Rings of modular forms and a splitting of ${{\,\mathrm{TMF}\,}}_0(7)$

Abstract

Among topological modular forms with level structure, ${{\,\mathrm{TMF}\,}}_0(7)$ at the prime 3 is the first example that had not been understood yet. We provide a splitting of ${{\,\mathrm{TMF}\,}}_0(7)$ at the prime 3 as ${{\,\mathrm{TMF}\,}}$-module into two shifted copies of ${{\,\mathrm{TMF}\,}}$ and two shifted copies of ${{\,\mathrm{TMF}\,}}_1(2)$. This gives evidence to a much more general splitting conjecture. Along the way, we develop several new results on the algebraic side. For example, we show the normality of rings of modular forms of level n and introduce cubical versions of moduli stacks of elliptic curves with level structure.

Appendices

Appendix A: Modular forms and q-expansions

The aim of this appendix is to review several different definitions of modular forms (complex-analytic, in the sense of Katz and via stacks) and compare them via explicit isomorphisms. Moreover, we will repeat this for modular forms with respect to the congruence subgroup $\Gamma _1(n)$ and the corresponding algebraic definition via the moduli stack of elliptic curves with level structure. We have no claim of originality here. The main reason for writing this appendix anyhow is the existence of two different versions of level structures, often called naive and arithmetic, whose precise relationship has confused at least the authors in the past. In particular, we will deduce a q-expansion principle for the naive level structure, namely Theorem A.19.

We have based our treatment on [17, 26, 50] and [51, Section 2], of which we recommend especially the first two as an introduction to modular forms. We also refer to [52] for a thorough treatment of the geometry on the analytic side.

1.1 Modular forms

In this section, we will give three definitions of modular forms and compare them.

1.1.1 Analytic definition of modular forms

We start by recalling the classical definition of modular forms. Let f be first any 1-periodic holomorphic function $\mathbb {H}\rightarrow \mathbb {C}$. Then there is a well-defined holomorphic function $g:{\mathbb {D}}{\setminus }\{0\} \rightarrow \mathbb {C}$ satisfying $f(z)=g(e^{2\pi i z})$, where ${\mathbb {D}}$ denotes the open unit disk. We say that f is holomorphic/meromorphic at $\infty$ if and only if g can be extended holomorphically/meromorphically to 0. In these cases, we call the Laurent expansion of g at 0 the classical q-expansion of f (at $\infty$).

Given a matrix $\gamma =\begin{pmatrix}a &{} b\\ c &{}d\end{pmatrix} \in {\text {GL}}_2(\mathbb {R})$ with positive determinant, an integer k and an arbitrary function $f:{\mathbb {H}}\rightarrow {\mathbb {C}}$, one defines a new function $f[\gamma ]_k$ as follows:

$$\begin{aligned} \begin{aligned} f[\gamma ]_k:&{\mathbb {H}}\rightarrow {\mathbb {C}}\\&z \mapsto (cz+d)^{-k}f\left( \frac{az+b}{cz+d}\right) . \end{aligned} \end{aligned}$$

By [17, Lemma 1.2.2], we have $(f[\gamma ]_k)[\gamma '_k] = f[\gamma \gamma ']_k$ for $\gamma , \gamma '\in {{\,\mathrm{SL}\,}}_2(\mathbb {Z})$ and the same proof works actually for arbitrary $\gamma , \gamma '\in {\text {GL}}_2(\mathbb {R})$ of positive determinant. We say that a holomorphic function $f:\mathbb {H}\rightarrow \mathbb {C}$ for a fixed k is holomorphic/meromorphic at all cusps if $f[\gamma ]_k$ is 1-periodic and holomorphic/meromorphic at $\infty$ for all $\gamma \in {{\,\mathrm{SL}\,}}_2(\mathbb {Z})$.

Let $\Gamma _1(n) \subset {{\,\mathrm{SL}\,}}_2(\mathbb {Z})$ be the subgroup of matrices that reduce to a matrix of the form $\begin{pmatrix}1&{}*\\ 0 &{} *\end{pmatrix}$ modulo n. Note that for $n=1$, we obtain $\Gamma _1(1) = {{\,\mathrm{SL}\,}}_2(\mathbb {Z})$. We denote by ${{\,\mathrm{mf}\,}}_k(\Gamma _1(n); {\mathbb {C}})$ the set of holomorphic functions $f:\mathbb {H}\rightarrow \mathbb {C}$ that satisfy

$$\begin{aligned} f\left( \frac{az+b}{cz+d}\right) =(cz+d)^kf(z) \quad \text { for every } z\in \mathbb {H}\text { and } \begin{pmatrix} a &{} b\\ c&{} d\end{pmatrix}\in \Gamma _1(n) \end{aligned}$$

(A.1)

and are holomorphic at all cusps. Note that it is automatic that $f[\gamma ]_k$ is 1-periodic for all $\gamma \in {{\,\mathrm{SL}\,}}_2(\mathbb {Z})$ as $\gamma \begin{pmatrix}1&{}1\\ 0&{}1\end{pmatrix}\gamma ^{-1} \in \Gamma _1(n)$. Elements of ${{\,\mathrm{mf}\,}}_k(\Gamma _1(n); {\mathbb {C}})$ are called holomorphic modular forms of weight k for $\Gamma _1(n)$. If we instead require f to be meromorphic at all cusps, we speak of meromorphic modular forms of weight k and denote the set of these by ${{\,\mathrm{MF}\,}}_k(\Gamma _1(n);{\mathbb {C}})$.

For a subring $R_0 \subset \mathbb {C}$, we denote by ${{\,\mathrm{MF}\,}}_k(\Gamma _1(n); R_0)$ the subset of ${{\,\mathrm{MF}\,}}_k(\Gamma _1(n); {\mathbb {C}})$ of modular forms with coefficients of classical q-expansion of f lying in $R_0$ and we use the notation ${{\,\mathrm{mf}\,}}_k(\Gamma _1(n);R_0)$ analogously.

We note that the multiplication of functions induces a multiplication on the direct sum ${{\,\mathrm{MF}\,}}(\Gamma _1(n);R_0) = \bigoplus _{k\in \mathbb {Z}} {{\,\mathrm{MF}\,}}_k(\Gamma _1(n); R_0)$, making it into a graded ring of modular forms. The q-expansion defines a ring homomorphism ${{\,\mathrm{MF}\,}}(\Gamma _1(n); R_0) \rightarrow R_0((q))$.

An important example of a modular form is the modular discriminant $\Delta \in {{\,\mathrm{mf}\,}}_{12}({{\,\mathrm{SL}\,}}_2(\mathbb {Z});\mathbb {Z})$ with q-expansion $q -24q^2 + \cdots$. From the q-expansion we see that for every meromorphic modular form $f\in {{\,\mathrm{MF}\,}}_k({{\,\mathrm{SL}\,}}_2(\mathbb {Z}); R_0)$, there is a $k>0$ such that $\Delta ^kf$ is a holomorphic modular form. Moreover, $\Delta$ vanishes nowhere on the upper half-plane [17, Corollary 1.4.2] so that $\Delta ^{-1}$ is a meromorphic modular form over $\mathbb {Z}$ again. We see that ${{\,\mathrm{mf}\,}}({{\,\mathrm{SL}\,}}_2(\mathbb {Z});R_0)[\Delta ^{-1}] \rightarrow {{\,\mathrm{MF}\,}}({{\,\mathrm{SL}\,}}_2(\mathbb {Z});R_0)$ is an isomorphism. As $\Delta [\gamma ]_{12} = \Delta$ for all $\gamma \in {{\,\mathrm{SL}\,}}_2(\mathbb {Z})$, we can repeat the argument above to see that ${{\,\mathrm{mf}\,}}(\Gamma _1(n);R_0)[\Delta ^{-1}] \rightarrow {{\,\mathrm{MF}\,}}(\Gamma _1(n);R_0)$ is an isomorphism for all n and similarly for other congruence subgroups of ${{\,\mathrm{SL}\,}}_2(\mathbb {Z})$.

1.1.2 Algebro-geometric definitions of modular forms

For the algebro-geometric definitions of modular forms, we will concentrate in this part on the situation without level, i.e. the one corresponding to modular forms for ${{\,\mathrm{SL}\,}}_2(\mathbb {Z})$. We denote for a (generalized) elliptic curve $p:E \rightarrow T$ the quasi-coherent sheaf $p_*\Omega ^1_{E/T}$ by $\omega _E$. For the definition of a generalized elliptic curve see [1, Definition 1.12].

Proposition A.2

([1, Proposition II.1.6]) Let $p:E\rightarrow T$ be a generalized elliptic curve, and denote its chosen section by $e:T\rightarrow E$. Then the sheaf $\omega _E = p_*\Omega ^1_{E/T}$ is a line bundle on T. Moreover, the adjunction counit

$$\begin{aligned} p^*p_*\Omega ^1_{E/T}\rightarrow \Omega ^1_{E/T} \end{aligned}$$

is an isomorphism and thus $p_*\Omega ^1_{E/T} \cong e^*\Omega ^1_{E/T}$.

