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### Modular Forms and Elliptic Curves: Taniyama-Shimura

```
Date: 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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