Fermat's Theorem Reprise

In recent years, the work of Andrew Wiles has been written about in the science news media and celebrated for his solution of the unforgettable Fermat equation Xⁿ+Yⁿ=Zⁿ. Now there are new insights into solutions to the generalized Fermat equation, X^p+Y^q=Z^r; these equations amazingly have solutions sometimes, for example, 33⁸+1549034²=15613³ or the easy solutions 1^p+2³=3².

Fermat of course claimed there are no solutions in integers X, Y, Z which are relatively prime for any integer n greater than 2. The margin, he claimed, of his copy of the ancient Greek Diophantus' Arithmetic was too small to contain his miraculous proof. Whatever proof he may have had in mind it must have had a flaw for it took mathematicians (quite a few superb ones too) another 300 years to find the key to unlock the mystery. Trying to solve the equation for each n is not the way to go; been there, tried that. By 1980 the equation had been checked for no solutions for n in the millions. Each value of n gives a very tough equation to understand. Euler and Fermat, two of the greats of the past, were able to write down a proof for n=3 or 4 but not much more. Algebraic number theory was born with the help of Kummer and others in the 19th century. Group theory came to the aid and helped to show that properties of the class group of the number field generated by the nth roots of 1 would lead to unsolvable Fermat equations. But it only handled one n at a time.

The first big breakthrough came with the work of Faltings' on his Finiteness Theorem. He won the Fields medal in 1986 for that work. An equation like the Fermat equation has as its solutions, a curve. For example the familiar Pythagorean equation X²+Y²=Z² can be regarded when Z=1 as describing the unit circle in the plane. This equation studied since Babylonian times for astronomical trigonometric calculations lead to the rational solutions X=t²-1, Y=2t, Z=1+t²; X/Z, Y/Z are rational forms of the cos(t), sin(t) trig functions. For larger values of n, rational functions can not be found to do the trick of parameterizing the Fermat curve. This was proved in the 19th century. (The genus is too big.) Falting's Theorem however, says that there can only be finitely many integer solutions for any one of these curves. His techniques used properties of the solution curve, not just with X, Y, Z real but even complex numbers. With complex numbers the solutions form a surface. A topological property of the surface is its genus or number of holes. This invariant is too coarse. The surface also has a geometrical quantity called its curvature; for the Fermat equations the curvature is negative. Of course having a surface with constant negative curvature is no small feat. Mathematicians bumped in that problem for a few thousand years before it broke through the clouds under the influence of Lobachevski, Bolyai and Gauss. They of course found that model of geometry, as studied by Euclid, without having the fifth postulate-the unique parallel to a given line through a given point. Their model of geometry did not have unique parallels, it had constant negative curvature and infinitely many parallels. The flat plane has curvature zero; the standard sphere has constant positive curvature. The Fermat equation for n=2 is the positive curvature case; n=3 is the (flat) zero curvature case. Infintely many integer solutions can only occur in the flat or positive curvature case; Falting's showed that negative curvature leads to only finitely many solutions. Falting's results are quite general and the Fermat equation was not his main point of study.

For Wiles, the Fermat equation was all important. Cloistered in his attic for seven years (research-wise) he emerged with a proof based on yet a different view of Fermat's equation. As Frey had shown an integer point on the Fermat curve leads to a modular elliptic curve. Elliptic curves are another case of curvature being zero and in this case also their equations have infinitely many solutions; but modular was the crucial point. Elliptic curves come up when you evaluate integrals of dx/f for f a polynomial of degree 3 or 4. In calculus we learn how to do the case of linear or quadratic f. The integrals are important in studying differential equations which arise in physics. An elliptic curve however when you realize its complex points is the surface of a torus or doughnut. Alternatively, an elliptic curve (torus) can be described by tesselating the plane with the translates of a fixed parallelogram; when you identify the opposite side of the parallelogram it is a torus. The set of all elliptic curves is the important concept for Wiles. These are just in correspondence with the parallelograms, and fixing one side say of length one, the other side is described by a point in the plane or just a complex number with positive imaginary part. This is the stuff of complex variables, linear fractional transformations and the like. The parameter space of elliptic curves is the upper half plane, it is in fact a complete space of constant curvature -1. It is the model of hyperbolic geometry that eluded geometers for 2000 years. The lines in this geometry are either ordinary lines perpendicualr to the x-axis or circles perpendicular to the x-axis. Out of a particular point on the axis (not a part of the geometry) there are infinitely many parallel lines which meet there. Now this hyperbolic plane admits an action of the modular group, all integer 2 by 2 matrices of determinant 1; they act as linear fractional transformations. The quotient is almost a sphere, one point is missing (from infinity). The subgroups of the modular group which are of finite index, just like in Galois theory, give coverings of this punctured sphere and these are therefore algebraic curves or Riemann surfaces. Curves which arise from the modular group in this way are called modular curves. They are defined over the rational numbers. (Incidentally, it turns out that one can reduce these curves mod p for most primes p and obtain some of the best error correcting codes.)

Wiles could prove the Fermat Theorem (using results of Ribet) if he could show that the Frey elliptic curve which has the extra technical condition of being semistable is also a modular curve. He announced this before an audience of specialists who over the next year or so mulled over his proof. All the details were slowly filled in as the experts required, save one. Wiles needed to show that a certain commutative ring, a Hecke algebra related to modular forms, has the property of being a complete intersection. He couldn't. Rings come up when you try to describe algebraic varieties. A complete intersection although technical sounding is just a restatement that the variety is an intersection of hypersurfaces, at least it is, locally. (The space curves from calculus are usually intersections, like the curve on the sphere of radius r which is the intersecton with the cylinder of diameter r.) He couldn't crack the complete intersection problem. The gap in his proof became a real annoyance. When it was offered, he turned down the opportunity to do an ad for the GAP. He had more important things to do; his immortality is longer than Warhol's standard of fifteen minutes.

Wiles sought help from his former student Taylor and together they filled the missing complete intersection problem. But now, the rest of the story. Actually, in mathematics, the story just keeps on going. But we'll stop soon. The property of semistable which Wiles used was a technical convenience it seemed. The result that described what was really going on had been conjectured by Taniyama, Shimura and Weil in the 60's: (TSW) every elliptic curve over the rationals is modular.

Just recently, Breuil, Conrad, Diamond and Taylor proved that TSW conjecture. So where to now, kimosabe. Consider the generalized Fermat curves X^p+Y^q=Z^r ; the correct form of negative curvature for these curves (and hence finitely many solutions) is 1/p+1/q+1/r<1. The finite index subgroups of the modular group which arise here are the well known (Hecke) triangle groups, generated by reflections in the sides of a triangle with angles p/p, p/q, p/r; you can do that in hyperbolic space; these triangles tesselate the hyperbolic plane. The generalized Fermat curves do have some solutions but not many by Faltings Theorem. But now we are varying p, q, r. So far the only solutions found, besides the ones in the first paragraph, are 2⁵+7²=3⁴, 7³+13²=2⁹, 2⁷+17³=71², 3⁵+11⁴=122², 17⁷+76271³=21063928² , 1414³+2213459²=65⁷ , 9262³+15312283²=113⁷ , 43⁸+96222³=30042907² . No doubt, to solve these problems, besides the TSW Theorem, new techniques will be needed to further our understanding of modular functions and L-functions.

Roger C. Alperin, 1/1/00