An invariant differential for E is a nowhere vanishing section of $\Omega ^1_{E/T}$ or equivalently a trivialization of $\omega _E$.

Our second definition of modular forms will define them as a certain kind of natural transformations. Fix a commutative ring $R_0$. For any $R_0$-algebra R, denote by ${{\,\mathrm{Ell}\,}}^1(R)$ the set of isomorphism classes of pairs $(E,\omega )$ consisting of an elliptic curve E over R together with an invariant differential. This defines (together with pullback of elliptic curves and of invariant differentials) a functor

$$\begin{aligned} {{\,\mathrm{Ell}\,}}^1(-):({\text {AffSch}}{/}{{\,\mathrm{Spec}\,}}(R_0))^{{\text {op}}} \rightarrow \hbox {Sets}. \end{aligned}$$

As in [50, Section 1.1], we can consider a notion of a modular form of level 1 and weight k over $R_0$ as the subset of the set of natural transformations $f\in {{\,\mathrm{Nat}\,}}^{R_0}({{\,\mathrm{Ell}\,}}^1(-), \Gamma (-))$ with the following scaling property: For any $R_0$-algebra R, elliptic curve with chosen invariant differential $(E,\omega )$ and any $\lambda \in R^{\times }$, we have

$$\begin{aligned} f(E,\lambda \omega )=\lambda ^{-k}f(E,\omega ). \end{aligned}$$

(A.3)

Denote the set of such natural transformations by ${{\,\mathrm{Nat}\,}}^{R_0}_{k}({{\,\mathrm{Ell}\,}}^1(-), \Gamma (-))$. Also here, the direct sum $\bigoplus _{k \in \mathbb {Z}}{{\,\mathrm{Nat}\,}}^{R_0}_{k}({{\,\mathrm{Ell}\,}}^1(-), \Gamma (-))$ carries a multiplication by multiplying values in the target. This multiplication gives again a definition of a graded ring of modular forms.

For the third definition, let $\mathcal {M}_{ell,R_0}$ be the moduli stack of elliptic curves over ${{\,\mathrm{Spec}\,}}(R_0)$ (see e.g. [1] or [53]). On its big étale site, one defines a line bundle $\underline{\omega }= \underline{\omega }_{R_0}$ as follows. For a morphism $t:T\rightarrow \mathcal {M}_{ell,R_0}$ from a scheme T, let $p:E\rightarrow T$ be the corresponding elliptic curve with unit section e. We associate with (T, t) the line bundle $\omega _E$ on T. To check that this actually defines a line bundle consider a cartesian square

with unit section $e':T' \rightarrow E'$. We obtain a chain of natural isomorphisms

$$\begin{aligned} f^*\omega _E \cong f^*e^*\Omega ^1_{E/T} \cong (e')^*{\tilde{f}}^*\Omega ^1_{E/T} \cong (e')^*\Omega ^1_{E'/T'} \cong \omega _{E'} \end{aligned}$$

(A.4)

as required.

The third definition of the meromorphic modular forms over $R_0$ of weight k is $H^0(\mathcal {M}_{ell,R_0};\underline{\omega }^{{{\,\mathrm{\otimes }\,}}k}_{R_0})$. Here, the direct sum $\bigoplus _{k \in \mathbb {Z}} H^0(\mathcal {M}_{ell,R_0};\underline{\omega }^{{{\,\mathrm{\otimes }\,}}k}_{R_0})$ carries a multiplication inherited from the tensor algebra $\bigoplus _{k\in \mathbb {Z}} \underline{\omega }^{{{\,\mathrm{\otimes }\,}}k}_{R_0}$, defining also here a graded ring of modular forms. Sometimes it is convenient to reinterpret this ring as $H^0(\mathcal {M}_{ell,R_0}^1, \mathcal {O}_{\mathcal {M}_{ell,R}^1})$, where $\mathcal {M}_{ell,R_0}^1$ is the relative spectrum of $\bigoplus _{i\in \mathbb {Z}}\underline{\omega }_{R_0}^{{{\,\mathrm{\otimes }\,}}i}$ [23, Section 12.1].

1.1.3 Comparision of definitions of modular forms

We start by comparing the two algebro-geometric definitions.

Proposition A.5

There is a natural isomorphism

$$\begin{aligned} \alpha :H^0(\mathcal {M}_{ell,R_0}, \underline{\omega }_{R_0}^{\otimes k}) \rightarrow {{\,\mathrm{Nat}\,}}^{R_0}_{k}({{\,\mathrm{Ell}\,}}^1(-), \Gamma (-)). \end{aligned}$$

Moreover, on the direct sum for all $k\in \mathbb {Z}$, the map $\alpha$ induces an isomorphism of graded rings.

Proof

There is an easy map

$$\begin{aligned} \alpha :H^0(\mathcal {M}_{ell,R_0}, \underline{\omega }_{R_0}^{\otimes k}) \rightarrow {{\,\mathrm{Nat}\,}}^{R_0}_{k}({{\,\mathrm{Ell}\,}}^1(-), \Gamma (-)), \end{aligned}$$

constructed as follows. Start with an element $f\in H^0(\mathcal {M}_{ell,R_0}, \underline{\omega }_{R_0}^{\otimes k})$, an $R_0$-algebra R and an elliptic curve E/R together with an invariant differential $\omega$. If E is classified by $\varphi :{{\,\mathrm{Spec}\,}}(R)\rightarrow \mathcal {M}_{ell,R_0}$, we have $\varphi ^*(\underline{\omega }_{R_0}^{\otimes k}) = \omega _E^{\otimes k}$. By pulling back, f defines an element in $\Gamma (\varphi ^*(\underline{\omega }_{R_0}^{\otimes k}))$, which via the isomorphism $\omega ^{\otimes k}$ from $\mathcal {O}_R^{\otimes k}$ to $\omega _E^{\otimes k}$ is identified with

$$\begin{aligned} \Gamma (\varphi ^*(\underline{\omega }_{R_0}^{\otimes k})) = \Gamma (\omega _E^{\otimes k})\cong \Gamma (\mathcal {O}_R^{\otimes k})\cong \Gamma (\mathcal {O}_R)=R. \end{aligned}$$

Define $\alpha (f)(E,\omega )$ to be the image in R of the element defined by f in the left-hand side. The naturality of $\alpha (f)$ is clear. Replacing $\omega$ by $\lambda \omega$ for $\lambda \in R^{\times }$ multiplies the chosen isomorphism above by $\lambda ^k$, so we obtain

$$\begin{aligned} \alpha (f)(E, \lambda \omega )=\lambda ^{-k}\alpha (f)(E,\omega ). \end{aligned}$$

Let us sketch why $\alpha$ is an isomorphism. By definition, the section f corresponds to a compatible choice of sections in $H^0(T;\omega _E^{{{\,\mathrm{\otimes }\,}}k})$ for all $T \rightarrow \mathcal {M}_{ell,R_0}$ classifying an elliptic curve E/T. As $\omega _E$ is locally trivial, f is uniquely determined by its values on those T where $\omega _E$ is already trivial and $T = {{\,\mathrm{Spec}\,}}R$ is affine and every coherent choice of values on such T induces a section of $\underline{\omega }^{{{\,\mathrm{\otimes }\,}}k}_{R_0}$. For such T, a section of $\omega _E^{{{\,\mathrm{\otimes }\,}}k}$ corresponds exactly to associating with each trivialization $\omega$ of $\omega _E$ an element $f(E, \omega )$ such that $f(E,\lambda \omega ) = \lambda ^{-k}f(E,\omega )$. This describes ${{\,\mathrm{Nat}\,}}^{R_0}_{k}({{\,\mathrm{Ell}\,}}^1(-), \Gamma (-))$. $\square$

Next, we exhibit the map which will turn out to be an isomorphism between the algebraic geometric definitions and the complex analytic ones.

Proposition A.6

For any subring $R_0$ of $\mathbb {C}$ define

$$\begin{aligned} \beta :{{\,\mathrm{Nat}\,}}^{R_0}_{k}({{\,\mathrm{Ell}\,}}^1(-), \Gamma (-))\rightarrow {{\,\mathrm{MF}\,}}_k({{\,\mathrm{SL}\,}}_2(\mathbb {Z}), R_0), \end{aligned}$$

as follows. For any $f\in {{\,\mathrm{Nat}\,}}^{R_0}_{k}({{\,\mathrm{Ell}\,}}^1(-), \Gamma (-))$ and any $\tau \in \mathbb {H}$, set

$$\begin{aligned} \beta (f)(\tau )=f(\mathbb {C}/\mathbb {Z}\oplus \mathbb {Z}\tau , dz) \in \mathbb {C}. \end{aligned}$$

Then $\beta$ is a natural isomorphism, and induces an isomorphism of graded rings on the direct sum for all $k\in \mathbb {Z}$.

