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From: Ludovicus on 26 Apr 2010 09:08
Whenever I look for a definition of modular (Forms, functions,
curves) ,
I found very advanced definitions and terms.
Is there a plain, intuitive, freshman level, definition of that
concept?
It's important for understand articles on the demonstration of Fermat
Last Theorem,
because all speak of Taniyama conjecture:" All elliptic curves are
modular".
From: Pubkeybreaker on 26 Apr 2010 11:05
On Apr 26, 9:08 am, Ludovicus <luir...(a)yahoo.com> wrote:
> Whenever I look for a  definition of modular (Forms, functions,
> curves) ,
>  I found very advanced definitions and terms.
> Is there a plain, intuitive, freshman level, definition of that
> concept?

No. You will first need to learn what a meromorphic function is.
And one needs to basic group theory as well.

This is not usually a freshman topic. Modular functions are functions
in the
upper half plane that satisfy certain symmetries under the action of
the
modular group. Do you know what SL(2,Z) is?

> It's important for understand articles on the demonstration of Fermat
> Last Theorem,
> because all speak of Taniyama conjecture:" All elliptic curves are
> modular".

Again, this is not a freshman topic. You first need to learn what a q-
series
(associated with an elliptic curve) is. Do you know that periodic
functions
have a representation as a Fourier series?
Do you know that an elliptic curve over C is topologically equivalent
as a
Riemann surface to a torus? Hence it is periodic and permits a
Fourier type
expansion.

"elliptic curves are modular" basically means that the q-series
associated with
the curve can be parameterized by a modular function.

A good intro to this subject is given in the book by Neil Koblitz.
From: Herman Rubin on 26 Apr 2010 11:13
On 2010-04-26, Ludovicus <luiroto(a)yahoo.com> wrote:
> Whenever I look for a definition of modular (Forms, functions,
> curves) ,
> I found very advanced definitions and terms.
> Is there a plain, intuitive, freshman level, definition of that
> concept?
> It's important for understand articles on the demonstration of Fermat
> Last Theorem,
> because all speak of Taniyama conjecture:" All elliptic curves are
> modular".

The various fields have quite unrelated uses of the
term "modular". Some are easy and some are not.


--
This address is for information only. I do not claim that these views
are those of the Statistics Department or of Purdue University.
Herman Rubin, Department of Statistics, Purdue University
hrubin(a)stat.purdue.edu Phone: (765)494-6054 FAX: (765)494-0558
From: Bacle on 26 Apr 2010 10:39
My knowledge is that a modular form can refer
to a matrix over Z, which is invertible over Z,
i.e., a matrix Q with determinant +/- 1.

One example of this matrix is the intersection form
on Cohomology( or its equiv. PD dual in homology),
which describes how surfaces (a s submanifolds)
intersect in 4-manifolds.

HTH
From: Gerry Myerson on 26 Apr 2010 18:54
In article
<025bf67b-1764-49c9-b722-3a094936a0cf(a)29g2000yqp.googlegroups.com>,
Ludovicus <luiroto(a)yahoo.com> wrote:

> Whenever I look for a definition of modular (Forms, functions,
> curves) ,
> I found very advanced definitions and terms.
> Is there a plain, intuitive, freshman level, definition of that
> concept?

A modular form f has the property that if a, b, c, and d
are integers with a d - b c = 1 then there is a simple relation
between f( ( a z + b) / (c z + d) ) and f(z).

There's more to it than that, but it quickly gets beyond
the freshman level. See pubkeybreaker's reply.

--
Gerry Myerson (gerry(a)maths.mq.edi.ai) (i -> u for email)
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