Understanding Wiles' Proof of Fermat's Last Theorem: A Beginner's Guide

[C0nfused]

Hi everybody,
I wonder what knowledge is required to study Wiles proof of the last theorem of Fermat. Of course, i don't mean fully understand it but just get to understand some of his thoughts and how he actually approached the problem and found the solution. I would really like to hear the experience of anyone that has studied it for a while.
Just make something clear: I wouldn't like very complicated explanations as I am really new in number theory. I have until now studied mostly analysis and linear algebra but i am now finding number theory a really interesting field.
Thanks

 

[arildno]

Well, you should start with Simon Singh's "Fermat's last theorem" book.

 

[mathwonk]

I have not read Wiles paper but have read Singh's book, and I recommend it highly.

I know a little about the proof though, as follows.

Wiles did not even look at Fermat's last theorem. He proved a theorem about elliptic curves, which had been conjectured by Frey, and proved by Ribet, to imply Fermat's theorem.

An elliptic curve is a curve with equation of form y^2 = (x-a)(x-b)(x-c). So you see there are three constants in there, a,b,c.

There are also three constants in Fermat's last theorem, p^n + q^n = r^n, namely p,q,r.

Frey realized that if p,q,r, satisfied Fermat's theorem, then a simple recombination of them, would give numbers a,b,c, that defined a very strange elliptic curve, one that did not arise from a "modular form".

Then Wiles proved no such elliptic curves could exist. Actually he assumed another condition, "stability", but that was enough to imply Fermat.

You can get an idea of how much mathematics is involved in Wiles proof by looking at his papers or at some of the books that have been written to explain the proof since then. There is a lot, but you might start by finding out something about elliptic curves.

 

[matt grime]

The idea of how he found the solution is simple.

Over the last 50 years various people proved small things that led to the Taniyama-Shimura conjecture, and its proof. (It states that every semi-stable elliptic curve is modular, or equivalently they give the same L-series, but that isn't important). The point is that he didn't sit down and prove "it" he proved a classification theorem which indicated something that someone else had shown would do something if it were to exist... That is he had an explicit problem to solve and lots of tools that had been developed over the years to solve it: he did what all research mathematics is, he stood on the shoulders of giants.


NB. If you want to understand anything more than that about the actual mathematics then you need to be familiar with:

Complex Analysis, such as holomorphic functions, upper half planes, laurent expansions, reimann surfaces might help, Fourier series type results. How these apply to Number Theory, via L-series, and so on, Modular forms, which are approximately functions of complex variables that take values on a torus. Elliptic curves, which are equations of the form y^2 = x^3 + ax +b, and their groups, and (galois) representations of them.

That ought to take you about 10 years to get a working knowledge of.

 

[mathwonk]

After reading Matt's catalog of topics, it occurred to me that a "beginning" book might be Serre's "Course of Arithmetic". That discusses L - functions, and modular forms, and their use in number theory. But you probably need complex analysis and some group theory first. You could try it though. The first page or so already has some nice stuff.

Here is an outline of some topics from my course from the more advanced part of Serre in 1997.
II) Holomorphic Functions
A. Taylor Series Expansions: Holomorphic versus Analytic
B. Order of Zeroes and Poles / Removable Singularities
C. Principle of Isolated zeroes / Analytic Continuation
D. Uniform Limits of Holomorphic Functions
E. Properties of Logarithmic and Exponential Functions

III) Group Characters
A. Dual group, Double Duality
B. Orthogonality Relations

IV) Dirichlet Series: Four Convergence theorems
A. Prop.6: Maximal Half plane of convergence
B. Prop.7: Convergence vs removable singularities
C. Prop.8: When ordinary D - series converge for Re(s) > 1
D. Prop.9: When ordinary D - series converge for Re(s) > 0

V) Zeta Function and L - Functions
A. Definitions and Euler Product Expansions
B. Prop. 10, z(s) has a simple pole at s=1
C. Prop.10, Cor 2: Behavior of ∑p p-s, ∑p,k≥2 1/kpks, s-->1+
D. Properties of LX = ∑ X(n)/ns, zm(s) = ∏X LX(s)
E. X≠1 implies LX(1) ≠ 0.

VI) Density and Dirichlet Theorem
A. Concept of (Analytic) Density of a set of primes
B. Lemmas 7,8: Behavior of fX(s) = ∑p X(p)/ps, as s-->1+
C. Lemma 9: ƒ(m)(∑p=a(mod m) p-s) = ∑X X(a-1) fX(s)
D. Density (Pa = {p : p=a(mod m)}) = 1/ƒ(m).

 

[matt grime]

You should not that mathwonk's (x-a)(x-b)(x-c) is equivalent to my x^3+ax+b, though the a's and b's are different. He has three constants, and whilst I have two I have "moved" the frame of reference.

 

[C0nfused]

Thanks for your answers. I understood that a person that doen't study mathematics(although i do have lots of mathematics in my studies-computer engineering is my subject) can't probably get to understand all these topics in order to reach Fermat because of time. I don't think that i will have much time to study all this staff while also having many other things to study. I also didn't know that number theory is so much related to other fields of mathematics. Anyway, thanks for your help- I will do as much as i can. Just a small question: are you mathematicians? I can't believe that this is just your hobby!

 

[matt grime]

Yes, I'm a mathematician.

Number theory can surprise: some important open questions in number theory have important implications in physics. See eg Riemann's Zeroes.

 

[mathwonk]

I am a college prof of math.

But math is also a hobby for people like me and Matt, i.e. we love it.

Look how many posts he has logged, helping people for free.

 

[matt grime]

As one of my Colleagues said to me recently (He's a professor in the English sense, which approximately means distinguished research professor in the US sense), "... I get paid to do something I'd've done for nothing."

 

[devious_]

"... I get paid to do something I'd've done for nothing."

Heh. That's pretty cool.

 

[Icebreaker]

What do we do with all the proofs of the special cases of FLT?

 

[matt grime]

What do you mean?

 

[DaveC426913]

"... I get paid to do something I'd've done for nothing."

Just hope his boss wasn't listening. Might take him up on it.