We will check (A.1) for $\beta (f)$. Let $\begin{pmatrix} a &{} b\\ c&{} d\end{pmatrix}\in {{\,\mathrm{SL}\,}}_2(\mathbb {Z})$ be given. Observe that we have a biholomorphism

$$\begin{aligned} \begin{aligned} \psi :\mathbb {C}/\left( \mathbb {Z}\cdot 1\oplus \mathbb {Z}\tau \right)&\rightarrow \mathbb {C}/\left( \mathbb {Z}\cdot 1\oplus \mathbb {Z}\frac{a\tau +b}{c\tau +d}\right) ,\\ {[z]}&\mapsto \left[ \frac{z}{c\tau +d}\right] \end{aligned} \end{aligned}$$

and by GAGA thus an isomorphism of the associated algebraic curves. Since f is well-defined on isomorphism classes, the scaling property implies

$$\begin{aligned} \begin{aligned} \beta (f)\left( \frac{a\tau +b}{c\tau +d}\right)&= f\left( \mathbb {C}/\left( \mathbb {Z}\cdot 1\oplus \mathbb {Z}\frac{a\tau +b}{c\tau +d}\right) , dz\right) \\&=(c\tau +d)^{k} f(\mathbb {C}/\mathbb {Z}\cdot 1\oplus \mathbb {Z}\tau , dz)=(c\tau +d)^{k} \beta (f)(\tau ). \end{aligned} \end{aligned}$$

We will come back to the question why $\beta (f)$ is holomorphic in the interior and meromorphic at the cusps and why $\beta$ is an isomorphism in “Appendices A.2.3 and A.4”.

1.2 Level structures

Throughout this section, let $R_0$ be a $\mathbb {Z}[\frac{1}{n}]$-algebra. While we gave the analytic definition of modular forms for $\Gamma _1(n)$ already above, there are two different corresponding algebro-geometric notions, based on naive and arithmetic level structures.

1.2.1 Naive level structures

Definition A.7

([1, Construction 4.8]) For an $R_0$-algebra R, let ${{\,\mathrm{Ell}\,}}^1_{\Gamma _1(n)}(R)$ denote the set of isomorphism classes of triples $(E, \omega , j)$, where E is an elliptic curve over R, further $\omega$ is a chosen trivialization of the line bundle $\omega _E$, and $j:{\mathbb {Z}}/n{\mathbb {Z}}_{R} \rightarrow E$ is a morphism of group schemes over ${{\,\mathrm{Spec}\,}}(R)$ and a closed immersion. This morphism j is called a $\Gamma _1(n)$-level structure.

Recall that ${\mathbb {Z}}/n{\mathbb {Z}}_{R}=\coprod _{{\mathbb {Z}}/n{\mathbb {Z}}}{{\,\mathrm{Spec}\,}}(R)$ as a scheme, with the obvious map to ${{\,\mathrm{Spec}\,}}R$ and group structure coming from the group structure on ${\mathbb {Z}}/n{\mathbb {Z}}$. The group structure on the elliptic curve is explained in [54, Section 2.1]. We can identify j with the image $P = j(1)\in E(R)$ since it determines j completely.

Remark A.8

We should remark that this variant of level structures is often called “naive” in the literature. Note also that the analogous definition in [26, Section 8.2], looks slightly different, but is equivalent by using that being closed immersion can be checked for proper schemes on geometric points.

Using again the scaling condition A.2 we can define ${{\,\mathrm{Nat}\,}}^{R_0}_k({{\,\mathrm{Ell}\,}}^1_{\Gamma _1(n)}(-), \Gamma (-))$ analogously to our definition without level in “Appendix A.1”.

We can also define a moduli stack $\mathcal {M}_1(n)$ classifying elliptic curves over $\mathbb {Z}[\frac{1}{n}]$-schemes with $\Gamma _1(n)$-level structure. We obtain a morphism $f_n:\mathcal {M}_1(n) \rightarrow \mathcal {M}_{ell}$ by forgetting the level structure. As in “Appendix A.1.3” we obtain a comparison isomorphism

$$\begin{aligned} \alpha :H^0(\mathcal {M}_1(n); \underline{\omega }^{{{\,\mathrm{\otimes }\,}}k}) \rightarrow {{\,\mathrm{Nat}\,}}^{R_0}_k({{\,\mathrm{Ell}\,}}^1_{\Gamma _1(n)}(-), \Gamma (-)); \end{aligned}$$

here and in the following we will abuse notation to denote the pullback of $\underline{\omega }$ to $\mathcal {M}_1(n)$ by $\underline{\omega }$ as well.

There are different ways to compare modular forms with and without level structure. The particular form of compatibility we want to use is expressed in the following commutative diagram.

We refer to [55, Example 4.40] for the fact that the quotient of an elliptic curve by a finite subgroup scheme is an elliptic curve again. Moreover, we will denote the right vertical morphism by $\beta _1$. The reason for our particular choice of $\beta _1$ might become clearer in the next subsection and even clearer when we discuss q-expansions. That $\beta _1$ actually lands in ${{\,\mathrm{MF}\,}}(\Gamma _1(n), R_0)$ will follow from “Appendices A.2.3 and A.4”.

Remark A.9

The group $(\mathbb {Z}/n)^\times$ acts on ${{\,\mathrm{Ell}\,}}^1_{\Gamma _1(n)}(-)$ by multiplication on the point of order n. Moreover, if we denote by $\Gamma _0(n)\subset {{\,\mathrm{SL}\,}}_2(\mathbb {Z})$ the subgroup of matrices $\begin{pmatrix}a&{}b\\ c&{}d\end{pmatrix}$ with c divisible by n, the quotient group $\Gamma _1(n){\setminus } \Gamma _0(n)$ acts on ${{\,\mathrm{MF}\,}}(\Gamma _1(n), \mathbb {C})$ as follows. For $g\in {{\,\mathrm{MF}\,}}_k(\Gamma _1(n), \mathbb {C})$ and $\gamma \in \Gamma _0(n)$, we define the action by $g. [\gamma ] = g[\gamma ]_k$ in the sense of “Appendix A.1.1”. The map

$$\begin{aligned} \Gamma _1(n){\setminus }\Gamma _0(n) \rightarrow (\mathbb {Z}/n)^\times , \quad \begin{pmatrix}a&{}\quad b\\ c&{}\quad d\end{pmatrix} \mapsto a \end{aligned}$$

is an isomorphism and under this isomorphism $\beta _1$ is equivariant.

To be compatible with [17, Section 5.2], we will actually work with the opposite convention though. This means that we will act with the inverse of an element of $(\mathbb {Z}/n)^\times$ on ${{\,\mathrm{Ell}\,}}^1_{\Gamma _1(n)}(-)$ and $\mathcal {M}_1(n)$ and use the identification

$$\begin{aligned} \Gamma _1(n){\setminus }\Gamma _0(n) \xrightarrow {\cong } (\mathbb {Z}/n)^\times , \quad \begin{pmatrix}a&{}\quad b\\ c&{}\quad d\end{pmatrix} \mapsto d. \end{aligned}$$

By the above, this makes $\beta _1$ into an equivariant map as well and this will be the equivariance we will use throughout this document.

1.2.2 Arithmetic level structures

Now we would like to discuss a different variant of level structures, called “arithmetic” in the literature.

Definition A.10

For an $R_0$-algebra R, let ${{\,\mathrm{Ell}\,}}^1_{\Gamma _{\mu }(n)}(R)$ denote the set of isomorphism classes of triples $(E, \omega , \iota )$, where E is an elliptic curve over R, again $\omega$ is a chosen trivialization of the line bundle $\omega _E$, and $\iota :\mu _{n,R} \rightarrow E$ is a morphism of group schemes over ${{\,\mathrm{Spec}\,}}(R)$ and a closed immersion. Here, $\mu _{n,R}$ is a group scheme given by the spectrum of the bialgebra $R[t]/(t^n-1)$ with comultiplication determined by $t\mapsto t\otimes t$. The morphism $\iota$ is called an arithmetic (or $\Gamma _{\mu }(n)$-) level structure on E.

For a $\mathbb {Z}\left[ \frac{1}{n}, \zeta _n\right]$-algebra R, the group schemes $\mu _{n,R}$ and ${\mathbb {Z}}/n{\mathbb {Z}}_R$ are isomorphic, but this is not true in general.

We can define the set of weight k modular forms with arithmetic level structure to be ${{\,\mathrm{Nat}\,}}^{R_0}_k({{\,\mathrm{Ell}\,}}^1_{\Gamma _{\mu }(n)}(-), \Gamma (-))$ with the same scaling condition as before. Likewise, we can define a moduli stack $\mathcal {M}_{\mu }(n)$ of elliptic curves with $\Gamma _{\mu }(n)$-level structure (over bases with n invertible). As before we obtain a comparison isomorphism

$$\begin{aligned} \alpha :H^0(\mathcal {M}_{\mu }(n); \underline{\omega }^{{{\,\mathrm{\otimes }\,}}k}) \rightarrow {{\,\mathrm{Nat}\,}}^{R_0}_k({{\,\mathrm{Ell}\,}}^1_{\Gamma _{\mu }(n)}(-), \Gamma (-)), \end{aligned}$$

where we abuse notation again to denote the pullback of $\underline{\omega }$ to $\mathcal {M}_{\mu }(n)$ by $\underline{\omega }$ as well.

