From: Wilbert Dijkhof <w.j.dijkhof@student.utwente.nl>
Subject: Modular equations, theta functions and solution of the equation of degree 5, 1/2
Date: Sun, 30 Apr 2000 15:59:18 +0200
Newsgroups: sci.math
Summary: [missing]

Hi all,

Modular equations and theta functions are (could be) used to 
solve the equation of degree 5. Properties of modular equations
and theta functions as well as the outline of the solution of the
equation of degree 5 are listed in the book "Pi and the AGM of 
Borwein and Borwein". A while now I'm trying to understand this
stuff, but I still don't understand most of it. So I hope there
are people who can help me. In this post I will give first some 
definitions and theorems that I do understand. 

For anyone who has this book, I don't understand much of the proof
of theorem 4.7 (about the modular equation for lambda).


Some definitions and theorems:

definition 1, page 33, basic theta functions)
The basic theta functions are defined for |q|<1 by

1) theta2(q) := sum( q^(n+1/2)^2, n=-oo..oo) with theta2(0) = 0

2) theta3(q) := sum( q^n^2, n=-oo..oo) with theta3(0) = 1

3) theta4(q) := sum( (-1)^n*q^n^2, n=-oo..oo) with theta4(0) = 1


definition 2, page 7, elliptic integrals)

The complete elliptical integral of the first kind is defined as

K := K(k) := int( (1-k^2*(sint)^2, t=0..Pi/2)

The complementary integral is defined as

K' := K'(k) := K(k') = K(sqrt(1-k^2))


theorem 1, page 41, nome)
Define k := k(q) = theta2(q)^2/theta3(q)^2 then it follows that

q = exp(-Pi*K'/K) is the unique solution in k=(0,1) to 
k = theta2(q)^2/theta3(q)^2, q is called the nome associated with k.

see theorem 2.1 page 35.

proof)
If someone is interested I will give the proof.


definition 3, page 113, modular group lambda)
The modular group lambda is the set of all transformations of the
form w = (at+b)/(ct+d) with a,d odd, b,c even and ad-bc=1.
It's a nice exercise to check that these transformations form indeed
a group under composition.


theorem 2, page 113, SL(2,Z))
These set of transformations w = (at+b)/(ct+d) are (could be made) 
isomorph to the set of matrices of the from [[a b],[c d]] under 
multiplication.

proof)
Nice exercise. 


theorem 3, page 117, generators of the lambda group)
Let q:=exp(i*Pi*t) let k(t) := k(q) and lambda(t):=(k(t))^2. 

It can be proven that (see page 115)

i) lambda(t+2) = lambda(t)
ii) lambda( t/(2t+1) ) = lambda(t)

or what is the same using our isomorphism from theorem 2:

i) lambda * S_lambda = lambda
ii) lambda * T_lambda = lambda

with S_lambda = [[1 2],[1 0]] and T_lambda = [[1 0],[2 1]]
and * means composition.

proof)
Nice exercise.


theorem 4, page 117, generators of the lambda group)
lambda is generated by S_lambda and T_lambda 

proof)
If someone is interested I will give the proof.
Pick an element w of the lambda-group, this theorem 
means that we can express w as a product of powers
of S_lambda and T_lambda and from theorem 3 follows
lambda * w = lambda.


definition 4, page 114, lambda-modular function)
A lambda-modular function is a function f which satisfies:

i) f is meromorphic in H:={Im(t)>0} (upper half plane)
ii) f(A(t)) = f(t) for all t in H* and A in lambda
iii) there's another one, but the above two are the most 
important ones


theorem 5, page 114, example of a lambda-modular function)
Above we gave a sketch of the following:
lambda(t) defined in theorem 3 is lambda-modular function.


Ok, this is enough for now. We need one important theorem
to formulate and start proving "Theorem 4.7", see next post.

