Modular Forms and Elliptic Curves: Taniyama-ShimuraDate: 10/30/97 at 01:26:38 From: Daniel Grech Subject: Modular Forms and Elliptic Curves Hi Dr. Math, I watched a PBS show on Fermat's last theorem, and they kept talking about modular forms and elliptic curves and how they are related. Could you tell me what each of them are and how they relate to one another? Thanks, Dan Date: 10/30/97 at 08:39:29 From: Doctor Jerry Subject: Re: Modular Forms and Elliptic Curves Hi Daniel, Modular forms and elliptic curves are very complex. You may wish to look at the comments in Eric's Treasure Trove of Mathematics at http://www.astro.virginia.edu/~eww6n/math/ModularForm.html You may be able to look up elliptic curves as well by looking around at this site. -Doctor Jerry, The Math Forum Check out our web site! http://mathforum.org/dr.math/ Date: 10/30/97 at 12:15:11 From: Doctor Anthony Subject: Re: Modular Forms and Elliptic Curves Elliptic curves are of the form y^2 = x^3 + ax^2 + bx + c where a, b, c are integers. The problem with elliptic curves is to find if they have integer solutions, and if so, how many. For example the equation y^2 = x^3 - 2 with a = b = 0 and c = -2 has only one set of integer solutions, namely x = 3, y = 5, but proving that there are no other solutions is extremely difficult. The problem is simplified by making the possible numbers finite, i.e. working in 'clock' arithmetic. So 5-clock arithmetic uses only 0, 1, 2, 3, 4 the 5 = 0 again. You may recognize this as 5 congruent 0 mod(5). It is then possible to make progress with determining the number of integer solutions of the elliptic curves. For a particular elliptic curve, the number of integer solutions in each clock arithmetic forms an L-series for that curve. Example: Elliptic curve x^3 - x^2 = y^2 + y L-series L1 = 1 number of solutions in 1-clock arithmetic L2 = 4 " 2-clock " L3 = 4 " 3-clock " L4 = 8 " 4-clock " L5 = 4 L6 = 16 L7 = 9 L8 = 16 ....... ....... This series can go on as far as you like. Because we cannot say how many solutions there are in normal number space, extending to infinity as it does, the L-series gives a great deal of information about the elliptic curve it describes. The idea is that studying the L-series you can learn all you want to know about its elliptic curve. A modular form is defined by two axes, x and y, but EACH axis has a real and imaginary part. In effect it is four dimensional (xr, xi, yr, yi) where xr means real part of x, xi means imaginary part of x, and similarly with yr and yi. The four-dimensional space is called hyperbolic space. The interesting thing about modular forms is that they exhibit infinite symmetry under transformations of the type az+b f(z) -> f[------] cz+d These are functions that remain unchanged when the complex variable z is changed according to the above transformation. Here the elements a, b, c, d, arranged as a matrix, form an algebraic group. There are infinitely many possible variations. They all commute with each other and the function f is invariant under the group of transformations. Modular forms come in various shapes and sizes, but each one is built from the same basic ingredients. What differentiates each modular form is the amount of each ingredient it contains. The ingredients of a modular form are labelled from one to infinity (M1,M2,M3,....) and a particular modular form might contain one lot of ingredient one (M1=1), three lots of ingredient two (M2=3), two lots of ingredient three (M3=2) and so on. So now we get an M-series M1 = 1 M2 = 3 M3 = 2 ...... ...... and so on. At this point you come to the work of Taniyama and Shimura who found a strange affinity between some elliptic curves and some modular forms. However far you took the L-series and the M-series for a particular elliptic curve and a particular modular form, the two matched exactly. This led to the Taniyama-Shimura conjecture that ALL elliptic curves are modular. It was in proving this conjecture that Andrew Wiles established the proof of Fermat's Last theorem. The reason they are connected is as follows. Gerhard Frey showed that IF there was a solution in integers to x^n + y^n = z^n, say A^n + B^n = C^n then we could get an elliptic curve of the form y^2 = x^3 + (A^n-B^n)x^2 - (A^n.B^n)x Another mathematician, Ken Ribet, showed that this equation could not be modular. So now we have the following chain of reasoning: (1) If the Taniyama-Shimura conjecture can be proved, then every elliptic curve is modular. (2) If every elliptic curve must be modular, then the Frey elliptic curve is forbidden to exist. (3) If the the Frey elliptic curve does not exist, then there can be no solutions to the Fermat equation. (4) Therefore Fermat's Last Theorem is true. The greatest difficulty was in proving that the Taniyama-Shimura conjecture was true. This is the contribution made by Andrew Wiles, and the final stage in establishing Fermat's Last theorem. -Doctor Anthony, The Math Forum Check out our web site! http://mathforum.org/dr.math/ |
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