Modular forms over $\mathbb{Z}$ vs modular forms with integral Fourier coefficients

Question

It is well known that the ring of modular forms over $\mathbb{C}$ is $$ \mathbb{C}[c_4,c_6] $$ where $$ c_4 = 1+240 q + \cdots,\qquad c_6 = 1-504 q - \cdots $$ are the standard Eisenstein series, and the discriminant $$ \Delta= q - 24 q^2 + \cdots $$ satisfies $$ c_4^3-c_6^2=1728 \Delta. $$

You can use elliptic curves over $\mathbb{Z}$ to define modular forms over $\mathbb{Z}$, and then the resulting ring of integral modular forms is known to be given by $$ \mathbb{Z}[c_4,c_6,\Delta]/(c_4^3-c_6^2-1728 \Delta), $$ cf. Deligne après Tate.

My question is as follows:

You can instead consider a subring of $\mathbb{C}[c_4,c_6]$ containing modular forms over $\mathbb{C}$ whose Fourier coefficients (in $q$) are all integers. Clearly this subring includes $\mathbb{Z}[c_4,c_6,\Delta]/(c_4^3-c_6^2-1728 \Delta)$. Is it known/shown/easy to see that they are actually equal?

Answer 1

Apparently it is a classic result, e.g. a paper of Igusa starts by stating this fact saying it's classic without citing any. A big more googling turned up that there's something called Victor Miller's basis which exactly does the job, implemented in Sage, see here and here.

Let me present the proof here, which proceeds by induction in the degree.

Let $M_k$ be the $\mathbb{C}$-vector space of modular forms of weight $k$, and $S_k$ be the $\mathbb{C}$-vector space of cusp forms of weight $k$ (i.e. those which vanish at $q=0$). It is a standard fact that the multiplication by $\Delta$ gives an isomorphism $M_k \simeq S_{k+12}$.

We would like to prove that any element of $M_k$ whose $q$-expansion coefficients are all integral is a $\mathbb{Z}$-linear combination of monomials of $c_4$, $c_6$ and $\Delta$.

For $0\le k < 12$ this can be proved by inspection.

Now let us assume $k\ge 12$, and say $f\in M_k$ is a modular form with integral $q$-coefficients. When $k$ is odd, $f=0$, and there is nothing to prove. When $k$ is even, there is a pair of nonnegative integers $(a,b)$ such that $c_4^a c_6^b \in M_k$. Writing $f=f_0 + f_1 q + f_2 q^2 + \cdots$, we find $f-f_0 c_4^a c_6^b$ is in $S_k$ and also has integral $q$-expansion coefficients. Dividing by $\Delta$, this gives an element $g\in M_{k-12}$ which has integral $q$-expansion coefficients such that $f=g\Delta + f_0 c_4^a c_6^b$. Applying the inductive hypothesis, we are done.