Note on Fourier expansions at cusps

TL;DR

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Contribution

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Abstract

This was originally an appendix to our paper `Fourier expansions at cusps' [arXiv:1807.00391]. The purpose of this note is to give a proof of a theorem of Shimura on the action of $\mathrm{Aut}(\mathbb{C})$ on modular forms for $\Gamma(N)$ from the perspective of algebraic modular forms. As the theorem is well-known, we do not intend to publish this note but want to keep it available as a preprint.

Equations

$$f|g(\tau)=\frac{1}{(C\tau+D)^{k}}f\Bigl{(}\frac{A\tau+B}{C\tau+D}\Bigr{)},\qquad f^{\sigma}(\tau)=\sum_{n}\sigma(a_{n})e^{2\pi in\tau/w}$$

$$(f|g{)}^{\sigma }=f^{\sigma }|g_{\lambda },$$

$$\beta :({\mu }_N{)}_R\times (\mathbb{Z}/N\mathbb{Z}{)}_R\stackrel{\cong }{\to }E[N]$$

$$F=F_{R^{\prime }}:\{\text{isomorphism classes of test objects of level N over R′}\}\to R^{\prime }$$

$$M_k^{\mathrm{alg}}(\Gamma (N);R)\hookrightarrow \mathbb{Z}((q^{1/N})){\otimes }_{\mathbb{Z}}R$$

$$F^{\mathrm{an}}(\tau)=F\Bigl{(}\frac{\mathbb{C}}{2\pi i\mathbb{Z}+2\pi i\tau\mathbb{Z}},dz,\beta_{\tau}\Bigr{)}$$

$$(f|_{k}g)(\tau)=F\Bigl{(}\frac{\mathbb{C}}{2\pi i(\mathbb{Z}+\tau\mathbb{Z})},dz,\beta^{\prime}_{\tau}\Bigr{)}$$

$${\beta }_{\tau }^{\prime }({\zeta }_N^m,n)={\beta }_{\tau }({\zeta }_N^{md+nb},mc+na).$$

$${\alpha }_{\tau }^{\prime }(a,b)={\alpha }_{\tau }((a,b)g).$$

$${\psi }_R:(\mathbb{Z}/N\mathbb{Z}{)}_R^2\to ({\mu }_N{)}_R\times (\mathbb{Z}/N\mathbb{Z}{)}_R$$

$$(g\cdot \alpha )(a,b)=\alpha ((a,b)g)((a,b)\in (\mathbb{Z}/N\mathbb{Z}{)}^2).$$

$$F_R^{\sigma }:\{\text{isomorphism classes of test objects of level N over R}\}$$

$$T$$

$$g\cdot {\beta }_{\mathrm{can}}^{\sigma }\cong (g_{\lambda }\cdot {\beta }_{\mathrm{can}}{)}^{\sigma }.$$

$$(\mathbb{Z}/N\mathbb{Z}{)}_R^2$$

$${\varphi }_{\sigma }({\psi }_R(a,b))={\varphi }_{\sigma }({\zeta }_{N,R}^b,a)=({\zeta }_{N,R^{\sigma }}^{{\lambda }^{-1}b},a)={\psi }_{R^{\sigma }}(a,{\lambda }^{-1}b)$$

$$\alpha^{\sigma}(a,b)=\phi_{\sigma}\circ\alpha\circ\gamma^{-1}(a,b)=\phi_{\sigma}\circ\alpha(a,\lambda b)=\phi_{\sigma}\circ\alpha\biggl{(}(a,b)\begin{pmatrix}1&0\\ 0&\lambda\end{pmatrix}\biggr{)}.$$

$$(g\cdot\alpha^{\sigma})(a,b)=\alpha^{\sigma}((a,b)g)=\phi_{\sigma}\circ\alpha\biggl{(}(a,b)g\begin{pmatrix}1&0\\ 0&\lambda\end{pmatrix}\biggr{)}.$$

$$(g_{\lambda}\cdot\alpha)^{\sigma}(a,b)=\phi_{\sigma}\circ(g_{\lambda}\cdot\alpha)\biggl{(}(a,b)\begin{pmatrix}1&0\\ 0&\lambda\end{pmatrix}\biggr{)}=\phi_{\sigma}\circ\alpha\biggl{(}(a,b)\begin{pmatrix}1&0\\ 0&\lambda\end{pmatrix}g_{\lambda}\biggr{)}.$$

