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From: A on 27 Apr 2010 15:41
On Apr 27, 3:39 pm, A <anonymous.rubbert...(a)yahoo.com> wrote:
> 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


Sorry, this should read ad-bc = 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)
>
>

From: master1729 on 27 Apr 2010 12:12
rubbertube :

> 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 ?

you didnt answer that. not familiar with that term ?


> >
> > 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).

what do you mean by " no requirement " ?

that it doesnt relate to FLT ?

that the equation doesnt make sense ?

that (it doesnt make sense because ? ) it cannot longer be periodic ?


>
> 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).

classical , sounds somewhat familiar , intresting ...
but not an answer to my question(s) ?


>
>
> >
> > > 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)
> >
> >
>

regards

tommy1729
From: A on 27 Apr 2010 17:11
On Apr 27, 4:12 pm, master1729 <tommy1...(a)gmail.com> wrote:
> rubbertube :
>
>
>
> > On Apr 27, 2:38 pm, master1729 <tommy1...(a)gmail.com>
> > wrote:
> > > Gerry Myerson wrote :
> > > > 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 ?
>
> you didnt answer that.  not familiar with that term ?
>
>


No, I don't think I've ever heard the term "semi-modular" before.


>
> > > 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).
>
> what do you mean by " no requirement " ?
>


That only the case when ad-bc = 1 is part of the definition of a
modular form. The reason why we only care about the case that ad-bc =
1 is supposed to be explained by some of the geometry of moduli spaces
of elliptic curves, which I said a little bit about; the integer
matrices satisfying ad-bc = 1 are precisely the elements of SL_2(Z),
and weight 0 modular forms are just functions (satisfying a growth
bound) on the complex upper half-plane modulo the action of SL_2(Z) on
the upper half-plane; in other words, weight 0 modular forms are
functions (satisfying a growth bound) on the upper half-plane such
that f((az+b)/(cz+d)) = z.



> that it doesnt relate to FLT ?
>
> that the equation doesnt make sense ?
>
> that (it doesnt make sense because ? ) it cannot longer be periodic ?
>
>
>
>
>
> > 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).
>
> classical , sounds somewhat familiar , intresting ...
> but not an answer to my question(s) ?
>
>
>
>
>
> > > > 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)
>
> regards
>
> tommy1729

From: master1729 on 27 Apr 2010 14:27
but i find ad - bc =/= 1

also intresting and fun looking

:)
From: Gerry Myerson on 27 Apr 2010 19:39
In article
<29a4b1d1-8ebe-48d1-ab68-7028a0f9cfa0(a)i37g2000yqn.googlegroups.com>,
Ludovicus <luiroto(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?

I'm going from memory, I'll probably get it wrong,
you should really look it up somewhere,
but it's something like:

A modular form of weight k satisfies
f( ( a z + b) / (c z + d) ) = |c z + d|^(- k) f(z).

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