We need to discuss a relation between $\Gamma _1(n)$- and $\Gamma _{\mu }(n)$-level structures. After base change to a $\mathbb {Z}\left[ \frac{1}{n}, \zeta _n\right]$-algebra R, the stacks $\mathcal {M}_{\mu }(n)$ and $\mathcal {M}_1(n)$ become equivalent over $\mathcal {M}_{ell, R}$ via the isomorphism $\mu _{n,R} \cong {\mathbb {Z}}/n{\mathbb {Z}}_R$. Less obviously, there is also a different equivalence between $\mathcal {M}_{\mu }(n)$ and $\mathcal {M}_1(n)$ that does not require any base change, but changes the underlying elliptic curve. To that purpose we recall the Weil paring [54, Section 2.8]

$$\begin{aligned} e_n:E[n](S)\times E[n](S) \rightarrow {\mathbb {G}}_{m,S}(S) \end{aligned}$$

for an elliptic curve E/S. Here, E[n] denotes the n-torsion $E\times _E S$, using the multiplication-by-n morphism $[n]:E \rightarrow E$ and the unit morphism $S\rightarrow E$ in the pullback. Using these ingredients, we add in the following lemma some details to the treatment in [51, Section 2.3].

Lemma A.11

There is an equivalence $\varphi :\mathcal {M}_{1}(n) \rightarrow \mathcal {M}_{\mu }(n)$ sending $(E\rightarrow S,P)$ to $(E/\langle P \rangle \rightarrow S, \delta )$, where $\delta$ can be described as follows: For $\zeta \in \mu _n(S)$, choose $Q\in E[n](S)$ such that $e_n(P,Q) = \zeta ^{-1}$. Then $\delta (\zeta ) = \pi (Q)$ for $\pi :E \rightarrow E/\langle P\rangle$.

Proof

With notation as in the statement of the lemma, we define $\delta :\mu _{n,S} \rightarrow E/\langle P \rangle$ as follows: As explained in [54, Section 2.8] there is a bilinear pairing

$$\begin{aligned} \langle -,-\rangle _{\pi }:\ker (\pi )\times \ker (\pi ^t) \rightarrow {\mathbb {G}}_{m,S} \end{aligned}$$

(A.12)

of abelian group schemes for $\pi :E \rightarrow E/\langle P\rangle$ the projection and $\pi ^t$ the dual isogeny. By [54, 2.8.2.1] and because $\ker (\pi ) = \langle P\rangle \cong (\mathbb {Z}/n)_S$, this induces a chain of isomorphisms

$$\begin{aligned} \ker (\pi ^t) \rightarrow {{\,\mathrm{Hom}\,}}_{S-\mathrm {gp}}(\ker (\pi ),{\mathbb {G}}_{m,S})\xrightarrow {{{\,\mathrm{ev}\,}}_P} \mu _{n,S}. \end{aligned}$$

(A.13)

The map $\delta$ is the composition of the inverse of this isomorphism with the natural inclusion $\ker (\pi ^t) \rightarrow E/\langle P \rangle$ composed with $[-1]$. The reasons for composing with $[-1]$ will be apparent in the example below.

An analogous construction dividing out $\mu _{n,S}$ provides an inverse of $\varphi$. To see this, we are using that in the situation above, $(E/\langle P \rangle )/\delta \cong E/E[n]$, and the isomorphism $E/E[n] \cong E$ induced by [n], the multiplication-by-n morphism. Thus, $\varphi :\mathcal {M}_1(n) \rightarrow \mathcal {M}_{\mu }(n)$ is an equivalence of stacks.

One can compute $\varphi$ in terms of the Weil pairing as follows: As $\pi \pi ^t = [n]$, we obtain from [54, 2.8.4.1] that $\langle P, \pi (Q) \rangle _{\pi }$ for $Q\in E[n](S)$ can be computed as $e_n(P,Q)$. Consider now $\zeta \in \mu _n(S)$. The inverse of the composition (A.13) sends $\zeta$ to $\pi (Q)$ for some $Q \in E[n](S)$ with $e_n(P,Q) = \zeta$. We obtain $e_n(P,-Q) = \zeta ^{-1}$ showing the result. $\square$

Example A.14

Let $E = \mathbb {C}/(\mathbb {Z}+n \tau \mathbb {Z})$ be an elliptic curve over ${{\,\mathrm{Spec}\,}}\mathbb {C}$ with chosen n-torsion point $\tau$. We claim that $\varphi (E,\tau ) = (\mathbb {C}/\mathbb {Z}+\tau \mathbb {Z}, \zeta _n\mapsto \frac{1}{n})$ with $\zeta _n = e^{\frac{2\pi i}{n}}$. Indeed, we have $e_n(\tau , \frac{1}{n}) = \zeta _n^{-1}$ by [54, 2.8.5.3] and thus $\zeta _n$ has to be send to $\frac{1}{n}$ as claimed by the preceding lemma.

The example implies directly the following lemma.

Lemma A.15

The following diagram commutes:

We will denote the diagonal arrow by $\beta _{\mu }$ and it will follow from “Appendices A.2.3 and A.4” that $\beta _{\mu }$ actually lands in ${{\,\mathrm{MF}\,}}(\Gamma _1(n); R_0)$.

1.2.3 Compactifications and comparison of algebraic and analytic theory

In this section we discuss the comparison of the algebraic and the analytic theory. The basic sources are [1, 52] and we will just give a short summary. We will use the compactifications $\overline{\mathcal {M}}_1(n)$ of $\mathcal {M}_1(n)$ as recalled in the beginning of Sect. 2. It is shown in [1, Section IV] that $\overline{\mathcal {M}}_1(n) \rightarrow {{\,\mathrm{Spec}\,}}\mathbb {Z}[\tfrac{1}{n}]$ is proper and smooth of relative dimension 1.

For $n\ge 5$, the stack $\overline{\mathcal {M}}_1(n)$ is representable by a projective scheme (see e.g. [4]). It is shown in [52, Thm. 2.2.2.1] that the Riemann surface associated with $\overline{\mathcal {M}}_1(n)_{\mathbb {C}}$ is isomorphic to a more classical construction, namely the compactification $X_1(n)$ of the quotient $Y_1(n)$ of the upper half-plane ${\mathbb {H}}$ by $\Gamma _1(n)$. Indeed, Conrad shows that both $\overline{\mathcal {M}}_1(n)_{\mathbb {C}}$ and $X_1(n)$ classify generalized elliptic curves over complex analytic spaces with $\Gamma _1(n)$-level structure. The family of elliptic curves $(\mathbb {C}/\mathbb {Z}+n\tau \mathbb {Z}, \tau )$ with $\Gamma _1(n)$-level structure over ${\mathbb {H}}$ descends to $Y_1(n)$ and extends to $X_1(n)$. (Indeed, Conrad considers the universal family $(\mathbb {C}/\mathbb {Z}+\tau \mathbb {Z}, \frac{1}{n})$ as in [52, Section 2.1.3], but the choice of $e^{2\pi i/n}$ as an nth root of unity allows us to consider the automorphism $\mathcal {M}_1(n)_{\mathbb {C}} \xrightarrow {\varphi } \mathcal {M}_{\mu }(n)_{\mathbb {C}} \simeq \mathcal {M}_1(n)_{\mathbb {C}}$ that carries one family of elliptic curves into the other as follows from Example A.14.) This specifies an isomorphism from $X_1(n)$ to the Riemann surface associated with $\overline{\mathcal {M}}_1(n)_{\mathbb {C}}$, and by restriction to the locus where the fibers of the universal generalized elliptic curve are smooth, also an isomorphism from $Y_1(n)$ to the Riemann surface associated with $\mathcal {M}_1(n)_{\mathbb {C}}$. More information about $Y_1(n)$ and $X_1(n)$ can be found in [52] and in [17, Chapter 2].

We will abuse notation again and denote by $\underline{\omega }$ the line bundle on $X_1(n)$ corresponding to the analytification of $\underline{\omega }$ on $\overline{\mathcal {M}}_1(n)_{\mathbb {C}}$ under the isomorphism above and likewise its restriction to $Y_1(n)$. By GAGA [56, Théorème 1], the morphism $H^0(\overline{\mathcal {M}}_1(n)_{\mathbb {C}}; \underline{\omega }^{{{\,\mathrm{\otimes }\,}}k}) \rightarrow H^0(X_1(n);\underline{\omega }^{{{\,\mathrm{\otimes }\,}}k})$ is an isomorphism. Given a section of $\underline{\omega }^{{{\,\mathrm{\otimes }\,}}k}$ on $Y_1(n)$ we can pull it back along $\pi :{\mathbb {H}} \rightarrow Y_1(n)$ and obtain a holomorphic function on ${\mathbb {H}}$ by trivializing $\pi ^*\underline{\omega }$ via dz. It is shown in [52, Lemma 1.5.7.2] that the image consists exactly of those holomorphic functions on ${\mathbb {H}}$ satisfying the transformation formula (A.1) for modular forms of weight k for $\Gamma _1(n)$. Moreover, Conrad shows that the image of $H^0(X_1(n);\underline{\omega }^{{{\,\mathrm{\otimes }\,}}k}) \hookrightarrow H^0(Y_1(n);\underline{\omega }^{{{\,\mathrm{\otimes }\,}}k})$ corresponds exactly to the holomorphic modular forms of weight k for $\Gamma _1(n)$.

