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From: Don Stockbauer on 26 Apr 2010 20:05
On Apr 26, 5:54 pm, Gerry Myerson <ge...(a)maths.mq.edi.ai.i2u4email>
wrote:
> In article
> <025bf67b-1764-49c9-b722-3a094936a...(a)29g2000yqp.googlegroups.com>,
>
>  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?
>
> 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.

My understanding is that modular means "occurring in modules." Like a
computer program which is built from modules. Or a hotel in downtown
San Antonio which is built from modules, each room being a module set
in place but built elsewhere.
From: Ludovicus on 27 Apr 2010 12:33
On Apr 26, 6:54 pm, Gerry Myerson <ge...(a)maths.mq.edi.ai.i2u4email>
wrote:
> 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).
>
> Gerry Myerson (ge...(a)maths.mq.edi.ai) (i -> u for email)

If f(z) means function of z, hitherto there is not problem.
But what simple relation is that?
Ludovicus
From: Pubkeybreaker on 27 Apr 2010 13:09
On Apr 27, 12:33 pm, Ludovicus <luir...(a)yahoo.com> wrote:
> On Apr 26, 6:54 pm, Gerry Myerson <ge...(a)maths.mq.edi.ai.i2u4email>
> wrote:
>
> > 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).
>
> > Gerry Myerson (ge...(a)maths.mq.edi.ai) (i -> u for email)
>
> If f(z) means function of z, hitherto there is not problem.
> But what simple relation is that?

Repeat after me: Google is my friend
From: master1729 on 27 Apr 2010 10:38
Gerry Myerson wrote :

> In article
> <025bf67b-1764-49c9-b722-3a094936a0cf(a)29g2000yqp.googl
> egroups.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

refresh my memory ; what is semi-modular again ?

and what if ad - bc =/= 1.

what do we call these ? are they identical ?



> 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)
From: A on 27 Apr 2010 15:39
On Apr 27, 2:38 pm, master1729 <tommy1...(a)gmail.com> wrote:
> Gerry Myerson wrote :
>
>
>
> > In article
> > <025bf67b-1764-49c9-b722-3a094936a...(a)29g2000yqp.googl
> > egroups.com>,
> >  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?
>
> > A modular form f has the property that if a, b, c,
> > and d
> > are integers with a d - b c = 1
>
> refresh my memory ; what is semi-modular again ?
>
> and what if ad - bc =/= 1.
>
> what do we call these ? are they identical ?


If ad-bc is not equal to 1, there is no requirement for how f((az+b)/
(cz+d)) should be related to f(z).

The set of 2*2 matrices

a b
c d

with integer entries and with ab-cd = 1 is the special linear group
SL_2(Z). The special linear group acts on the upper half-plane of the
complex plane by sending a complex number z to (az + b)/(cz + d).
What's really going on here is that the orbit space of this action of
SL_2(Z) on the complex upper half-plane is the moduli space for
nonsingular elliptic curves over the complex numbers; if you throw in
a point at infinity, this is a compactification of that moduli space,
and, if memory serves, it also happens to give you the moduli space of
elliptic curves which are allowed to have a nodal singularity (but not
a cusp). I seem to recall that an elliptic curve has a canonically-
associated one-dimensional vector space of invariant differentials,
and this gives rise to a vector bundle on the (compact) moduli stack
of (possibly nodal) elliptic curves. Call that vector bundle \omega;
then a "weight 2k modular forms" is just a global section of the kth
tensor power of omega. In more prosaic terms, it's a function on the
complex upper half-plane which respects the action of SL_2(Z) in an
appropriate way and which obeys a certain growth bound (if you remove
the growth bound part of the definition of modular form, you get the
global sections of the tensor powers of \omega on the moduli stack of
nonsingular elliptic curves instead).


>
> > 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 (ge...(a)maths.mq.edi.ai) (i -> u for
> > email)
>
>

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