularity To: " (Ray Van Raamsdonk) " X-Mailer: Mail*Link SMTP/MS 3.0.0 You write: |I am trying to understand bits and pieces of papers relating |to Fermat's theorem and Wiles proof. I was wondering if anyone |would care to discuss what an elliptic curve being modular means |and what the intuitive or geometric idea and significance of this is. |Also an example of an elliptic curve which is modular and one which |is not, and why, would be nice. Unfortunately for you, it is likely that there are no elliptic curves over the rationals which are not modular. Perhaps a good non-example, though, is the fact that if there had been a counterexample to Fermat's Last Theorem, a^n+b^n=c^n, then the elliptic curve given by y^2=x(x-a^n)(x+b^n) would have been a non-modular elliptic curve. That's a difficult theorem by Ken Ribet. The way Wiles proved FLT was by showing that *that* elliptic curve would have to be modular, which means there aren't any counterexamples. Oh, I suppose if we allow elliptic curves which are not defined over the rationals there will be "non-modular" ones, but so far as I know we only think about "modularity" when the curve is over the rationals (defined by polynomial equations with rational coefficients, Diophantine equations). If we take tau=i*pi below in the description of how we make an elliptic curve, probably it can't be modular. Maybe only rational elliptic curves are modular-- I haven't thought about it. I suspect it is so, because all the modular elliptic curves can be "extracted" by looking with the right tools at the modular curves X_0(N) (which are not generally elliptic, by the way). I don't have a modular parametrization of an elliptic curve handy either, I'm afraid. I know that it is possible to produce examples without too much trouble, but it requires some calculation. |terms: |modular? There are various equivalent definitions. Perhaps the easiest one is the one you give below, which I'll explain some more. |quotient space? If you have a topological space X and an equivalence relation ~ on it, then there is a quotient space X/~, whose points are equivalence classes under ~ of points of X, and whose open sets are those subsets S of X/~ such that the set of all the elements of X that are in one of the equivalence classes in S is an open set of X. Another way to define it is to say that it satisfies a certain "universal property"; there is a quotient map pi:X-->X/~ which has the property that if f:X-->Y is a continuous map and any two points of X equivalent under ~ get mapped to the same element of Y, then there is a unique map f~:X/~-->Y so that f= f~ composed with pi. So say for example that X is the unit sphere in space, and two points are equivalent under ~ if and only if they are equal or antipodal points. Then X/~ is a space where the opposite points of X have been "glued" together, leaving us with a projective plane. Or for an easier to picture example, take just two points of X and identify them. Say (x,y,z)~(x',y',z') for points of the sphere X if either (x,y,z)=(x',y',z') or (x,y,z) and (x',y',z') are (1,0,0) and (-1,0,0), not necessarily respectively. That's an equivalence relation and X/~ is essentially a sphere with two of its points attached together. The case you're probably interested in is of a quotient of a space by a group action. If G is a group acting on X, then the quotient of X by G is the quotient of X by the equivalence relation, x~x' if for some g in G g.x=x'. So the orbits of G on X are collapsed to points. The first example above is an example of that. A group of two elements can act on X by the identity doing nothing and the other element mapping each point to its antipodal point. The quotient spaces of key interest in this context are quotients of the upper half plane {z : Im z>0} by the action of subgroups of the 2 by 2 matricies with integer entries. |holomorphic? Differentiable, in the sense of complex functions. A function f is holomorphic on an open set of the complex plane when f maps to the complex plane also, and for each z in that open set, the limit as |w| tends to zero of f(z+w)/w exists, f', where w is another complex variable. If you want a good idea of what this is about, you probably need to study some complex analysis. Lots of ordinary functions are holomorphic. An equivalent definition is that it can be defined by a power series near each point in the region. If it's holomorphic in a region, it's differentiable there, and f' is also holomorphic. That means that (unlike in the case of a function of a real variable) being differentiable once is enough to make it differentiable an unbounded number of times. |compact Riemann surface? Well, it's a surface, and has a complex structure on it (so that stuff like holomorphic functions makes sense for it). It's defined a lot like "manifold". A Riemann surface is a topological space with a collection of "charts": homeomorphisms phi_i of open subsets of X to open subsets of the complex plane, such that the charts cover X, and wherever two charts overlap, the functions phi_i composed with {phi_j}^{-1} is a holomorphic function. The coordinate systems defined by the maps phi_i differ from one another by "nice" maps, so that if we define a function by a holomorphic function using one