Wilbert
==============================================================================

From: Wilbert Dijkhof <w.j.dijkhof@student.utwente.nl>
Subject: Modular equations, theta functions and solution of the equation of degree 5, 2/2
Date: Sun, 30 Apr 2000 22:38:38 +0200
Newsgroups: sci.math

I had promised we needed one important theorem to formulate and
proof "Theorem 4.7 of Pi and the AGM of Borwein and Borwein".


A transformation of order p is a matrix [[a b],[c d]], a,d odd,
b,c even and ad-bc=p<>0. These transformations form a group
which I will denote by GL(2,Z). We will assume that p is an odd 
prime. We say that M is equivalent to N mod G (M = N mod G) for a 
group of transformations G if there's an S in G so that M = S*N.


theorem 6, page 120)

a) Every M in GL(2,Z) is equivalent mod lambda to one of the p+1 
transformations of the set B, where

B := {B_p:=[[p 0],[0 1]], B_j:=[[1 2j],[0 p]], j=0,1,...,p-1}

Thus for every M in GL(2,Z) there's a B_j in B and L in the 
lambda-group such that M = L*B_j

b) The p+1 elements of B are pairwise inequivalent mod lambda.

proof)
Nice exercise, do this.


theorem 7, page 120)

c) The p+1 functions lambda(B_j(t)), j=0..p
are permuted by any element of the lambda-group.

proof)
We must show that {lambda * B_j * S}_j=1..p = {lambda * B_j}_j=1..p
where * means composition.

Suppose S is in lambda-group, then from theorem 6 we know that 
there's a B_k in B (B_k <> B_j) such that B_j*S = B_k mod lambda, 
because B_j*S is an element from GL(2,Z). That is there's a L in 
the lambda-group such that B_j*S = L*B_k for some k <> j 
(k in {0,1,..,p}).

Thus by modularity of lambda follows
lambda * B_j * S = lambda * L * B_k = lambda * B_k with k<>j.


Let's give an introduction to "Theorem 4.7, page 121":

The modular equation for lambda of order p is the polynomial

(a) W_p(x,lambda) := (x-lambda_0)*(x-lambda_1)*...*(x-lambda_p)

with lambda_j := lambda * B_j.

This is obviously of degree p+1 in x and has a root at each
lambda_j. Note that lambda_p(t) = lambda(pt) and 
lambda_j(t) = lambda( (t+2j)/p ), j<p.

Thus as functions of q (remember q:=exp(i*Pi*t)) we have
lambda_p(q) = lambda(q^p) and lambda_j(q) = lambda(a^j*q^(1/p))
with a^p = 1.

We now show that (independent of t) W_p is also a polynomial in 
lambda. This relies on two basic facts. First, any symmetric
polynomial in the lambda_j is lambda_modular an second, any
lambda-modular function is a rational function of lambda.
Ok, I can't follow the proof of the following theorem. So
I hope some people can help me.


Theorem 4.7, page 121, modular equation for lambda)

W_p(x,lambda) is polynomial of degree p+1 in x and lambda
with integer coefficients. The coefficients of x^(p+1) and
lambda^(p+1) are both 1.

proof)
We need three lemma's for this proof. I don't understand the
third one:

lemma 1, symmetric polynomials, page 356)
A polynomial in n variables is symmetric if it remains unchanged
by any permutation of the variables. An elementary symmetric polynomial
f_j in the variables x_1, ..., x_n is defined by

(y-x_1)*(y-x_2)*...*(y-x_n) = y^n - f_1*y^(n-1) + f_2*y^(n-2) - ...
(-1)^n*f_n

Show by induction that any symmetric polynomial in n variables with
integer
coefficients can be written as a polynomial with integer coefficients in
the
elementary symmetric functions.

proof lemma 1)
Not so nice exercise.
Note that f_n = x_1*...*x_n, ..., f_2 = x_1*x_2 + x_1*x_3 + ... +
x_(n-1)*x_n, 
f_1 = x_1 + x_2 + ... + x_n.


lemma 2, about lambda(q))
We have 
lambda(q) = 16*q*product[ (1+q^2n)^8/(1+q^(2n-1))^8, n=1..oo)

proof, given in page 64)
The proof relies on the definition of lambda(q) and Euler's indentity
product( 1+q^n, n=1..oo ) * product( 1-q^(2n-1), n=1..oo ) = 1


lemma 3, page 123, q-expansions)

Suppose that f(q) := sum( c_n*q^n, n=-h..oo ) with c_n real.
Show, for a:=exp(2*Pi*i/p) and p prime, that

sum( [f(a^n*q^(n/p)]^m, n=1..p ) with m integer

has no fractional powers in q in its expansion.

