Lecture 15: Modular curves over Z

$$ \DeclareMathOperator{\Frac}{Frac} \DeclareMathOperator{\Proj}{Proj} \newcommand{\sm}{\mathrm{sm}} \newcommand{\GL}{\mathrm{GL}} \newcommand{\bZ}{\mathbf{Z}} \newcommand{\fM}{\mathfrak{M}} \newcommand{\bA}{\mathbf{A}} \newcommand{\fh}{\mathfrak{h}} \newcommand{\bP}{\mathbf{P}} \newcommand{\bQ}{\mathbf{Q}} \newcommand{\bG}{\mathbf{G}} \let\ol\overline \newcommand{\bF}{\mathbf{F}} \newcommand{\bC}{\mathbf{C}} \newcommand{\cA}{\mathcal{A}} \newcommand{\cO}{\mathcal{O}} \newcommand{\id}{\mathrm{id}} $$

This lecture covers three topics: the coarse moduli space, compactifying modular curves via generalized elliptic curves, and defining modular curves over all over $\bZ$. I also briefly discuss the fibers in bad characteristic.

Coarse spaces

Let $\fM_0(N)$ be the stack over $\bZ[1/N]$ assigning to a scheme $S$ the groupoid of pairs $(E, G)$ where $E/S$ is an elliptic curve and $G \subset S$ is a cyclic subgroup of order $N$. We saw last time that $\fM_0(N)$ is a Deligne--Mumford stack. (I called it $Y_0(N)$ last time, but it will be better to not use that notation from now on.) Recall that this means it has an étale cover by a scheme. Explicitly, suppose $p \nmid N$ is a prime, and let $Y(S)$ be the set of isomorphism classes of data $(E,G,(P,Q))$, where $(E,G)$ is as above and $(P,Q)$ is a $\Gamma(p)$-structure on $E$. Then $Y$ is a representable by a scheme over $\bZ[1/pN]$, and this provides an étale cover of $\fM_0(N)$ over $\bZ[1/pN]$. In fact, $\fM_0(N)[1/p]$ is the quotient stack $[Y/\GL_2(\bZ/p\bZ)]$.

One can also consider the quotient scheme $Y/\GL_2(\bZ/p\bZ)$. It is not hard to show that this represents the sheafification of the presheaf $S \mapsto |\fM_0(N)(S)|$ on $\bZ[1/pN]$-schemes, where $|\cdot|$ denotes the set of isomorphism classes. Using different values of $p$ and patching, we see that $S \mapsto |\fM_0(N)(S)$ is represented by a $\bZ[1/N]$-scheme, which we denote by either $M_0(N)$ or $Y_0(N)$. This is called the <em>coarse space</em> of $\fM_0(N)$, and is the universal scheme to which $\fM_0(N)$ maps. When $N=1$ then $Y_0(N)$ is the familiar $j$-line $\cong \bA^1$. Furthermore, the set of complex points of $Y_0(N)$ is identified with $\fh/\Gamma_0(N)$. It is affine and smooth over $\bZ[1/N]$, as can be seen from its description as $Y/\GL_2(\bZ/p\bZ)$ when some prime $p \nmid N$ is inverted.


Level 1

Recall that, over the complex numbers, $Y(1)=\fh/\Gamma(1)$ is not compact, and that we compactified it by adding cusps; precisely, the compactification $X(1)$ was defined as $\fh^*/\Gamma(1)$, where $\fh^*$ is the union of $\fh$ and $\bP^1(\bQ)$. The group $\Gamma(1)$ acts transitively on $\bP^1(\bQ)$, and so $X(1)$ has a unique cusp.

We would like to give a moduli-theoretic meaning to the cusp. The valuative crtierion for properness suggests that this should be related to the problem of extending elliptic curves over DVR's, which is something we know a lot about. In particular, the semi-stable reduction theorem tells us that, after a possible base change, every elliptic curve over $K=\Frac(A)$ ($A$ a DVR) extends to either an elliptic curve over $A$ or a nodal cubic over $A$. This suggests that the cusp should correspond to a nodal cubic.

We now give some definitions. Let $n \ge 1$ be an integer and let $k$ be a field. The <em>standard $n$-gon</em> over $k$, denoted $C_n$, is the quotient of $(\bP^1)_k \times \bZ/n\bZ$ where $(\infty, i)$ is identified with $(0, i+1)$. When $n=1$, this gives the nodal cubic; in general, it has $n$ irreducible components. Note that the smooth locus $C_n^{\sm}$ of $C_n$ is just $\bG_m \times \bZ/n\bZ$, and is a group. Furthermore, the action of $C_n^{\sm}$ on itself extends to an action of $C_n^{\sm}$ on all of $C_n$: the $\bG_m$ part fixes the singular points.

