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](https://doi.org/10.1007/BFb0097583).


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?



Modular forms over $\mathbb{Z}$ vs modular forms with integral Fourier coefficients
Apparently it is a classic result, e.g. [a paper of Igusa](https://doi.org/10.2307/2373943) 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](https://doc.sagemath.org/html/en/reference/modfrm/sage/modular/modform/vm_basis.html) and [here](https://wstein.org/books/modform/modform/level_one.html#the-miller-basis).


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.


