I am self studying analytic number theory from Tom M Apostol Modular functions and Dirichlet series in number theory.
I am struck on proving this theorem and I need help
Theorem is - Two pairs ( a, b) and ( c, d) are equivalent iff there exists a 2× 2 matrix
$$
\begin{matrix}
p & q \\
r & s\\
\end{matrix}
$$
with integer entries and determinant of matrix equal + 1 or - 1 such that $ (c, d ) ^T $ =
$$
\begin{matrix}
p & q \\
r & s \\
\end{matrix}
$$
× $ ( a, b)^T $ .
Definition- Two pairs of complex numbers ( a, b) and ( c, d) are called equivalent iff if they generate same lattice of periods.
Lattice of periods of (a,b) is set of all linear combinations of ax+by where a, b belongs from complex numbers such that a/b is complex number and x, y are integers.
Please give some hints . I am unable to prove any of the side.
Edit ---> I proved one side, assuming equivalence I proved matrix exists.
But assuming such a matrix exists I have no clue how to prove equivalence . Please give some hint.