Dear discussion groups,

In the proof of Fermat's last theorem a new kind

of number theory was used called "modular forms"

and it's connection to elliptic equations. I'm working

at learning this new area where it's hard to find good books.

It going to take a while to get this new stuff!

Starting on the learning curve!

Modular forms inversion?

From

http://mathworld.wolfram.com/ModularForm.html
f((a*t+b)/(c*t+d))=(c*t+d)^k*f(t)

If we let

x=(a*t+b)/(c*t+d)

which gives:

t=(-d*x+b)/(c*x-a)

Substitution gives:

f(x)=((b*c-a)/(c*x-a))^k*f((-d*x+b)/(c*x-a))

Which can be rearranged to the same form as is started from .

That gives a symmetry!

matrix

m0={{a,b},{c,d}} det=a*b-c*d trace a+d

implies the matrix:

m1={{-d,b},{c,-a}} det =a*d-c*b trace=-(a+d)

also the factor in the power gives:

c---> c/(b*c-a)

d---> -a/(b*c-a)

Are these relationships relevant to the gamma groups?

The inversion of Poincare transforms like this is one of

the first properties I learned about these bilinear transforms!

The implications to the groups are:

Gamma(N)=mod({1,0},{0,1}},N)---> mod({{-1,0},{0,-1}},N)

Gamma0(N)=mod({x,x},{0,x}},N)---> mod({{-x,x},{0,-x}},N)

Gamma1(N)=mod({1,x},{0,1}},N)---> mod({{-1,x},{0,-1}},N)

I can tell this is going to take a while to learn!

Respectfully,

Roger L. Bagula

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