# When does the module of Katz modular forms contain a basis for the vector space of classical modular forms?

Let $\Gamma\le SL(2,\mathbb{Z})$ be a congruence subgroup of level $N$. Let $R$ be a $\mathbb{Z}[1/N]$-algebra.

Let $\mathcal{Y}(\Gamma)_R$ denote the moduli stack over $R$ of elliptic curves equipped with a $\Gamma$-structure. For an integer $k$, let $\mathcal{Y}(\Gamma,\Omega^k)_R$ denote the moduli stack of triples $(E/S,\alpha,\omega)$, where $f : E\rightarrow S$ is an elliptic curve over an $R$-scheme $S$, $\alpha$ a $\Gamma$-structure on $E/S$, and $\omega$ is a basis of the $\mathcal{O}_S$-module $(f_*\Omega^1_{E/S})^{\otimes k}$.

There is a natural map $p_k : \mathcal{Y}(\Gamma,\Omega^k)_R\rightarrow\mathcal{Y}(\Gamma)_R$ given by forgetting the differential $\omega$. My understanding is that a Katz modular form of weight $k$ is just a section of $p_k$ (perhaps up to 2-isomorphism?). Is this correct? (Here I want to consider weakly holomorphic modular forms - that is, modular forms "holomorphic on the upper half plane, meromorphic at the cusps")

The set of such sections ("weakly holomorphic" Katz modular forms of weight $k$) form an $R$-module $KM^k(\Gamma,R)$. My question is, if we pick an embedding $R\hookrightarrow\mathbb{C}$, any section of $p_k$ can be pulled back to a section of $p_{k,\mathbb{C}} : \mathcal{Y}(\Gamma,\Omega^k)_\mathbb{C}\rightarrow\mathcal{Y}(\Gamma)_{\mathbb{C}}$, and over $\mathbb{C}$, I can more or less see that the $\mathbb{C}$-vector space $KM^k(\Gamma,\mathbb{C})$ is the "same" as the space of weakly holomorphic modular forms for $\Gamma$ defined classically.

My question is - is $KM^k(\Gamma,R)\otimes_R\mathbb{C}\cong KM^k(\Gamma,\mathbb{C})$?

If this is not always true, then under what circumstances will it be true? (perhaps some condition like the Tate curve over $\mathbb{Z}((q))\otimes_\mathbb{Z} R$ admitting "all $\Gamma$-structures"?)

Also, if $\Gamma$ is torsion free, so that $\mathcal{Y}(\Gamma)_R$ is a scheme, then does $p_k$ (over $R$) always admit a section? Equivalently, if $f : E\rightarrow S$ is an elliptic curve equipped with a "fine level structure" (e.g. a $\Gamma(N)$-structure for $N\ge 3$, a $\Gamma_1(N)$-structure for $N\ge 4$,...etc), then is $f_*\Omega^1_{E/S}\cong \mathcal{O}_S$?

I apologize if this is obvious.

I do not think that your definition of Katz modular form is exactly correct. A (Katz) weight-$k$ modular form for $\Gamma$ is a section of the line bundle on $\mathcal{Y}(\Gamma)_R$ that evaluates on every scheme $S$ classifying an elliptic curve $f\colon E\to S$ with $\Gamma$-level structure as $(f_*\Omega^1_{E/S})^{\otimes k}$. Your definition looks similar, but you seem only to allow nowhere vanishing sections. I will use $KM^k(\Gamma,R)$ in the following in my sense (which is, I think, also Katz's sense).
Sections of lines bundles commute with flat base change (as global sections are computed as a kernel). Thus $KM^k(\Gamma, R) \cong KM^k(\Gamma, \mathbb{Z}[\frac1n]) \otimes R$ for every torsionfree $\mathbb{Z}[\frac1n]$-algebra $R$ and your first question has a positive answer.
The answer to your second question is also positive, at least for $\Gamma_1(n)$. See e.g. Lemma 4.10 of https://arxiv.org/abs/1609.09264
• I'm confused - is it really possible that $KM^k(\Gamma,\mathbb{Z}[1/n])\otimes\mathbb{C}$ is always equal to $KM^k(\Gamma,\mathbb{C})$? Isn't $KM^k(\Gamma,\mathbb{Z}[1/n])$ essentially the ring of weight $k$ modular forms which have Fourier coefficients in $\mathbb{Z}[1/n]$? For example, I know there are definitely modular curves which don't admit models over $\mathbb{Q}$, so when you take $k = 0$, for these curves I don't think it's possible that $KM^0(\Gamma,\mathbb{Q})\otimes_\mathbb{Q}\mathbb{C} = KM^0(\Gamma,\mathbb{C})$... Oct 24 '17 at 15:39
• I think, it is correct what I wrote, but one has to be careful with the interpretation. For example, for me a $\Gamma(n)$-level structure would mean that one fixes an isomorphism of the $n$-torsion $E[n]$ with $(\mathbb{Z}/n)^2$. Over $\mathbb{C}$, we de not get classical modular forms for $\Gamma(n)$, but rather $\phi(n)$ copies of it. But if you want to $\mathcal{Y}(\Gamma(n))_R$ to be defined for all $\mathbb{Z}[\frac1n]$ you do not have much of a choice; to single out the "right" component, you have to restrict to $\mathbb{Z}[\frac1n, \zeta_n]$-algebras. Oct 24 '17 at 19:54
• I'm having some trouble understanding how to actually define global sections of a sheaf (e.g. line bundle) on a DM stack. A lot of sources define sheaves on a stack $\mathcal{X}$ by defining them on the lisse-etale site or the small etale-site of $\mathcal{X}$. However, the objects of these sites are schemes (over $\mathcal{X}$), and hence these sites do not have final objects, and a sheaf on such a site cannot simply be 'evaluated' at the final object to yield global sections. Do you know of a good reference which defines "global sections" of a line bundle on an algebraic stack? Oct 26 '17 at 23:53
• Abstractly, you define them as the limit over the evaluation at all schemes over $\mathcal{X}$. More concretely: Let $\mathcal{F}$ be a sheaf on the etale site of $\mathcal{X}$. Pick an etale cover $U\to \mathcal{X}$ by a scheme $U$. Then $U\times_{\mathcal{X}}U$ is a scheme as well (as the diagonal of $\mathcal{X}$ is representable by assumption). Then $\mathcal{F}(X) = \mathrm{ker}(pr_1^* - pr_2^*: \mathcal{F}(U) \to \mathcal{F}(U\times_{\mathcal{X}}U))$. Oct 27 '17 at 8:03