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1The algebraic Katz sheaf

Recall on Riemann surfeca $X(\Gamma )$ there xists a line bundle ${\omega }_k$ whose section is the modular form of weight $k$. We want an algebraic model of ${\omega }_k$.

For simplicity, let $\Gamma ={\Gamma }_1(N)$, $N\ge 5$. Let $\pi :E^{univ}\to X(\Gamma )$ be the universal generalized elliptic curve. Consider ${\Omega }_{E^{univ}/X(\Gamma )}^1$ the sheaf of relative differentials on $E^{univ}$.

定义 1.1. $\omega ={\omega }_{X(\Gamma )}:={\pi }_{\ast }{\Omega }_{E^{univ}/X(\Gamma )}^1$.

Facts:

•	$\omega$ is a line bundle on $X(\Gamma )$.
•	Let $e:X(\Gamma )\to E^{univ}$ be the zero section, then $\omega =e^{\ast }{\Omega }_{E^{univ}/X(\Gamma )}^1$.
•	${\omega }^{\otimes k}$ is an algebraic model of the Katz sheaf ${\omega }_k$.

定义 1.2. For any ring $R$ over $\mathbb{Z}[1/N]$, a modular form of weight $k$, level $\Gamma$ and coefficient in $R$ is a global section of ${\omega }_R^{\otimes k}$.

The space of such forms is denoted by $M_k(\Gamma ;R)$.

Similarly, define $S_k(\Gamma ;R):=H^0(X(\Gamma {)}_R,{\omega }_R^{\otimes k}-(cusp))$.

When $R=\mathbb{C}$, we have $M_k(\Gamma ,\mathbb{C})=M_k(\Gamma )$.

注 1.3. Such definition does not apply to ${\Gamma }_0(N)$. There is no “$\omega$” on $X_{{\Gamma }_0(N)}$. But one may work over stacks.

•	There is a natural isomorphism ${\omega }_{X(\Gamma )}^{\otimes 2}\simeq {\Omega }_{X(\Gamma )}^1(cusp)$, the latter is the sheaf of differentials allowing simple poles along the cusps. (Kodaira–Spencer isomorphism).

•	Let $R$ be a $\mathbb{Z}[1/N]$-algebra, and $S$ an $R$-algebra. Suppose $k\ge 2$, then there is a natural isomorphism$$M_k(\Gamma ;S)\simeq M_k(\Gamma ;R){\otimes }_{R}S$$

2Katz’s alternative interpretation of modular forms

定义 2.1. Let $E\to S$ be an elliptic curve over $S$. Define ${\omega }_{E/S}={\pi }_{\ast }{\Omega }_{E/S}^1$, the sheaf of invariant differentials.

Let us take $\Gamma =S{L}_2(\mathbb{Z})$ for now.

定义 2.2. $R$ a fixed base ring. A Katz modular form of weight $k$ (and level SL_2(Z)) with coefficients in $R$ is a rule $f$ wich assigns to any elliptic curve $E$ over an $R$-scheme $S$ a section $f(E/S)$ of ${\omega }_{E/S}^{\otimes k}$ such that the following are satisfied

1.	$f(E/S)$ only depends on the $S$-isomorphic class of $E/S$.
2.	The formation of $f(E/S)$ commutes arbitrary bas echange $g:S^{\prime }\to S$ in the sense that $f(E_{S^{\prime }}/S^{\prime }=g^{\ast }f(E/S)$

The space of such rules is denoted by $M_k^{Katz}(S{L}_2(\mathbb{Z});R)$.

注 2.3. Insteaf of considering all $E/S$, we could consider only those $S=\mathrm{Spec}(R^{\prime })$ for some $R$-algebra $R^{\prime }$. More precisely,

定义 2.4. A Katz modular form of weight $k$ wutg coefficients in $R$ is a rule $f$ wich assigns to any pair $(E/R^{\prime },\omega )$ where $R^{\prime }$ is an $R$-algebra, $E/R^{\prime }$ an elliptic curve, and $\omega$ is a nowhere vanishing section of ${\Omega }_{(}^{1}E/R^{\prime })$ over $E$, with an element $f(E/R^{\prime },\omega )\in R^{\prime }$ such that

1.	$f(E/S)$ only depends on the $R^{\prime }$-isomorphic class of $(E/R^{\prime },\omega )$.
2.	$f$ is “homogeneous of degree $k$”, i.e., for all $\lambda \in (R^{\prime }{)}^{\times }$, $f(E/R^{\prime },\lambda \omega )={\lambda }^{-k}f(E/R^{\prime },\omega )$.
3.	The formation of $f(E/R^{\prime },\omega )$ commutes with arbitrary base extension $R^{\prime }\to R^{\prime \prime }$, namely $$f(E_{R^{\prime \prime }}/R^{\prime \prime },\omega {\otimes }_{R^{\prime }}{R}^{\prime \prime })=g^{\ast }f(E/R^{\prime },\omega )$$

$$f(E/\mathrm{Spec}(R^{\prime }))=f(E/R^{\prime },\omega )\cdot {\omega }^{\otimes k}.$$

注 2.5. We can even restrict to those $R^{\prime }$ such that ${\omega }_{E/R^{\prime }}$ is free.