Modular FunctionsDate: 06/23/97 at 06:22:05 From: Mart de Graaf Subject: Modular functions Hi, I hope you can help me with this problem. Recently I saw a documentary on the proof of Fermat's Last Theorem. There I encountered "Modular functions." They said there were functions over the complex area with an incredible amount of symmetry. Can you tell me some more about modular functions? I got really curious then, but I couldn't find any answers. Thanks, Mart Date: 06/23/97 at 09:00:46 From: Doctor Anthony Subject: Re: Modular functions Dear Mart, Modular funcions are functions with super-symmetry, which means they can be transfomed in an infinity of different ways and yet remain unaltered. They cannot be represented graphically because they exist in hyperbolic space - they are complex, but with a real and imaginary component along the x-axis and a real and imaginary component along the y-axis. A simple example of the type of transformation involved is: If q = e(pi*i*w) q is a function of w, and if now w is replaced by w' where aw + b w' = -------- ..(1) a, b, c, d integers such that ad-bc = 1 cw + d The infinity of transformations (1) forms a group G and then with the aid of functions F(0), F(1), F(2), F'(1) being themselves functions of q we can construct further functions that remain unaltered for G or for some subgroup of G. In the proof of Fermat's Last Theorem, use is made of the conjecture that every elliptic equation is related to a modular form. If an elliptic equation is found that cannot be related to a modular form, then that equation has no solutions. Elliptic equations are of the form y^2 = x^3 + ax^2 + bx + c with a, b, c integers, and we require integer solutions for x and y. It was shown that if A^n + B^n = C^n with A, B, C integers was a solution of the Fermat equation, then this could be transformed into an elliptic equation: y^2 = x^3 + (A^n-B^n)x^2 - A^n*B^n It turns out that this equation can never be modular, so since all elliptic equations are modular, we cannot have A^n + B^n = C^n What Andrew Wiles accomplished was to prove the Taniyama-Shimura conjecture that every elliptic equation must be modular. From there the rest of Fermat's Last Theorem falls into place. -Doctor Anthony, The Math Forum Check out our web site! http://mathforum.org/dr.math/ |
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