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/