In summary, we obtain an isomorphism $\psi :H^0(\overline{\mathcal {M}}_1(n)_{\mathbb {C}};\underline{\omega }^{{{\,\mathrm{\otimes }\,}}k}) \cong {{\,\mathrm{mf}\,}}_k(\Gamma _1(n);\mathbb {C})$. Unraveling the definitions from [52, Section 1.5.1 and 1.5.2] shows that this is compatible with our comparison map

$$\begin{aligned} \beta _1\alpha :H^0(\mathcal {M}_1(n)_{\mathbb {C}}; \underline{\omega }^{{{\,\mathrm{\otimes }\,}}k} )\xrightarrow {\cong } {{\,\mathrm{Nat}\,}}^{\mathbb {C}}_k({{\,\mathrm{Ell}\,}}_{\Gamma _1(n)}(-),\Gamma (-)) \rightarrow \{\text {functions on }\mathbb {H}\}. \end{aligned}$$

We will argue why $\beta _1\alpha$ actually takes values in ${{\,\mathrm{MF}\,}}_k(\Gamma _1(n);\mathbb {C})$ (as claimed before) and why with this target $\beta _1\alpha$ becomes an isomorphism.

The modular form $\Delta \in {{\,\mathrm{mf}\,}}_{12}({{\,\mathrm{SL}\,}}_2(\mathbb {Z});\mathbb {Z}) \subset {{\,\mathrm{mf}\,}}_{12}(\Gamma _1(n);\mathbb {C})$ (see “Appendix A.1.1”) corresponds to a holomorphic section of $\underline{\omega }^{{{\,\mathrm{\otimes }\,}}12}$ on $X_1(n)$ with simple zeros at all cusps, i.e. at all those points in the complement of $Y_1(n)$; this can be seen by considering the q-expansion of $\Delta$ and the construction of the $X_1(n)$. Thus, $H^0(X_1(n); \underline{\omega }^{{{\,\mathrm{\otimes }\,}}*})[\Delta ^{-1}]$ corresponds exactly to those holomorphic sections of $\underline{\omega }^{{{\,\mathrm{\otimes }\,}}*}$ on $Y_1(n)$ that can be meromorphically extended to $X_1(n)$. This in turn corresponds exactly to the (algebraic) sections of $\underline{\omega }^{{{\,\mathrm{\otimes }\,}}*}$ on $\mathcal {M}_1(n)_{\mathbb {C}}$. This implies an identification $H^0(\mathcal {M}_1(n)_{\mathbb {C}}; \underline{\omega }^{{{\,\mathrm{\otimes }\,}}*}) \cong H^0(\overline{\mathcal {M}}_1(n)_{\mathbb {C}}; \underline{\omega }^{{{\,\mathrm{\otimes }\,}}*})[\Delta ^{-1}]$. Under this identification, $\beta _1\alpha$ can be written as

$$\begin{aligned} H^0(\overline{\mathcal {M}}_1(n)_{\mathbb {C}};\underline{\omega }^{{{\,\mathrm{\otimes }\,}}*})[\Delta ^{-1}] \underset{\psi }{\xrightarrow {\cong }} {{\,\mathrm{mf}\,}}_*(\Gamma _1(n); \underline{\omega }^{{{\,\mathrm{\otimes }\,}}*})[\Delta ^{-1}] \cong {{\,\mathrm{MF}\,}}_*(\Gamma _1(n);\mathbb {C}), \end{aligned}$$

(followed by the inclusion into $\{\text {functions on }\mathbb {H}\}$). This shows our claims.

For $n<5$, $\overline{\mathcal {M}}_1(n)$ is no longer a scheme. In this case, one can analogously use a GAGA theorem for stacks as, for example, proven in [57]. In our situation the proof should be considerably simplified though as $\overline{\mathcal {M}}_1(n)_{\mathbb {C}}$ has a finite faithfully flat cover by a scheme (e.g. by $\overline{\mathcal {M}}_1(5n)_{\mathbb {C}}$) and one should be able to deduce a sufficiently strong GAGA theorem just by descent from the scheme case.

1.3 The Tate curve

In this section, we will discuss the Tate curve, which will give us an algebraic way to define q-expansions of modular forms. For an alternative treatment we refer e.g. to [54, Section 8.8]. We first discuss the situation over the complex numbers.

Theorem A.16

([28, Theorem V.1.1]) For any $q,u\in \mathbb {C}$ with $|q|<1$, define the following quantities:

$$\begin{aligned} \begin{aligned} \sigma _k(n)&=\sum _{d|n} d^k,\\ s_k(q)&=\sum _{n\ge 1} \sigma _k(n)q^n=\sum _{n\ge 1} \frac{n^kq^n}{1-q^n},\\ a_4(q)&= -5s_3(q),\\ a_6(q)&=-\frac{5s_3(q)+7s_5(q)}{12},\\ X(u,q)&=\sum _{n\in \mathbb {Z}} \frac{q^nu}{(1-q^nu)^2}-2s_1(q),\\ Y(u,q)&=\sum _{n\in \mathbb {Z}} \frac{(q^nu)^2}{(1-q^nu)^3}+s_1(q). \end{aligned} \end{aligned}$$

• (1) Then the equation

  $$\begin{aligned} y^2+xy=x^3+a_4(q)x+a_6(q) \end{aligned}$$

  (A.17)

  defines an elliptic curve $E_q$ over $\mathbb {C}$, and X, Y define a complex analytic isomorphism

  $$\begin{aligned} \begin{aligned} \mathbb {C}^{\times }/q^{\mathbb {Z}}&\rightarrow E_q\\ u&\mapsto {\left\{ \begin{array}{ll} (X(u,q), Y(u,q)), &{}\quad \text{ if } u\notin q^{\mathbb {Z}},\\ O,&{}\quad \text{ if } u \in q^{\mathbb {Z}} \end{array}\right. } \end{aligned} \end{aligned}$$

• (2) The power series $a_4(q)$ and $a_6(q)$ define holomorphic functions on the open unit disk ${\mathbb {D}}$.

• (3) As power series in q, both $a_4(q),a_6(q)$ have integer coefficients.

• (4) The discriminant of $E_q$ is given by

  $$\begin{aligned} \Delta (q)=q\prod _{n\ge 1} (1-q^n)^{24} \in {\mathbb {Z}}\llbracket q\rrbracket . \end{aligned}$$

• (5) Every elliptic curve over $\mathbb {C}$ is isomorphic to $E_q$ for some q with $|q|<1$.

Let ${{\,\mathrm{Conv}\,}}\subset \mathbb {Z}((q))$ be the subset of “convergent” Laurent series, i.e. those that define meromorphic functions on ${\mathbb {D}}$ that are holomorphic away from 0; in particular, $a_4, a_6\in {{\,\mathrm{Conv}\,}}$. As $\Delta (q)$ is non-vanishing for $q\ne 0$ in ${\mathbb {D}}$, it defines an invertible element in ${{\,\mathrm{Conv}\,}}$ and thus we can use the Weierstraß equation (A.17) to define an elliptic curve ${{\,\mathrm{Tate}\,}}(q)$ over ${{\,\mathrm{Conv}\,}}$. For our computations in Section 3 it will be convenient to consider the analogously defined ring ${{\,\mathrm{Conv}\,}}_{q^n} \subset \mathbb {Z}((q^n))$ with the Tate curve ${{\,\mathrm{Tate}\,}}(q^n)$ defined by $a_4(q^n)$ and $a_6(q^n)$ over it.

Let $q_0\in {\mathbb {D}}$ be a nonzero point and consider the morphism ${{\,\mathrm{ev}\,}}_{q_0}:{{\,\mathrm{Conv}\,}}\rightarrow \mathbb {C}$. By the theorem above, we see that the analytic space associated with ${{\,\mathrm{ev}\,}}_{q_0}^*{{\,\mathrm{Tate}\,}}(q)$ is isomorphic to $\mathbb {C}^\times /q_0^\mathbb {Z}$. The invariant differential $\eta ^{can}$ associated to the Weierstraß equation corresponds under this isomorphism to $\frac{du}{u}$, as can be shown by elementary manipulations using [28, Section V.1].