coordinate system, then it will still be holomorphic if we change to another coordinate system. A Riemann surface is a compact Riemann surface if it is compact as a topological space. For example, the complex plane is a Riemann surface. Just have one coordinate chart, the identification of the complex plane with itself. That's the simplest one. Any open set of the complex plane is also a Riemann surface. By the Riemann mapping theorem, though, there is an isomorphism between any two bounded simply connected open subsets of the complex plane; there's a 1-1 map from one to the other which has an inverse which is also holomorphic. None of those is compact, though. You can "compactify" the complex plane by adding one point, called infinity, and a second chart, which takes z-->1/z and infinity to zero. Then there are two overlapping charts; one covers everything but infinity, and the other covers everything but zero. On the overlap, the overlap map is z-->1/z of course, and that's holomorphic away from zero. That compact Riemann surface is the simplest one, and it's called the Riemann sphere. You think of a sphere sitting on the complex plane at the origin, and each point is identified with the point on the sphere which is collinear with the given point and the top of the sphere. That way every point is identified with a point on the sphere, but the point on top is not identified with anything on the plane; it's the point "infinity". Some things get easier to deal with on the Riemann sphere. Rational functions are holomorphic maps from the sphere to itself; the points where the denominator goes to zero but the numerator doesn't get mapped to infinity, and infinity gets mapped to the limit of the function as you approach infinity. One can check this extension makes the function holomorphic. Also those are essentially all the holomorphic functions on the Riemann sphere. Other more "exotic" compact Riemann surfaces are topologically surfaces with varying numbers of holes. The ones with one hole (genus 1) are the elliptic curves. The projective plane I mentioned is not a Riemann surface. You make it by identifying points under a map, but the map is not holomorphic, so the Riemann structure gets lost. The map is z--> 1/z-bar, where (a+bi)-bar is a-bi. That's not differentiable as a complex function. If you identify points of the Riemann sphere under the action of a finite group acting on it by holomorphic maps, you get a Riemann surface again, yes, but it's the Riemann sphere again. That's a little too simple. |Nick Halloway wrote the following: |An elliptic curve is _moduler_ if for some integer N, there is a |holomorphic map (isomorphism?) from X_0(N) onto the complex |points of the curve. It doesn't have to be an isomorphism. Often it isn't. X_0(N) is a quotient of the upper half plane, {z: Im(z)>0}. That's a non-compact Riemann surface. It becomes compact if you take the quotient by a suitable group acting on it. Rather than just give the group Gamma_0(N), where X_0(N) is the quotient of the upper half plane by Gamma_0(N), let me motivate it a bit. One can represent elliptic curves as quotients of the complex plane by shifts under a lattice. For example, if Z[i] is the set of Gaussian integers, m+ni where m and n are integers, then Z[i] acts nicely on the complex plane by addition: z-->z+m+ni. In the quotient space we identify two points which differ by a Gaussian integer. This "leaves" us essentially just a square, with the opposite sides attached together. That's a doughnut-shaped surface, compact. It's an elliptic curve, in fact. You can get all elliptic curves that way, but with different numbers in place of i. Instead of the Gaussian integers, you take {m+n*tau} where tau is some complex number. Now to classify elliptic curves, this is a good start. We have gotten them all (as Riemann surfaces, that is). However, we're being redundant. The elliptic curve with tau=i+1, for instance, is the same as the one with tau=i. That's because the set of shifts m+n(i+1) is the same as the set of m+ni; just use a different n. It's also true that tau=1/2+i/2 gives essentially the same curve. If you draw the lattice of points given by m+n(1/2+i/2), it's still a square lattice, but rotated by 45 degrees. So we really want to identify different tau's which yield the same elliptic curve. It turns out that tau and tau' give the same curve by this recipe if tau' = (a*tau+b)/(c*tau+d) where a,b,c,d are integers, and ad-bc=+-1. We can throw out the case of ad-bc=-1, because that just lets us flip tau below the real line to above it. tau-->-tau flips tau across to the other side. So we restrict to the upper half plane. A tau on the real line doesn't give us an elliptic curve, because you'd just be taking the quotient by horizontal shifts, not a lattice. Okay, so consider just the ad-bc case, and tau in the upper half plane. The functions a*tau+b/c*tau+d are rational, so they're holomorphic. They are 1-1 on the upper half plane. They compose like matricies; if we apply one such transformation a*tau+b/c*tau+d, and then another one, the result we get is what you'd get if instead you wrote them out as matricies and multiplied them first. (a' b')(a b) (a'a+b'c a'b+b'd) (c' d')(c d)= (c'a+d'c