Note that $C_n^{\sm}[n]$ has order $n^2$. In fact, there is a natural short exact sequence

$$ 0 \to \mu_n \to C_n^{\sm}[n] \to \bZ/n\bZ \to 0, $$

where the $\mu_n$ sits in the identity component of $C_n^{\sm}$.

A generalized elliptic curve over a base scheme $S$ is, roughly, a curve over $S$ whose fibers are either elliptic curves or $n$-gons. More precisely, it is a tuple $(E,+,e)$, where $E/S$ is a proper flat curve, $e \in E(S)$, and $+$ is a map $E^{\sm} \times E \to E$ such that: (1) $+$ (with $e$) gives $E^{\sm}$ the structure of a group and defines aan action on $E$; (2) the geometric fibers of $E$ are elliptic curves or $n$-gons.

Define $\ol{\fM}(1)(S)$ to be the groupoid of generalized elliptic curves $E/S$ such that the fibers are either elliptic curves or 1-gons. Then one has the following result:

Theorem. $\ol{\fM}(1)$ is a proper smooth Deligne--Mumford stack over $\bZ$.

This can be proved similarly to how we handled the $\fM(1)$ case. Note that in the valuative criterion for properness for stacks, one is allowed to make an extension of the DVR. Thus the properness in this theorem corresponds exactly to the semi-stable reduction theorem.

Higher level

Let $E/S$ be a generalized elliptic curve. A $\Gamma_0(N)$-structure on $E$ is a cyclic subgroup $G$ of order $N$ inside of $E^{\sm}$. The definition of $\ol{\fM}_0(N)$ should clearly be related to $\Gamma_0(N)$-structures on generalized elliptic curves, but there is some subtlety.

To explain this, we should first familiarize ourselves more with the cusps on $X_0(N)$. For simplicity, let's suppose that $N$ is prime. The set of cusps if $\bP^1(\bQ)/\Gamma_0(N)$, which is easily seen to be isomorphic to $\bP^1(\bF_N)/G$, where $G \subset \GL_2(\bZ/N\bZ)$ is the group of upper triangular matrices. The group $G$ has two orbits on $\bP^1(\bF_n)$, namely, that of 0 (which is all of $\bA^1(\bF_N)$) and that of $\infty$ (which is fixed by $G$). Thus $X_0(N)$ has two cusps, which are denoted 0 and $\infty$, as they are the images of $0,\infty \in \bP^1(\bQ)$.

Now, the 1-gon only admits one $\Gamma_0(N)$-structure, namely $\mu_N \subset \bG_m$. Thus if we only considered 1-gons with $\Gamma(N)$-structure we would only by adding 1 point to $\fM_0(N)(\bC)$. This is insufficient since there are two cusps.

We can get more $\Gamma_0(N)$-structures by allowing $N$-gons. Up to isomorphism, it has two $\Gamma_0(N)$ structures, namely, the subgroups $\mu_N$ and $\bZ/N\bZ$. This therefore looks like the right thing to use. However, for technical reasons (that we'll soon see), we need our level structure to meet all the irreducible components. We therefore take $(C_1, \mu_N)$ and $(C_N, \bZ/N\bZ)$ to be the cusps, which we call 0 and $\infty$. Note that 0 is distinguished from $\infty$ by the fact that its $\Gamma_0(N)$-structure lives in the identity component.

Here is the definition of the moduli problem, for any $N$. We let $\ol{\fM}_0(N)(S)$ be the groupoid of pairs $(E, G)$, where $E/S$ is a generalized elliptic curve and $G \subset E^{\sm}$ is a cyclic subgroup of order $N$ such that in each fiber, $G$ meets each irreducible component of $E$. We then have the following result:

Theorem. $\ol{\fM}_0(N)$ is a proper smooth Deligne--Mumford stack over $\bZ[1/N]$.

Maps between moduli spaces

Over the complex numbers, $X_0(N)$ is identified with $\fh^*/\Gamma_0(N)$. Thus if $N' \mid N$ then the inclusion $\Gamma_0(N) \subset \Gamma_0(N')$ induces a map $X_0(N) \to X_0(N')$. How can we see this in terms of the moduli problem?

For $\fM_0(N)$ it is not hard: given $(E, G)$ with $G$ a cyclic subgroup of order $N$, there is a unique subgroup $H \subset G$ of order $N'$. The map $\fM_0(N) \to \fM_0(N')$ takes $(E, G)$ to $(E, H)$.