Next, we want to describe a group homomorphism $\iota :\mu _{n,{{\,\mathrm{Conv}\,}}[\frac{1}{n}]} \rightarrow {{\,\mathrm{Tate}\,}}(q)_{\mathbb {Z}[\frac{1}{n}]}$ for $n\ge 2$. We first define a morphism $\iota _{\zeta }:\mu _{n,{{\,\mathrm{Conv}\,}}[\frac{1}{n}, \zeta _n]} \rightarrow {{\,\mathrm{Tate}\,}}(q)_{\mathbb {Z}[\frac{1}{n}, \zeta _n]}$. As $\mu _n$ is isomorphic to $\mathbb {Z}/n$ over $\mathbb {Z}[\frac{1}{n},\zeta _n]$, it suffices to specify an n-torsion point in ${{\,\mathrm{Tate}\,}}(q)({{\,\mathrm{Conv}\,}}[\frac{1}{n}, \zeta _n])$ as the image of $\zeta _n$; we take $(X(\zeta _n,q), Y(\zeta _n, q))$. As X and Y have integer coefficients, we see that for every ring automorphism $\sigma$ of $\mathbb {Z}[\frac{1}{n},\zeta _n]$, we have $\iota _{\zeta }(\sigma (\zeta _n)) = \sigma (\iota _{\zeta }(\zeta _n))$. Thus, Galois descent implies that $\iota _{\zeta }$ descends to a morphism $\iota :\mu _{n,{{\,\mathrm{Conv}\,}}[\frac{1}{n}]} \rightarrow {{\,\mathrm{Tate}\,}}(q)_{\mathbb {Z}[\frac{1}{n}]}$. Note that we can check that this is indeed a group homomorphism into the n-torsion by evaluating at infinitely many points in ${\mathbb {D}}$. For a nonzero $q_0\in {\mathbb {D}}$, this $\iota$ corresponds under the isomorphism of ${{\,\mathrm{ev}\,}}_{q_0}^*{{\,\mathrm{Tate}\,}}(q)$ with $\mathbb {C}^\times /q_0^\mathbb {Z}$ exactly to the composite $\mu _n(\mathbb {C}) \rightarrow \mathbb {C}^\times \rightarrow \mathbb {C}^\times /q_0^\mathbb {Z}$. Note that $\iota$ defines a $\Gamma _{\mu }(n)$-structure on ${{\,\mathrm{Tate}\,}}(q)_{\mathbb {Z}[\frac{1}{n}]}$.

As a last point, we mention that for a subring $R\subset \mathbb {C}$ containg $\zeta _n$, the n-torsion ${{\,\mathrm{Tate}\,}}(q^n)_R[n]$ is isomorphic to $(\mathbb {Z}/n)_{{{\,\mathrm{Conv}\,}}_{q^n,R}}^2$ as it has rank $n^2$ over ${{\,\mathrm{Conv}\,}}_R$ [54, Theorem 2.3.1] and we can specify $n^2$ points by $(X(\zeta _n^aq^b, q^n), Y(\zeta _n^aq^b, q^n))$, where $0\le a, b \le n-1$.

1.4 q-expansions

Our goal in this subsection is to define the q-expansion both in the holomorphic and in the algebraic context, to compare them and to obtain a q-expansion principle.

Consider a modular form f in ${{\,\mathrm{MF}\,}}(\Gamma _1(n);\mathbb {C})$ for $n\ge 1$. We recall that f factors through a meromorphic function $g:{\mathbb {D}} \rightarrow \mathbb {C}$ on the open unit disk with only possible pole in 0; more precisely, we have $g(q) = f(z)$, where $q = q(z) = e^{2\pi i z}$. Taylor expansion of g at 0 yields the classical q-expansion

$$\begin{aligned} \Phi ^{hol}:{{\,\mathrm{MF}\,}}_k(\Gamma _1(n);\mathbb {C}) \rightarrow \mathbb {C}((q)). \end{aligned}$$

Let us fix for the whole subsection a $\mathbb {Z}[\frac{1}{n}]$-subalgebra $R_0\subset \mathbb {C}$. On the algebraic side, we obtain a map

$$\begin{aligned} \Phi ^{\mu , R_0}:{{\,\mathrm{Nat}\,}}^{R_0}_k({{\,\mathrm{Ell}\,}}^1_{\Gamma _{\mu }(n)}(-), \Gamma (-)) \rightarrow R_0((q)) \end{aligned}$$

by evaluating the natural transformation at the pullback to $R_0((q))$ of the Tate curve $({{\,\mathrm{Tate}\,}}(q), \eta _{can}, \iota )$ from the last section.

We want to show that $\Phi ^{hol}$ and $\Phi ^{\mu ,\mathbb {C}}$ correspond to each other under $\beta _{\mu }$. Both have actually image in ${{\,\mathrm{Conv}\,}}_{R_0} \subset \mathbb {C}((q))$. Thus we can check the agreement of $\Phi ^{hol}\beta _{\mu }$ with $\Phi ^{\mu ,\mathbb {C}}$ after postcomposing these two maps with ${{\,\mathrm{ev}\,}}_{q_0}:{{\,\mathrm{Conv}\,}}_{R_0} \rightarrow \mathbb {C}$ for infinitely many $q_0 \in {\mathbb {D}} {\setminus } \{0\}$.

To that purpose, choose $h \in {{\,\mathrm{Nat}\,}}^{R_0}_k({{\,\mathrm{Ell}\,}}^1_{\Gamma _{\mu }(n)}(-), \Gamma (-))$ and $\tau _0\in \mathbb {H}$ with $e^{2\pi i \tau _0} = q_0$. The exponential defines an isomorphism

$$\begin{aligned} \left( \mathbb {C}/(\mathbb {Z}+\tau _0\mathbb {Z}), dz, \zeta _n\mapsto \frac{1}{n}\right) \cong \left( \mathbb {C}^\times /q_0^\mathbb {Z}, \frac{du}{u}, \iota ^{can}\right) , \end{aligned}$$

of elliptic curves with invariant differential and arithmetic level structure, where $\iota ^{can}$ denotes the composition $\mu _n(\mathbb {C}) \rightarrow \mathbb {C}^\times \rightarrow \mathbb {C}^\times /q_0^\mathbb {Z}$. This implies

$$\begin{aligned} {{\,\mathrm{ev}\,}}_{q_0}\Phi ^{hol}\beta _{\mu }(h) = h\left( \mathbb {C}^\times /q_0^\mathbb {Z}, \frac{du}{u},\iota ^{can}\right) . \end{aligned}$$

On the other hand, ${{\,\mathrm{ev}\,}}_{q_0}\Phi ^{\mu ,\mathbb {C}}(h)$ is by definition the evaluation of h at

$$\begin{aligned} ({{\,\mathrm{ev}\,}}_{q_0}^*{{\,\mathrm{Tate}\,}}(q), {{\,\mathrm{ev}\,}}_{q_0}^*\eta ^{can}, {{\,\mathrm{ev}\,}}_{q_0}^*\iota ) \end{aligned}$$

and we have seen in the last section that this triple is isomorphic to $(\mathbb {C}^\times /q_0^\mathbb {Z}, \frac{du}{u}, \iota ^{can})$. Thus, the following triangle commutes indeed:

We obtain the q-expansion morphism

$$\begin{aligned} \Phi ^{1,R_0}:{{\,\mathrm{Nat}\,}}^{R_0}_k({{\,\mathrm{Ell}\,}}^1_{\Gamma _1(n)}(-), \Gamma (-)) \rightarrow {{\,\mathrm{Conv}\,}}_{R_0} \end{aligned}$$

as the composition $\Phi ^{\mu ,R_0}\alpha (\varphi ^*)^{-1}\alpha ^{-1}$, where $\varphi$ is as in Sect. A.2.2.

Lemma A.18

Assume that $R_0 \subset \mathbb {C}$ and let $q_0\ne 0$ be a point in the open unit disk. Evaluating at $q_0$ yields a morphism ${{\,\mathrm{ev}\,}}_{q_0}:{{\,\mathrm{Conv}\,}}_{R_0} \rightarrow \mathbb {C}$. Then

$$\begin{aligned} {{\,\mathrm{ev}\,}}_{q_0}\Phi ^{1,R_0}(h) = h\left( \mathbb {C}^\times /q_0^{n\mathbb {Z}}, \frac{du}{u}, q_0\right) \end{aligned}$$

for every $h \in {{\,\mathrm{Nat}\,}}^{R_0}_k({{\,\mathrm{Ell}\,}}^1_{\Gamma _1(n)}(-), \Gamma (-))$ and thus $\Phi ^{1,R_0}(h)$ is the Taylor expansion of

$$\begin{aligned} q \mapsto h\left( \mathbb {C}^\times /q^{n\mathbb {Z}}, \frac{du}{u}, q\right) \end{aligned}$$

at 0.

Proof

It suffices to show that

$$\begin{aligned} \varphi \left( \mathbb {C}^\times /q_0^{n\mathbb {Z}}, \frac{du}{u}, q_0\right) = \left( \mathbb {C}^\times /q_0^\mathbb {Z}, \frac{du}{u}, \iota ^{can}\right) . \end{aligned}$$

This follows from Example A.14. $\square$

Note that these discussions show that $\beta _1$ and $\beta _{\mu }$ actually have target in the ring ${{\,\mathrm{MF}\,}}(\Gamma _1(n); R_0)$, i.e. that the classical q-expansion of $\beta _1$ of a modular form over $R_0$ actually has coefficients in $R_0$ and similarly for $\beta _{\mu }$.