Full text

The second author was partially funded by the DFG-Forschergruppe 1920 and the LOEWE research unit “Uniformized Structures in Arithmetic and Geometry”

In this note we give a new proof of the following theorem which is originally due to Shimura, see [5, Theorem 8] and [6, Lemma 10.5]. It gives the interaction between the $\mathrm{SL}_{2}(\mathbb{Z})$-action and the $\operatorname{Aut}(\mathbb{C})$-action on spaces of modular forms on the group $\Gamma(N)$. These actions on a modular form $f(\tau)=\sum_{n}a_{n}e^{2\pi in\tau/w}$ of weight $k\geq 1$ are defined as follows:

[TABLE]

for $g=\left(\begin{smallmatrix}A&B\\ C&D\end{smallmatrix}\right)\in\mathrm{SL}_{2}(\mathbb{Z})$ and $\sigma\in\operatorname{Aut}(\mathbb{C})$. For any integer $N\geq 1$ we denote $\zeta_{N}=e^{2\pi i/N}$.

[TABLE]

where $g_{\lambda}$ is any lift in $\mathrm{SL}_{2}(\mathbb{Z})$ of the matrix $\left(\begin{smallmatrix}A&\lambda B\\ \lambda^{-1}C&D\end{smallmatrix}\right)\in\mathrm{SL}_{2}(\mathbb{Z}/N\mathbb{Z})$.

This theorem immediately implies that if a modular form $f$ of level $N$ has Fourier coefficients in a field $K_{f}$, then the Fourier coefficients of $f|\left(\begin{smallmatrix}A&B\\ C&D\end{smallmatrix}\right)$ for any $\left(\begin{smallmatrix}A&B\\ C&D\end{smallmatrix}\right)\in\mathrm{SL}_{2}(\mathbb{Z})$ will lie in $K_{f}(\zeta_{N})$. In [1] we obtain improved results if $f$ is a modular form for $\Gamma_{0}(N)$ or $\Gamma_{1}(N)$ and in the case of newforms on $\Gamma_{0}(N)$, we determine the number field generated by the Fourier coefficients of $f|\left(\begin{smallmatrix}A&B\\ C&D\end{smallmatrix}\right)$ explicitly.

We recall the theory of algebraic modular forms, in order to give a new proof of Theorem 1. For more details on this theory, see [3, Chap. II] and the references therein.

[TABLE]

satisfying $e_{N}(\beta(\zeta,0),\beta(1,1))=\zeta$ for every $\zeta\in(\mu_{N})_{R}$. Here $\mu_{N}=\operatorname{Spec}\mathbb{Z}[t]/(t^{N}-1)$ is the scheme of $N$-th roots of unity, and $e_{N}$ is the Weil pairing on $E[N]$ 111Our definition of the Weil pairing is the reciprocal of Silverman’s definition [7, III.8]. With our definition, we have $e_{N}(1/N,\tau/N)=e^{2\pi i/N}$ on the elliptic curve $\mathbb{C}/(\mathbb{Z}+\tau\mathbb{Z})$ with $\mathrm{Im}(\tau)>0$..

If $\phi:R\to R^{\prime}$ is a ring morphism, we denote by $T_{R^{\prime}}=(E_{R^{\prime}},\omega_{R^{\prime}},\beta_{R^{\prime}})$ the base change of $T$ to $R^{\prime}$ along $\phi$.

The isomorphism classes of test objects over $\mathbb{C}$ are in bijection with the set of lattices $L$ in $\mathbb{C}$ endowed with a symplectic basis of $\frac{1}{N}L/L$ [3, 2.4]. Another example is given by the Tate curve $\mathrm{Tate}(q)=\mathbb{G}_{m}/q^{\mathbb{Z}}$ [2, §8]. It is an elliptic curve over $\mathbb{Z}((q))$ endowed with the canonical differential $\omega_{\mathrm{can}}=dx/x$ and the level $N$ structure $\beta_{\mathrm{can}}(\zeta,n)=\zeta q^{n/N}\text{ mod}\,{q^{\mathbb{Z}}}$. The test object $(\mathrm{Tate}(q),\omega_{\mathrm{can}},\beta_{\mathrm{can}})$ is defined over $\mathbb{Z}((q^{1/N}))$.

