Deligne, elliptic curves and modular forms

Question

I'm trying to understand an argument of Deligne (in Courbes elliptiques: Formulaire d'après J. Tate), but I'm not familiar enough with algebraic geometry, so I'm getting quite confused. So even in my statement I might say things that don't make any sense, please correct me if so.

An elliptic curve is a smooth proper morphism $p : E \rightarrow S$ of schemes, such that the geometric fibers are connected curves of genus 1, along with a section $e: S \rightarrow E$. The sheaf of differentials $\Omega_{E/S}$ can be pushed forward to get a sheaf $\omega_{E/S} = p_{*}\Omega_{E/S}$ on $S$.

Next, he shows how $E$ can be embedded in projective space $\mathbb{P}_S^2$, in non-homogeneous coordinates: $ (*)\space\space\space\space y^2 + a_1 xy + a_3 y = x^3 + a_2 x^2 + a_4 x + a_6$.

Then later on, he defines modular forms of weight $n$ as being a law that associates to each elliptic curve $E \rightarrow S$ as above, a section of $\omega_{E/S}^{\otimes n}$ in a manner which is compatible with base change. He next says: applying the definition to the equation above (*), we see that any modular form of weight $n$ is a polynomial of degree $n$ in the $a_i$'s. I sort of see how that might make sense, but I'd appreciate it if someone could explain a bit more to me how to explicitly derive that conclusion (that they're polynomials in the $a_i$'s). Thanks in advance!

Edit: extra question: for a general scheme $S$, what do the $a_i$'s mean? if $S = Spec(R)$, I expect them to be elements of $R$, but what are they in the case of a more complicated scheme?

Answer 1

Let $\Delta$ denote the discriminant of the cubic curve given by the formula ($*$). (This is a somewhat complicated expression in the $a_i$s which I won't write down here; but let me note that there are standard expressions $c_4$ and $c_6$ which are certain polynomials in the $a_i$s, such that $1728 \Delta = c_4^3 - c_6^2$.) The formula ($*$) then defines an elliptic curve over $S:= $Spec $\mathbb Z[a_1,a_2,a_3,a_4,a_6,\Delta^{-1}].$ Also the invertible sheaf $\omega$ over $S$ attached to this elliptic curve is canonically trivialized, because it admits the global section $dx/(2y +a_1x + a_3) = dy/(3x^2 + 2 a_2 x + a_4 - a_1y).$

Thus any modular form of weight $n$, when evaluated on ($*$), gives rise to a global section of the $\mathcal O_S$, which is to say an element of $\mathbb Z[a_1,\cdots,a_6,\Delta^{-1}].$

Furthermore, as Deligne shows and as you noted, given any ellptic curve $E$ over any base $S'$, we may cover $S'$ by open sets $U$ such that $E_{| U}$ is the pull-back of ($*$) via a map $U \to S$. Thus the value of the modular form on $E$ is determined by its value on $(*)$.

In short, any modular form is determined by giving a certain element of $\mathbb Z[a_1,\ldots,a_6,\Delta^{-1}]$.

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Note that not every element of this ring is actually a modular form, because the maps $U \to S$ discussed above are not unique. Thus modular forms are those elements of $\mathbb Z[a_1,\ldots,a_6,\Delta^{-1}]$ which are invariant under the automorphisms of this ring which are induced by "change of Weierstrass equation". Deligne discusses this, and concludes that the ring of modular forms is exactly $\mathbb Z[c_4,c_6,\Delta^{\pm 1}]$. (There is also a question of holomorphicity of the cusps which I am ignoring here; probably Deligne addresses it by allowing certain singular curves as well, and hence working over $\mathbb Z[a_1,\ldots,a_6]$ rather than $\mathbb Z[a_1,\ldots,a_6,\Delta^{-1}].$ This will give the correct answer of $\mathbb Z[c_4,c_6,\Delta]$, i.e. he doesn't allow $\Delta^{-1}$ as a modular forms, since while this is well-defined on true elliptic curves, it is not well-defined on singular cubic curves.)

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Note that one can also replace $\mathbb Z$ by another ring $R$, and restrict attention to $R$-schemes, and hence define the ring of modular forms over $R$. E.g. if you take $R = \mathbb F_2$, you get the ring of modular forms mod $2$. You can check, using Deligne's formlas, that $a_1$ is in variant under change of Weierstrass equation in char. $2$, and thus defines a modular form mod $2$, called the Hasse invariant.

Similarly, you can check that $b_2$ is a well-defined modular form mod $3$ (but only mod $3$). This is the mod $3$ Hasse invariant.

These are good examples to think about to practice using Deligne's (actually Tate's) formulas for the change of Weierstrass equation to define modular forms.