First off, I can't cope with anything much beyond high school math, but given that restriction, can anyone enlighten me as to what a modular form is and why they are marvelous mathematical objects?

 

I managed to read all of Simon Singh's book on Fermat's Last Theorem without gaining any insight into what a modular form is. Funnily enough, I just looked on Amazon and the first reviewer, who is clearly a mathematician, enjoyed the book but complains about precisely this point- if you don't know what a modular form is then you miss the main idea behind the proof.

 

(P.S. Such is my ignorance that I don't even know whether this is the best subforum for this question. Mods please shift it if necessary)

I came away from the book with exactly the same questions, and persued it with my maths teacher.

 

A modular form is effectively four dimensional, made from both real and imaginary x and y axes. As you already know, they're very interesting because you can do quite a lot to them and they'll exhibit infinite symmetry.

 

Other than this, I know they're very much related to elliptic equations and the functions sn(u, k) and cs(u, k) (which are also elliptic functions). I'll try and find some more out about these, but for you to have a look at in the meantime:

 

http://mathworld.wolfram.com/JacobiEllipticFunctions.html

 

Have fun

dave

 

I think that's one of things called an 'answer' without being an 'explanation' ;-)

 

I had already directly looked up Modular Forms and still came away clueless. It may be the unfortunate truth that the explanation requires more of the basics than I have, but maybe it doesn't.

Unfortunately, modular forms are (on the mathematical scale of things) quite advanced. I'll freely admit that I know nothing about them at all other than their relationship to the proof of Fermat's Last Theorem. On the undergraduate scale, elliptic equations are covered in either the third or fourth year as an optional module (at Warwick and most other universities I've seen). I'm not quite sure where modular forms fit in.

 

Sorry I can't be of moe help.