[TABLE]

satisfying the following properties:

1. (1)

$F(E,\lambda^{-1}\omega,\beta)=\lambda^{k}F(E,\omega,\beta)$ for every $\lambda\in(R^{\prime})^{\times}$; 2. (2)

$F$ is compatible with base change: for every morphism of $R$-algebras $\psi:R^{\prime}\to R^{\prime\prime}$ and for every test object $T$ of level $N$ over $R^{\prime}$, we have $F_{R^{\prime\prime}}(T_{R^{\prime\prime}})=\psi(F_{R^{\prime}}(T))$.

We denote by $M_{k}^{\mathrm{alg}}(\Gamma(N);R)$ the $R$-module of algebraic modular forms of weight $k$ and level $N$ over $R$.

Evaluating at the Tate curve provides an injective $R$-linear map

[TABLE]

called the $q$-expansion map. The $q$-expansion principle states that if $R^{\prime}$ is a subring of $R$, then an algebraic modular form $F\in M_{k}^{\mathrm{alg}}(\Gamma(N);R)$ belongs to $M_{k}^{\mathrm{alg}}(\Gamma(N);R^{\prime})$ if and only if the $q$-expansion of $F$ has coefficients in $R^{\prime}$.

Algebraic modular forms are related to classical modular forms as follows. To any algebraic modular form $F\in M_{k}^{\mathrm{alg}}(\Gamma(N);\mathbb{C})$, we associate the function $F^{\mathrm{an}}:\mathcal{H}\to\mathbb{C}$ defined by

[TABLE]

with $\beta_{\tau}(\zeta_{N}^{m},n):=[2\pi i(m+n\tau)/N]$.

We now interpret the action of $\mathrm{SL}_{2}(\mathbb{Z})$ on modular forms in algebraic terms. Let $F\in M_{k}^{\mathrm{alg}}(\Gamma(N);\mathbb{C})$ with $f=F^{\mathrm{an}}$, and let $g=\left(\begin{smallmatrix}a&b\\ c&d\end{smallmatrix}\right)\in\mathrm{SL}_{2}(\mathbb{Z})$. A simple computation shows that

[TABLE]

where the level $N$ structure $\beta^{\prime}_{\tau}$ is given by

[TABLE]

Let $\psi:(\mathbb{Z}/N\mathbb{Z})^{2}\to\mu_{N}(\mathbb{C})\times\mathbb{Z}/N\mathbb{Z}$ be the isomorphism defined by $\psi(a,b)=(\zeta_{N}^{b},a)$. Let us identify the level structure $\beta_{\tau}$ (resp. $\beta^{\prime}_{\tau}$) with the map $\alpha_{\tau}=\beta_{\tau}\circ\psi$ (resp. $\alpha^{\prime}_{\tau}=\beta^{\prime}_{\tau}\circ\psi$). Then (4) shows that

[TABLE]

What we have here is the right action of $\mathrm{SL}_{2}(\mathbb{Z})$ on the row space $(\mathbb{Z}/N\mathbb{Z})^{2}$, which induces a left action on the set of level $N$ structures. As we will see, all this makes sense algebraically. For any $\mathbb{Z}[\zeta_{N}]$-algebra $R$, we denote by $\zeta_{N,R}$ the image of $\zeta_{N}=e^{2\pi i/N}$ under the structural morphism $\mathbb{Z}[\zeta_{N}]\to R$.

Let $R$ be a $\mathbb{Z}[\zeta_{N},1/N]$-algebra. We have an isomorphism of $R$-group schemes

[TABLE]

given by $\psi_{R}(a,b)=(\zeta_{N,R}^{b},a)$. The group $\mathrm{SL}_{2}(\mathbb{Z})$ acts from the right on the row space $(\mathbb{Z}/N\mathbb{Z})^{2}_{R}$ by $R$-automorphisms, and for $\alpha:(\mathbb{Z}/N\mathbb{Z})^{2}_{R}\xrightarrow{\cong}E[N]$ we define

[TABLE]

Using $\psi_{R}$, we transport this to a left action of $\mathrm{SL}_{2}(\mathbb{Z})$ on the set of level $N$ structures of an elliptic curve over $R$. Given a test object $T=(E,\omega,\beta)$ over $R$, we define $g\cdot T:=(E,\omega,g\cdot\beta)$. For any $F\in M_{k}^{\mathrm{alg}}(\Gamma(N);R)$, we define $F|g\in M_{k}^{\mathrm{alg}}(\Gamma(N);R)$ by the rule $(F|g)(T)=F(g\cdot T)$ for any test object $T$ over any $R$-algebra $R^{\prime}$. The computation (3) then shows that the right action of $\mathrm{SL}_{2}(\mathbb{Z})$ on $M_{k}^{\mathrm{alg}}(\Gamma(N);\mathbb{C})$ corresponds to the usual slash action on $M_{k}^{!}(\Gamma(N))$.

