From: rusin@washington.math.niu.edu (Dave Rusin)
Newsgroups: sci.math
Subject: Re: Elliptic curve and modular:Fermat
Date: 24 Aug 1995 04:49:41 GMT

In article <17400109S86.RVANRAAM@bcsc02.gov.bc.ca>, Ray Van Raamsdonk
 <RVANRAAM@bcsc02.gov.bc.ca> wrote:
>I am trying to understand bits and pieces of papers relating
>to Fermat's theorem and Wiles proof. I was wondering if anyone
>would care to discuss what an elliptic curve being modular means
>and what the intuitive or geometric idea and significance of this is.

OK, I'll chance this.

Geometrically, modularity means that the curve  C  can be parameterized
by means of an algebraic map by one of the particular Riemann surfaces
X_0(N) where  N  is an integer. This surface may be described as
the result of taking the upper half plane  H  of the complex plane
and glueing together any two points of  H  which are translates of
each other under the action of the modular group  Gamma_0(N). I'm
not sure how much of that means anything to you. You may be familiar with
the process of taking the plane and glueing together points whose
coordinates differ by integers, resulting in the torus; well, this
replaces the straight-edge fundamental domain of the group  Z+Z by
the loopy-edged fundamental domains of the Gamma_0(N)'s. (Oh yeah, you
have to plug up little holes -- "cusps" -- which result from these glueings)

The algebraist in me prefers to think about a Riemann surface like  X_0(N)
as floating around in  C^2, that is, it's the zero-set of some
equation in two complex variables  z  and  w. Taking polynomials in
z and  w  gives a ring of functions defined on  X_0(N); its quotient
field is called, unimaginatively, the function field of the curve.
If you had a surjection from this curve to another one  C, then any
function defined on  C  could be pulled back to a function defined on
X_0(N), giving an injection of the function field on  C  to that of
X_0(N). Turns out the converse is true too, so that modularity can
be expressed in this way:  a curve  C  is modular iff its function field
embeds in the function field of  X_0(N) for some  N.

So, for example, you view a curve  C  as the solution set to some cubic
y^2=x^3+Ax+B, and see that its function field is  a quadratic extension
of the field   Q(x)  (that is, it's  Q(x)[y]/(y^2-(x^3+Ax+B)). )
Then you stare at each of the function fields of the  X_0(N)
(they're all of transcendence degree  1  over  Q) and try to figure out if
there's a copy of your function field in there. 

Actually this gives me a chance to make the big conjecture plausible.
A classic result of algebraic number theory is the Kronecker-Weber
theorem, which asserts that every field extension of the rational
number field Q  with an Abelian Galois group embeds in a cyclotomic
field  Q[zeta_N]  for some  N. Very roughly speaking, a lot of results
on (function fields of) elliptic curves can be thought of as genus-1 
equivalents of results about field extensions of the ground field. 
So you might think that all elliptic curves over  Q  have function
fields that embed in some nice coutable list, like those of the
curves  X_0(N), that is, you might think all curves over  Q  are
modular. Well, others have thought that too, and they'd be darn happy to 
have you prove it.

dave (still, himself, trying to learn about this stuff)