57.1.1 The Unimodular Transformations
Definition 57.1.10 Suppose f is a nonconstant elliptic function and the module of
periods is of the form
where a,b are integers and w1∕w2 is not real. Then
by analogy with linear algebra,
is referred to as a basis. The unimodular
transformations will refer to matrices of the form
where all entries are integers and
These linear transformations are also called the modular group.
The following is an interesting lemma which ties matrices with the fractional linear
Lemma 57.1.11 Define
z if and only if
where I is the identity matrix and c≠0. Also if f
, then f−1
exists if and
only if ad − cb≠
0. Furthermore it is easy to find f−1.
Proof: The equation in 57.1.3 is just a simple computation. Now suppose
Then letting A
and so az + b = cz2 + dz. Since this is to hold for all z it follows c = 0 = b and a = d. The
other direction is obvious.
Consider the claim about the existence of an inverse. Let ad−cb≠0 for f
exists and equals
which shows f−1
exists and it is easy to find.
Next suppose f−1 exists. I need to verify the condition ad−cb≠0. If f−1 exists, then
from the process used to find it, you see that it must be a fractional linear
transformation. Letting A =
it follows there exists a matrix B
However, it was shown that this implies BA is a nonzero multiple of I which requires
that A−1 must exist. Hence the condition must hold.
Theorem 57.1.12 If f is a nonconstant elliptic function with a basis
module of periods, then
is another basis, if and only if there exists a unimodular
A such that
is a basis, there exist integers,
such that 57.1.4
It remains to show the transformation determined by the matrix is unimodular. Taking
is also given to be a basis, there exits another matrix having all
However, since w1′∕w2′ is not real, it is routine to verify that
and so det
But the two matrices have all integer entries
and so both determinants must equal either 1 or −
is unimodular. I need to verify that
is a basis. If
w ∈ M,
exist integers, m,n
which is an integer linear combination of
It only remains to verify that
is not real.
Claim: Let w1 and w2 be nonzero complex numbers. Then w2∕w1 is not real if and
Proof of the claim: Let λ = w2∕w1. Then
Thus the ratio is not real if and only if
0 if and only if w1w2 −w1w2≠
Now to verify w2′∕w1′ is not real,
This proves the theorem.