Theorem A.19

(q-expansion principle) Let $R_0$ be a subring of $\mathbb {C}$. The morphisms

$$\begin{aligned} \beta _{\mu }:{{\,\mathrm{Nat}\,}}^{R_0}_k({{\,\mathrm{Ell}\,}}^1_{\Gamma _{\mu }(n)}(-), \Gamma (-)) \rightarrow {{\,\mathrm{MF}\,}}(\Gamma _1(n); R_0) \end{aligned}$$

and

$$\begin{aligned} \beta _1:{{\,\mathrm{Nat}\,}}^{R_0}_k({{\,\mathrm{Ell}\,}}^1_{\Gamma _1(n)}(-), \Gamma (-)) \rightarrow {{\,\mathrm{MF}\,}}(\Gamma _1(n); R_0) \end{aligned}$$

are isomorphisms. In other words: If the coefficients of the q-expansion of a complex modular form are in $R_0$, it is actually already defined over $R_0$.

Proof

By the considerations above, it suffices to show the first statement. For $R_0 = \mathbb {C}$, this was discussed in Sect. A.2.3. The general case follows by the q-expansion principle as stated in [26, Theorem 12.3.4]. $\square$

1.5 Summary

Let R be any $\mathbb {Z}[\frac{1}{n}]$-algebra. We can define holomorphic modular forms for $\Gamma _1(n)$ of weight k over R as $H^0(\overline{\mathcal {M}}_1(n)_R; \underline{\omega }^{{{\,\mathrm{\otimes }\,}}k})$ and meromorphic modular forms as $H^0(\mathcal {M}_1(n)_R;\underline{\omega }^{{{\,\mathrm{\otimes }\,}}k})$. We have a morphism ${{\,\mathrm{Spec}\,}}\mathbb {C}\rightarrow \mathcal {M}_1(n)$ classifying the elliptic curve $\mathbb {C}/\mathbb {Z}+n\tau \mathbb {Z}$ with chosen point $\tau$ of order n. Pulling $f \in H^0(\mathcal {M}_1(n);\underline{\omega }^{{{\,\mathrm{\otimes }\,}}k})$ back to ${{\,\mathrm{Spec}\,}}\mathbb {C}$ and using the trivialization $\underline{\omega }^{{{\,\mathrm{\otimes }\,}}k}$ induced by the choice of differential dz, defines a holomorphic function of $\tau \in {\mathbb {H}}$ that is a meromorphic modular form for $\Gamma _1(n)$ in the classical sense. This defines isomorphisms

$$\begin{aligned} \beta _1:H^0(\mathcal {M}_1(n)_{\mathbb {C}}; \underline{\omega }^{{{\,\mathrm{\otimes }\,}}k}) \rightarrow {{\,\mathrm{MF}\,}}_k(\Gamma _1(n);\mathbb {C}) \end{aligned}$$

and

$$\begin{aligned} \beta _1:H^0(\overline{\mathcal {M}}_1(n)_{\mathbb {C}}; \underline{\omega }^{{{\,\mathrm{\otimes }\,}}k}) \rightarrow {{\,\mathrm{mf}\,}}_k(\Gamma _1(n);\mathbb {C}). \end{aligned}$$

The q-expansion of $\beta _1(f)$ lies in $R\subset \mathbb {C}$ if and only if f is in the image of the injection

$$\begin{aligned} H^0(\mathcal {M}_1(n)_R; \underline{\omega }^{{{\,\mathrm{\otimes }\,}}k}) \rightarrow H^0(\mathcal {M}_1(n)_{\mathbb {C}}; \underline{\omega }^{{{\,\mathrm{\otimes }\,}}k}). \end{aligned}$$

As a last point, we consider the ${\mathbb {G}}_m$-torsor $\overline{\mathcal {M}}_1^1(n) \rightarrow \overline{\mathcal {M}}_1(n)$ that is the relative ${{\,\mathrm{Spec}\,}}$ of the quasi-coherent algebra $\bigoplus _{k\in \mathbb {Z}} \underline{\omega }^{{{\,\mathrm{\otimes }\,}}k}$. By construction,

$$\begin{aligned} H^0(\overline{\mathcal {M}}_1^1(n)_{R};\mathcal {O}_{\overline{\mathcal {M}}_1^1(n)}) \cong \bigoplus _{k\in \mathbb {Z}}H^0(\overline{\mathcal {M}}_1(n)_{R}; \underline{\omega }^{{{\,\mathrm{\otimes }\,}}k}) \cong \bigoplus _{k\in \mathbb {Z}} {{\,\mathrm{mf}\,}}_k(\Gamma _1(n); R). \end{aligned}$$

Appendix B: The invertible summand in ${{\,\mathrm{Tmf}\,}}_0(7)$ (joint with Martin Olbermann)

We recall from Theorem 8.6 that ${{\,\mathrm{Tmf}\,}}_0(7)_{(3)}$ splits as a ${{\,\mathrm{Tmf}\,}}$-module as

$$\begin{aligned} {{\,\mathrm{Tmf}\,}}_{(3)} \oplus \Sigma ^4{{\,\mathrm{Tmf}\,}}_1(2)_{(3)} \oplus \Sigma ^8{{\,\mathrm{Tmf}\,}}_1(2)_{(3)}\oplus L, \end{aligned}$$

where $L \in {{\,\mathrm{Pic}\,}}({{\,\mathrm{Tmf}\,}}_{(3)})$. The goal of this appendix is to determine L. The necessary computations of $\pi _*{{\,\mathrm{Tmf}\,}}_0(7)$ were obtained by Martin Olbermann. It turns out that for the purposes of this article, we need only a small part of these computations, which the authors of the main part of this article extracted from Olbermann’s computations.

We recall from [13] that ${{\,\mathrm{Pic}\,}}({{\,\mathrm{Tmf}\,}}_{(3)}) \cong \mathbb {Z}\oplus \mathbb {Z}/3$. More precisely, their computation shows that the morphisms

$$\begin{aligned} \mathbb {Z}/72 \rightarrow {{\,\mathrm{Pic}\,}}({{\,\mathrm{TMF}\,}}_{(3)}), \quad [k] \mapsto \Sigma ^k{{\,\mathrm{TMF}\,}}\end{aligned}$$

and

$$\begin{aligned} \mathbb {Z}\rightarrow {{\,\mathrm{Pic}\,}}({{\,\mathrm{Tmf}\,}}_{\mathbb {Q}}),\quad k \mapsto \Sigma ^k {{\,\mathrm{Tmf}\,}}_{\mathbb {Q}} \end{aligned}$$

are isomorphisms and moreover that

$$\begin{aligned} {{\,\mathrm{Pic}\,}}({{\,\mathrm{Tmf}\,}}_{(3)}) \rightarrow \mathbb {Z}/72 \times _{\mathbb {Z}/24} \mathbb {Z}\subset {{\,\mathrm{Pic}\,}}({{\,\mathrm{TMF}\,}}_{(3)}) \times {{\,\mathrm{Pic}\,}}({{\,\mathrm{Tmf}\,}}_{\mathbb {Q}}) \end{aligned}$$

is an isomorphism as well. (While this last fact is not explicitly stated in [13], it is clearly visible in the proof of their Theorem B.)

As described in Theorem 8.6, we obtain L as the global sections of an invertible $\mathcal {O}^{top}$-module ${\mathcal {L}}$ on $\overline{\mathcal {M}}_{ell,(3)}$ with $\pi _0{\mathcal {L}} \cong \underline{\omega }^{{{\,\mathrm{\otimes }\,}}(-6)}$. As $\pi _0\Sigma ^{12}\mathcal {O}^{top} \cong \underline{\omega }^{{{\,\mathrm{\otimes }\,}}(-6)}$ as well and global sections define an equivalence

$$\begin{aligned} \Gamma :\mathrm {QCoh}(\overline{\mathcal {M}}_{ell, (3)},\mathcal {O}^{top}) \rightarrow {{\,\mathrm{Tmf}\,}}_{(3)}\mathrm {-mod} \end{aligned}$$

of $\infty$-categories [14], we see that the image of L in ${{\,\mathrm{Pic}\,}}({{\,\mathrm{Tmf}\,}}_{\mathbb {Q}})$ is $\Sigma ^{12}{{\,\mathrm{Tmf}\,}}_{\mathbb {Q}}$. We will show:

Proposition B.1

The image $L[\Delta ^{-1}]$ of L in ${{\,\mathrm{Pic}\,}}({{\,\mathrm{TMF}\,}}_{(3)})$ is $\Sigma ^{36}{{\,\mathrm{TMF}\,}}_{(3)}$ and hence $L \simeq \Sigma ^{36}\Gamma ({\mathcal {J}}^{{{\,\mathrm{\otimes }\,}}(-1)})$ in the notation from [13, Construction 8.4.2].