c'b+d'd) whereas, [a'*[(a*tau+b)/(c*tau+d)]+b']/[c'*[(a(tau+b)/(c*tau+d)]] = [(a'a+b'c)*tau+(a'b+b'd)]/[(c'a+d'c)*tau+(c'b+d'd)]. So we finally settle on the idea of taking the quotient of the upper half plane by this group of 2x2 matricies with the determinant ad-bc=1, known as SL_2(Z), acting on the upper half plane by this formula tau-->(a*tau+b)/(c*tau+d). The result is a space which classifies elliptic curves (as Riemann surfaces; it's possible to have more arithmetic structure on them, but to define modularity it isn't necessary). In a tidy way, it turns out that the quotient of the upper half plane by SL_2(Z) is isomorphic to the Riemann sphere, if you add a "point at infinity" for the limit as the imaginary part of tau goes to infinity. If not you just get a complex plane. The upper half plane gets rolled up and the bottom folded together, and the point at infinity in the imaginary direction rounds it out at the top. There is then a mapping j from the upper half plane to the Riemann sphere, such that j(tau)=j((a*tau+b)/(c*tau+d)) if a,b,c,d are integers and ad-bc=1. As im(tau) goes to infinity j(tau) does too. It's called the j-invariant of the associated elliptic curve. OK, that's essentially X_0(1). It doesn't give us any modular elliptic curves, though, because you can't map from the Riemann sphere to any elliptic curve holomorphically. X_0(N) is a more complicated moduli space. We call a space which classifies some structure like X_0(1) classifies elliptic curves a moduli space. Now instead of classifying elliptic curves themselves, we want to classify elliptic curves together with a certain set of points on them. You get an elliptic curve (as a Riemann surface) by wrapping up the complex plane. Now there is an addition law for complex numbers, and it carries over into Elliptic curves too. If we take the elliptic curve I described before, the plane quotiented out by {m+ni}, m,n integers, then for two points z, z', there's also the sum z+z'. If z is identified with z2, and z' with z2', then z+z' is identified with z2+z2'. So it all works out. The additional structure you add to the elliptic curve is to choose a cyclic subgroup of order N. That is a set of N points which are multiples of a point P, 0, P, 2P, ..., (N-1)P, where N*P=0 again. For example, on our example, the point 1/N generates the subgroup {0,1/N,2/N,...,(N-1)/N}. Or you could take the subgroup {0,1/N+i/N,...,(N-1)/N+(N-1)i/N}. Now to classify an elliptic curve *along with* it's subgroup of order N, you need a bit more information. It's still enough to give a complex number tau in the upper half plane. You can always get an elliptic curve with a subgroup by rotating your lattice so that the subgroup is along the real line, and letting tau be somewhere off of it. Now if we want to keep the information about this subgroup, we can't transform tau in all of the ways we could before. The group of transformations on tau which leave both the elliptic curve *and* the subgroup of order N is Gamma_0(N). It consists of 2x2 matricies with integer entries a,b,c,d, where ad-bc=1, and also N divides c. That last condition is what makes Gamma_0(N) smaller than SL_2(Z). I suppose at this point I can note that the transformation tau-->(-tau)/(-1) doesn't actually do anything. So we can consider the quotient of SL_2(Z) by the matrix (-1 0) (0 -1) instead, if we like, and that quotient is called PSL_2(Z). So really we want Gamma_0(N) as a subgroup of PSL_2(Z). If we take the quotient of the upper half plane by Gamma_0(N), we get X_0(N). One has to be sure to include certain limiting points, called cusps (kind of like including the point at infinity). X_0(N) is a compact Riemann surface. In fact there is more detail to add to its structure; you want to be able to write it out as a solution to some Diophantine equations too, but that's a much longer story. If we can map X_0(N) to an elliptic curve E, we call E modular. N is called the conductor. N describes some things about the arithmetic of E. If instead of classifying elliptic curves with subgroups of order N, we classify elltiptic curves with a specified point of order N, we get instead X_1(N), the quotient of the upper half plane by Gamma_1(N), the set of 2x2 matricies where a,b,c,d are integers, ad-bc=1, and N divides c, and also N divides a-1 and d-1. That's not the same as a group of order N, because a group is generated by various elements in it. {0,1/N,...,(N-1)/N} is also generated by (N-1)/N, in the quotient under {m+ni, m,n integers}. If we add also the condition that N divides b, we get a group called simply Gamma(N), and the quotient by Gamma(N) is called X(N). X(N) classifies elliptic curves with all of their points of order N "pegged down" as it were. I realize that this is quite a bit of complicated stuff that I'm writing all together. Don't be surprised if it takes you a good while to get familiar with all of this, if you are studying it. Naturally you would want to study complex analysis, and then perhaps study some of a textbook about elliptic curves, such as Silverman's book, _The Arithmetic of Elliptic Curves_. That book describes what I have been saying here (and more carefully). Keith Ramsay Ramsay-MT@hermes.bc.edu .AD RVANRAAM * * * Regards, Ray Van Raamsdonk (389-3725) BC Systems QUIT