I am reading Zagier's lectures "The 1-2-3 of Modular Forms".
Before formulate the question let me introduce some notations.
Let $z$ be the coordinate on the upper-half plane $\mathbb{H}$ and $q=\text{exp}(2\pi iz)$. Theta function $\theta(z)$ is defined as follows
$$\theta(z)=\sum\limits_{n\in\mathbb{Z}}q^{n^2}=1+2q+2q^4+2q^9+2q^{16}+...\,.$$ On page 27 Zagier defined two variants of this function:
$$\theta_M(z)=\sum\limits_{n\in\mathbb{Z}}(-1)^nq^{n^2}=1-2q+2q^4-2q^9...\,,$$
$$\theta_F(z)=\sum\limits_{n\in\mathbb{Z}+1/2}q^{n^2}=2q^{1/4}+2q^{9/4}+2q^{25/4}+...\,.$$
Then he states that there is an identity
$$\theta(z)^4=\theta_M(z)^4+\theta_F(z)^4.$$
To see this he makes the following claim:
by the $q$-expansion principle all we have to do to prove it is to verify the equality of a finite number of coefficients (here just one).
Can anobody explain me what does he mean?
Thank you.
First, it is not at all clear to me that "the $q$-expansion principle" genuinely refers to the causal mechanism that he wants to invoke. Maybe it means different things to different people... In any case, the point is that a space of holomorphic modular forms of a given level is finite-dimensional... and, in fact, there are various not-so-hard estimates of how many coefficients one must check before being sure of equality. If one knows (for some other reason) that the dimension is $1$, for example, checking a single coefficient suffices.