The map is more subtle over the compactified space, since the objects are different: $\fM_0(N)$ includes $N$-gons while $\fM_0(N')$ does not. The construction is as follows. Let $(E,G)$ be a generalized elliptic curve with $\Gamma_0(N)$-structure. Again, there is a unique $H \subset G$ which is cyclic of order $N'$. We now contract the components of $E$ which do not meet $H$. This can be done canonically as follows. Let $f \colon E \to S$ be the structure map. Let $\cA$ be the sheaf of graded rings on $S$ given by $\bigoplus_{n=0}^{\infty} f_*(\cO_E(nH))$. Then the contraction is $\Proj(\cA)$.

Working over Z

A problem

We would now like to study moduli problems over $\bZ$, as opposed to $\bZ[1/N]$. Of course, this means that we'll have to include the case of elliptic curves in characteristic $N$.  This causes problems with the definitions of level structure due to the lack of points.

More precisely, suppose we want to study $\fM(N)$. Our definition of a $\Gamma(N)$-structure on an elliptic curve $E$ is a pair $(P,Q)$ of $N$-torsion points which form a basis for $E[N]$. However, if $E/\bF_N$ is supersingular, then there are no non-zero $N$-torsion points, and so therefore no $\Gamma(N)$-structures (under this definition)! This will clearly prevent the moduli space from being proper.

This problem can be solved by using the notion of a Drinfeld level structure. Fortunately, we will not need to go down this road.

The case of Γ_0(N)

We are primarily interested in the case of $\Gamma_0(N)$-structures, with $N$ squarefree, where a simpler definition can be given over $\bZ$. Namely, if $E/S$ is a generalized elliptic curve, a $\Gamma_0(N)$-structure on $E$ is a closed subgroup $G \subset E$ which is finite and flat over $S$ of order $N$. We let $\ol{\fM}_0(N)(S)$ be the groupoid of pairs $(E,G)$ as above. We have the following result:

Proposition. $\ol{\fM}_0(N)$ is a flat Deligne--Mumford stack over $\bZ$.

Remark. ne can define $\ol{\fM}_0(N)$ for any $N$. When $N$ is not squarefree, however, one needs an appropriate definition of cyclic. In general, the non-compactified space $\fM_0(N)$ is a Deligne--Mumford stack. However, if $p^2 \mid N$ then $\mu_p$ can appear in the automorphism group of a generalized elliptic curve with $\Gamma_0(N)$-structure, which implies that $\ol{\fM}_0(N)$ cannot be a Deligne--Mumford stack; it is an Artin stack, however.

The fiber in bad characteristic

Let us now consider the space $\ol{\fM}_0(N)_{\bF_p}$, where $p \mid N$. Suppose $k$ is an algebraically closed field of characteristic $p$ and $E/k$ is an elliptic curve. If $E$ is supersingular, then $E[p] \cong \alpha_{p^2}$ has a unique subgroup of order $p$, namely $\alpha_p$. If $E$ is ordinary, then $E[p] \cong \mu_p \times (\bZ/p\bZ)$. In this case, $E[p]$ has exactly two subgroups of order $p$, namely $\bZ/p\bZ$ and $\mu_p$. The cuspidal points admit exactly two structures as well, $\mu_p$ and $\bZ/p\bZ$.

Let $N'=N/p$. We can then define two maps $f,g \colon \ol{\fM}_0(N')_{\bF_p} \to \ol{\fM}_0(N)_{\bF_p}$, as follows. Let $E/S$ be a generalized elliptic curve, where $S$ is an $\bF_p$-scheme, with a $\Gamma_0(N')$-structure $G$. Then we have the relative Frobenius map $F \colon E \to E^{(p)}$ and Verscheibung $V \colon E^{(p)} \to E$. We define $f(E,G)=(E,G,\ker(F))$ and $g(E,G)=(E^{(p)}, V^{-1}(G))$.

We can also define two maps $f',g' \colon \ol{M}_0(N)_{\bF_p} \to \ol{M}_0(N')_{\bF_p}$, as follows. Let $E/S$ be an elliptic curve, let $G$ be a $\Gamma_0(N')$-structure on $E$, and let $H$ be a $\Gamma_0(p)$-structure on $E$. We define $f'(E,G,H)=(E,G)$ and $g'(E,G,H)=(E/H, \textrm{image of$G$in$E/H$})$.

It is clear that $f'f=\id$. It is also clear that $g'g=\id$, since $E^{(p)}/\ker(V)$ is canonically isomorphic to $E$. All these maps induce maps on the coarse spaces, and so we see:

Theorem. $M_0(N)_{\bF_p}$ is obtained by taking two copies of $M_0(N')_{\bF_p}$ and glueing the supsersingular loci (a finite set of points) by the Frobenius map.

The picture is especially nice when $N'=1$, as then $M_0(N')$ is just $\bP^1$.