I am self studying analytic number theory from Tom M. Apostol Modular Functions and Dirichlet Series in Number Theory and I am stuck on this theorem on page 40.
Theorem 2.8. Every rational function of $J$ is a
modular function. Conversely, every modular function
can be expressed as a rational function of $J$.PROOF. The first part is clear. To prove the second,
suppose $f$ has zeros at $z_1,z_2,\dots,z_n$ and poles
at $p_1,p_2,\dots,p_n$ with the usual conventions about
multiplicities. Let $$g(\tau) = \prod_{k=1}^n \frac{ J(\tau) – J( z_k) } { J(\tau) – J( p_k) } $$
where a factor $1$ is inserted whenever $ z_k $ or $ p_k $ is $ \infty $ . Then $g$ has the same zeros
and poles as $f$ in the closure of $R_\Gamma$, each
with proper multiplicity. Therefore, $\,f/g\,$ has no
zeros or poles and must be constant, so $\,f\,$ is a
rational function.
I am not able to understand "where a factor $1$ is inserted whenever $ z_k $ or $ p_k $ is $ \infty $".
I am not able to understand what purpose introducing this factor solves.
The proof continues but except the above line. I have no doubts in proof.
Can someone please explain. I have thought a lot about it but I can't get it.
Also, I have no help as I am self studying and the university in which I am studying doesn't have a number theorist.
Best Answer
Your specific question was
The explanation goes back to the beginning of of Section $2.4$ on Modular Functions on page $34$ where he defines what it means for a modular function to have a pole of order $\,m\,$ at $\,i\infty\,$ using the leading term in its Fourier expansion. By the way, the statement of Theorem $2.8$ should have $\,\infty\,$ replaced with $\,i\infty\,$ instead.
The key observation is that you have to distinguish between an ordinary pole or zero and a pole or zero at $\,i\infty.\,$ A similar situation holds for rational functions defined on the extended complex plane Riemann sphere. Each ordinary zero or pole at $\,w\,$ is associated with factors of $\,F_w(z):=(z-w)\,$ raised to an integer power whose absolute value is the multiplicity of the zero or pole. But $\,F_\infty(z) = (z-\infty)\,$ is not a valid function.
However, all of these $\,F_w(z)\,$ factors have a similar behavior as $\,z\to\infty,\,$ in that they are asymptotically equivalent. Because of this behavior at $\,\infty,\,$ we can define the order of a zero or pole at $\,\infty\,$ so that each factor of $\,(z-w)\,$ is regarded as a pole at $\,\infty\,$ and further define $\,F_\infty(z):=1.\,$ This is essentially the projective viewpont. Using this convention, we can now state that any non-constant rational function has a equal number of zeros and poles (up to multiplicity), but some of the zeros and poles may be at $\,\infty.\,$
For example, the rational function $\,F_w(z)\,$ is said to have a simple zero at $\,w\,$ and a simple pole at $\,\infty.\,$ Thus, we can now write $\,z-w = F_w(z)/F_\infty(z).\,$ Any products and quotients of such factors would have an equal number of $\,F\,$ factors in the numerator and denominator. A similar situation arises in the case of modular functions with the point $\,\infty\,$ replaced by $\,i\infty\,$ and $\,F_w(z)\,$ replaced with $\,J(z)-J(w).\,$