We now interpret the action of $\operatorname{Aut}(\mathbb{C})$ in algebraic terms (see [4, p. 88]). Let $\sigma\in\operatorname{Aut}(\mathbb{C})$. For any $\mathbb{C}$-algebra $R$, we define $R^{\sigma}:=R\otimes_{\mathbb{C},\sigma^{-1}}\mathbb{C}$, which means that $(ax)\otimes 1=x\otimes\sigma^{-1}(a)$ for all $a\in\mathbb{C}$, $x\in R$. We endow $R^{\sigma}$ with the structure of a $\mathbb{C}$-algebra using the map $a\in\mathbb{C}\mapsto 1\otimes a\in R^{\sigma}$. We denote by $\phi_{\sigma}:R\to R^{\sigma}$ the map defined by $\phi_{\sigma}(x)=x\otimes 1$. The map $\phi_{\sigma}$ is a ring isomorphism, but one should be careful that $\phi_{\sigma}$ is not a morphism of $\mathbb{C}$-algebras, as it is only $\sigma^{-1}$-linear. For any test object $T$ over $R$, we denote by $T^{\sigma}$ its base change to $R^{\sigma}$ using the ring morphism $\phi_{\sigma}$.

Let $F\in M_{k}^{\mathrm{alg}}(\Gamma(N);\mathbb{C})$ be an algebraic modular form. For any $\mathbb{C}$-algebra $R$, we define

[TABLE]

One may check that the collection of functions $F^{\sigma}_{R}$ satisfies the conditions (1) and (2) above, hence defines an algebraic modular form $F^{\sigma}\in M_{k}^{\mathrm{alg}}(\Gamma(N);\mathbb{C})$. Moreover, since the Tate curve is defined over $\mathbb{Z}((q))$, one may check that the map $F\mapsto F^{\sigma}$ corresponds to the usual action of $\operatorname{Aut}(\mathbb{C})$ on the Fourier expansions of modular forms: for every $F\in M_{k}^{\mathrm{alg}}(\Gamma(N);\mathbb{C})$ and every $\sigma\in\operatorname{Aut}(\mathbb{C})$, we have $(F^{\sigma})^{\mathrm{an}}=(F^{\mathrm{an}})^{\sigma}$.

We finally come to the proof of Theorem 1.

[TABLE]

For any scheme $X$ over $R$, let $X^{\sigma}$ denote its base change to $R^{\sigma}$ along $\phi_{\sigma}$. Since $\phi_{\sigma}$ is a ring isomorphism, the canonical projection map $X^{\sigma}\to X$ is an isomorphism of schemes, and we also denote by $\phi_{\sigma}:X\to X^{\sigma}$ the inverse map.

Put $E=\mathrm{Tate}(q)$ and $\beta=\beta_{\mathrm{can}}$. Let $\alpha=\beta\circ\psi_{R}:(\mathbb{Z}/N\mathbb{Z})^{2}_{R}\xrightarrow{\cong}E[N]$. By functoriality, the level structure $\beta^{\sigma}$ is given by the following commutative diagram

[TABLE]

Let us compute the dotted arrow $\gamma$. Since $\phi_{\sigma}$ is $\sigma^{-1}$-linear, we have $\phi_{\sigma}(\zeta_{N,R})=\zeta_{N,R^{\sigma}}^{\lambda^{-1}}$. It follows that

[TABLE]

so that $\gamma(a,b)=(a,\lambda^{-1}b)$. We may thus express $\alpha^{\sigma}$ in terms of $\alpha$ by

[TABLE]

Let us make explicit both sides of (7). By (6) and (10), the left hand side is given by

[TABLE]

Let us now turn to the right hand side of (7). By (6), we have $(g_{\lambda}\cdot\alpha)(a,b)=\alpha((a,b)g_{\lambda})$. Applying the commutative diagram (8) with $\alpha$ replaced by $g_{\lambda}\cdot\alpha$, we get

[TABLE]

Finally, we note that $g\left(\begin{smallmatrix}1&0\\ 0&\lambda\end{smallmatrix}\right)=\left(\begin{smallmatrix}1&0\\ 0&\lambda\end{smallmatrix}\right)g_{\lambda}$.

∎