Modular Form Ingredients in Fermat's Last Theorem
Date: 12/13/2000 at 19:04:59 From: Dan Subject: Modular forms and Fermat's Last Theorem Dear Dr. Math, In one of your responses concerning the proof of Fermat's Last Theorem you discussed how the L-series of elliptic curves and the M-series of modular forms correspond, and this led Taniyama and Shimura to state the conjecture that elliptic curves are modular. I do not understand what the M-series is, and how modular forms are composed of different "ingredients." It would be very helpful if you could provide a better description of the ingredients of a modular form. Thank you.
Date: 12/14/2000 at 08:01:23 From: Doctor Anthony Subject: Re: Modular forms and Fermat's Last Theorem The mathematics of modular forms is very abstract and difficult to express in terms that are familiar to any but a specialist in the subject. If you have studied matrix theory you will know that a transformation matrix will have associated eigenvalues and eigenvectors. The eigenvector is a vector that remains invariant (except perhaps in magnitude) under the operation of the transformation matrix. The eigenvectors are used amongst other things in diagonalizing a matrix, that is expressing a matrix in a form where any power of the matrix can be calculated in a few seconds. The 'ingredient' that you mention is a property of these eigenvectors and what are called Hecke operators. Below is a brief definition of these but I won't promise that you will feel much the wiser after reading it. Hecke Operators: The space of modular forms M = M(N) is finite dimensional. There are special operators called Hecke operators (denoted by T(p), one for each prime p) on this space. M is simultaneously diagonalizable by the T(p): it has a basis of modular forms that are eigenvectors for each T(p). Relation with Elliptic Curves: Start with E of conductor N. Then there is a modular form f = S a(n)e(zn) that is a T(p) eigenvector for each p such that a(p) = p#E. [matching of L and M series] Note: This means that we can calculate the number of points on E mod p without using any information about E whatsoever. - Doctor Anthony, The Math Forum http://mathforum.org/dr.math/
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