In the following, we will leave the localization at 3 for the moduli stacks, rings of modular forms and variants of ${{\,\mathrm{TMF}\,}}$ implicit to avoid clutter in the notation. We already know from the discussion above that $L[\Delta ^{-1}] \simeq \Sigma ^k{{\,\mathrm{TMF}\,}}$ for $k=12, 36$ or 60. Moreover, the descent spectral sequence for $L[\Delta ^{-1}]$ embeds into that of ${{\,\mathrm{TMF}\,}}_0(7)$ as a summand. Recall that the latter has $E_2$-term $H^*(\mathcal {M}_0(7);\underline{\omega }^{{{\,\mathrm{\otimes }\,}}*})$. Since $\mathcal {M}_0(7)$ has the $(\mathbb {Z}/7)^\times$-Galois cover $\mathcal {M}_1(7)$, we use the definition of Čech cohomology to identify this $E_2$-term with $H^*((\mathbb {Z}/7)^\times ; {{\,\mathrm{MF}\,}}_1(7))$, where ${{\,\mathrm{MF}\,}}_1(7)$ is used as our abbreviation for ${{\,\mathrm{MF}\,}}(\Gamma _1(7);\mathbb {Z}[\frac{1}{7}])$. Actually, as

$$\begin{aligned} \mathcal {O}^{top}(\mathcal {M}_1(7)^{\times _{\mathcal {M}_0(7)}k}) \simeq \prod _{((\mathbb {Z}/7)^{\times })^{\times k}}{{\,\mathrm{TMF}\,}}_1(7) \end{aligned}$$

and $\mathcal {M}_1(7)$ is affine, we obtain even an identification of the cosimplicial objects defining the descent spectral sequence for ${{\,\mathrm{TMF}\,}}_0(7)$ and the homotopy fixed point spectral sequence for ${{\,\mathrm{TMF}\,}}_1(7)^{h(\mathbb {Z}/7)^{\times }} \simeq {{\,\mathrm{TMF}\,}}_0(7)$, and hence an isomorphism of these spectral sequences.

Moreover, the descent spectral sequence for $L[\Delta ^{-1}]$ has $E_2$-term isomorphic to $H^0(\mathcal {M}_{ell}; \underline{\omega }^{{{\,\mathrm{\otimes }\,}}(* - 6)})$. Under these identifications, the embedding of descent spectral sequences sends $1 \in H^0(\mathcal {M}_{ell}; \underline{\omega }^{{{\,\mathrm{\otimes }\,}}(6-6)})$ to $\sigma _3^2 \in H^0((\mathbb {Z}/7)^\times ; {{\,\mathrm{MF}\,}}_1(7)_6)$. This follows after identification of source and target with the primitive elements in the $({\widetilde{A}},{\widetilde{\Gamma }})$ comodules ${\widetilde{A}}$ and $S_{{\widetilde{A}}}$ from Proposition 7.6. As $d_5(\Delta ) = \alpha \beta ^2$ and $d_5(\Delta ^2) = -\Delta \alpha \beta ^2$, while $d_5(1) = 0$ in the descent spectral sequence for ${{\,\mathrm{TMF}\,}}$ itself [46], it suffices to show the following lemma.

Lemma B.2

In the descent spectral sequence for ${{\,\mathrm{TMF}\,}}_0(7)$, the class $\Delta \sigma _3^2 \in H^0(\mathcal {M}_0(7); \underline{\omega }^{{{\,\mathrm{\otimes }\,}}(-6)})$ has a trivial $d_5$-differential.

Proof

Our first tool is the map of descent spectral sequences from that for TMF to that for $TMF_0(7)$, which on the 0-line of the $E_2$-term is a map

$$\begin{aligned} H^0(\mathcal {M}_{ell}; \underline{\omega }^{\otimes *}) \rightarrow H^0(\mathcal {M}_0(7); \underline{\omega }^{\otimes *}). \end{aligned}$$

(B.3)

Recall from above that $H^0(\mathcal {M}_0(7); \underline{\omega }^{\otimes *}) \cong H^0((\mathbb {Z}/7)^{\times }; {{\,\mathrm{MF}\,}}_1(7))$. Proposition 2.7 implies that we can express every element in the invariants as a polynomial in $\sigma _1, \sigma _3$ and $p=z_1^2z_2+z_2^2z_3+z_3^2z_1$. As every element in $H^0(\mathcal {M}_{ell}; \underline{\omega }^{\otimes *})$ is a polynomial in the $a_i$, Proposition 3.5 and Theorem 3.13 give us a concrete way to calculate the map (B.3). In particular, we obtain

$$\begin{aligned} \Delta \mapsto -\sigma _3^3p-8\sigma _3^4. \end{aligned}$$

By the splitting from Theorem 1.5 and the known differentials from the descent spectral sequence of ${{\,\mathrm{TMF}\,}}$, we see that there are no differentials shorter than a $d_5$ in the descent spectral sequence for ${{\,\mathrm{TMF}\,}}_0(7)$. In particular, we obtain that $\Delta$ is a $d_i$-cycle for $i<5$ in the descent spectral sequence for ${{\,\mathrm{TMF}\,}}_0(7)$, but $d_5(\Delta ) = \alpha \beta ^2$ (where we use the same notation for the images of $\Delta$ and $\alpha \beta ^2$ in the descent spectral sequence for ${{\,\mathrm{TMF}\,}}_0(7)$ as in that for ${{\,\mathrm{TMF}\,}}$).

Our second tool is the transfer

$$\begin{aligned} {{\,\mathrm{Tr}\,}}:{{\,\mathrm{MF}\,}}_1(7) = H^0(\{e\}; {{\,\mathrm{MF}\,}}_1(7)) \rightarrow H^0((\mathbb {Z}/7)^\times ; {{\,\mathrm{MF}\,}}_1(7)), \quad x \mapsto \sum _{g \in (\mathbb {Z}/7)^\times } gx. \end{aligned}$$

We have ${{\,\mathrm{Tr}\,}}(x)y = {{\,\mathrm{Tr}\,}}(x {{\,\mathrm{res}\,}}(y)) = 0$ for all $x \in {{\,\mathrm{MF}\,}}_1(7)$ and $y \in H^*((\mathbb {Z}/7)^\times , {{\,\mathrm{MF}\,}}_1(7))$ with $*> 0$ by [58, Formula V.3.8]. In particular, these elements act trivially on $H^*((\mathbb {Z}/7)^\times , {{\,\mathrm{MF}\,}}_1(7))$ for $*> 0$. As 3 is in the image of ${{\,\mathrm{Tr}\,}}$, we see in particular that $H^*((\mathbb {Z}/7)^\times , {{\,\mathrm{MF}\,}}_1(7))$ for $*> 0$ is 3-torsion.

Moreover we claim that all elements in the image of the transfer ${{\,\mathrm{Tr}\,}}$ are permanent cycles in the homotopy fixed point spectral sequence (or, equivalently, the descent spectral sequence) converging to $\pi _*{{\,\mathrm{TMF}\,}}_0(7) = \pi _*{{\,\mathrm{TMF}\,}}_1(7)^{h(\mathbb {Z}/7)^\times }$. Indeed: Consider the $(\mathbb {Z}/7)^\times$-equivariant map $a:(\mathbb {Z}/7)^{\times }_+ \wedge {{\,\mathrm{TMF}\,}}_1(7) \rightarrow {{\,\mathrm{TMF}\,}}_1(7)$ induced by ${{\,\mathrm{id}\,}}_{{{\,\mathrm{TMF}\,}}_1(7)}$, where the action on the source is only on $(\mathbb {Z}/7)^{\times }_+$. On homotopy groups, this induces the map $(x_g)_{g\in (\mathbb {Z}/7)^{\times }} \mapsto \sum _{g\in (\mathbb {Z}/7)^{\times }} gx_g$. Thus, the map that a induces on homotopy fixed point spectral sequences agrees in the 0-line exactly with the transfer ${{\,\mathrm{Tr}\,}}$ under the identification ${{\,\mathrm{MF}\,}}_1(7) \cong H^0((\mathbb {Z}/7)^{\times }; \bigoplus _{(\mathbb {Z}/7)^{\times }} \pi _*{{\,\mathrm{TMF}\,}}_1(7))$. As the homotopy fixed point spectral sequence for $(\mathbb {Z}/7)^{\times }_+\wedge {{\,\mathrm{TMF}\,}}_1(7)$ is concentrated in the 0-line, this implies that every element in the image of ${{\,\mathrm{Tr}\,}}$ is a permanent cycle. In particular, $\sigma _3p = {{\,\mathrm{Tr}\,}}(\frac{z_1^3z_2^2z_3}{2})$ implies that $d_5(\sigma _3p) = 0$.

Taken together, these tools imply the following computation:

$$\begin{aligned} \alpha \beta ^2&= d_5(\Delta ) \\&= d_5(-\sigma _3^3p-8\sigma _3^4) \\&= -\sigma _3^2d_5(\sigma _3p+8\sigma _3^2)-d_5(\sigma _3^2)(\sigma _3p+8\sigma _3^2)\\&= -8\sigma _3^2d_5(\sigma _3^2)-8\sigma _3^2d_5(\sigma _3^2)\\&= -\sigma _3^2d_5(\sigma _3^2). \end{aligned}$$

In total, we obtain $\sigma _3^{2}d_5(\sigma _3^2) = -\alpha \beta ^2$ and deduce

$$\begin{aligned} d_5(\Delta \sigma _3^2)&= \alpha \beta ^2 \sigma _3^2 +\Delta d_5(\sigma _3^2) \\&= \alpha \beta ^2 \sigma _3^2-8\sigma _3^4d_5(\sigma _3^2) \\&= 9\alpha \beta ^2 \sigma _3^2=0. \end{aligned}$$

$\square$