They give the following hint (page 208) if w is primitive p-th root of
unity
then sum( w^jk, j=0..p-1 ) = p if k = 0 mod p else sum( w^jk, j=0..p-1 )
= 0.

proof lemma 3)
I doubt whether this is correct. Let me prove a version which is correct
(I think).
Suppose that f(q) := sum( c_v*q^v, v=-h..oo ) with c_v real. 
Take m=1 (m=0 is trivial):

Look at sum( f(a^n*q^(1/p), n=0..p-1 ) , this equals
sum( f(a^n*q^(1/p), n=0..p-1 ) = 
sum( sum( c_v*a^(nv)*q^(v/p), v=-h..oo ) , n=0..p-1 ) = 
sum( c_v * q^(v/p) * sum( a^(nv), n=0..p-1 ) , v=-h..oo ) = (using our
hint)
sum( p * c_v * q^(v/p) , over all v = 0 mod p and v>=-h ) 

which has indeed no fractional powers in q in its expansion.

I can see that this is also right for m=2,3,... . I doubt whether this
is
true for m=-1,-2,..., but I don't think you need that.

Ok after seeing the lemma's, let continue proving Theorem 4.7.

Borwein and Borwein consider 
W'_p(x,lambda) := (y-lambda'_0)*(y-lambda'_1)*...*(y-lambda'_p) (see
(a)) with
lambda'_j := 16/lambda_j and y := 16/x. They say it's convenient to work
with, but
I don't see why.

Note that 16^(p+1)*W_p(x,lambda) = [x^(p+1)*(lambda_0 * ...
*lambda_p)]*W'_p(y,lambda).

Now by theorem 6 and lemma 1 any symmetric polynomial in 1/(lambda_0),
1/(lambda_1), 
..., 1/(lambda_p) is left invariant by any element of the lambda-group. 
Check: let f_t be the coefficient of y^(p+1-t) in W'_p(y,lambda). Thus
we have f_0 = 1,
f_1 = 1/(lambda_0) + ... + 1/(lambda_p), f_2 = 1/(lambda_0) *
1/(lambda_1) + ... ,
f_(p+1) = 1/(lambda_0) * ... * 1/(lambda_p) which are indeed left
invariant by any
element of the lambda-group. Thus f_t is lambda-modular for t=1..(p+1).

I don't see where in the proof we actually use that f_t is
lambda-modular, but ok
let continue, from lemma 2:

lambda(q) = 16*q*product[ (1+q^2v)^8/(1+q^(2v-1))^8, v=1..oo)

and we have integers c_v such that

lambda' := 16/lambda = 1/q + sum( c_v*q^v, v=0..oo), that's easy to see.

Similar we have

lambda'_j = 1/(a^j*q^(1/p)) + sum( c_v*a^(vj)*q^(v/p), v=0..oo) with j<p
and a^p = 1.

and lambda'_p = 1/(q^p) + sum( c_v*q^(vp), v=0..oo).

Using lemma 3 we show that there are integers e(t)_j such that

f_t = sum( e(t)_j*q^j, j=-(p+1)..oo).

Let's look at lemma 3 with m = 1 and h = 1:

We have f_1 = sum( p * c_(vp) * q^(v) , v=0..oo ) + 1/(q^p) + sum(
c_v*q^(vp), v=0..oo)
= sum( e_v * q^(v) , v=-1..oo ) with c_v, p*c_(vp) and thus e_v
integers.

Funny I understand more than before I wrote these two posts ago. I will
post the rest
tomorrow.

Wilbert