Lecture 14: Modular curves over Q

$$ \DeclareMathOperator{\Spec}{Spec} \DeclareMathOperator{\div}{div} \newcommand{\GL}{\mathrm{GL}} \newcommand{\bC}{\mathbf{C}} \newcommand{\bZ}{\mathbf{Z}} \newcommand{\bQ}{\mathbf{Q}} \let\ol\overline \newcommand{\rH}{\mathrm{H}} \DeclareMathOperator{\Aut}{Aut} \DeclareMathOperator{\Spec}{Spec} \newcommand{\cF}{\mathcal{F}} \let\wt\widetilde \DeclareMathOperator{\End}{End} $$

This is the first lecture on the arithmetic moduli theory of elliptic curves. I began by explaining why the natural moduli problem for elliptic curves is not representable by a scheme. I then proved (in a fair amount of detail) that the moduli problem of elliptic curves with full level 3 structure is representable by a scheme. Using this, I deduced representability for full level N structure, for N>3. I then returned to the level 1 case and briefly explained the concept of Deligne--Mumford stack and how it applies here.

The main issue with representability

We have previously defined $Y(1)$ as a Riemann surface via an analytic construction, and saw that its points correspond bijectively to elliptic curves over $\bC$. We now want to do this theory algebraically, over any field (or base scheme). As with any moduli problem, we first approach $Y(1)$ by defining its functor of points. If $S$ is some scheme, then a map $S \to Y(1)$ should correspond to a family of elliptic curves over $S$. We already have a good notion of this: an elliptic curve over $S$ is a smooth proper scheme $E \to S$ equipped with a section $0 \in E(S)$ such that each geometric fiber is a geometrically connected genus 1 curve.

We are therefore lead to the following functor: for a scheme $S$, let $F_{\Gamma(1)}(S)$ denote the set of isomorphism classes of elliptic curves over $S$. We would like to define $Y(1)$ to be the scheme (over $\bZ$) that represents $F_{\Gamma(1)}$. Unfortunately, $F_{\Gamma(1)}$ is not representable!

The reason for this is the same reason that the $j$-invariant does detect isomorphism classes over a non-algebraically closed field. If $X$ is any scheme over $\bQ$ then the map $X(\bQ) \to X(\ol{\bQ})$ is injective. Since the map $F_{\Gamma(1)}(\bQ) \to F_{\Gamma(1)}(\ol{\bQ})$ is not injective (because of this problem with the $j$-invariant), $F_{\Gamma(1)}$ cannot be representable. In fact, this shows that $F_{\Gamma(1)}$ is not even a sheaf.

It is worthwhile to examine the situation a little more closely. Suppose $E$ and $E'$ are elliptic curves over a field $k$ that become isomorphic over the separable closure $k^s$ --- we say that $E$ and $E'$ are twisted forms. Choose an isomorphism $\varphi \colon E \to E'$ over $k^s$. If $\sigma$ is an element of the absolute Galois group of $k$, then $\varphi^{\sigma}$ defines a (possibly different) isomorphism $E \to E'$ over $k^s$. Thus $\psi_{\sigma}=\varphi^{\sigma} \varphi^{-1}$ is an automorphism of $E$ over $k^s$. It is not difficult to verify that $\sigma \mapsto \psi_{\sigma}$ satisfies the 1-cocycle condition, and that $\varphi$ can be chosen to be defined over $k$ if and only if $\psi$ is a 1-coboundary. This construction in fact defines a bijection

$$ \{ \textrm{twisted forms of $E$ (up to isom.)} \} \to \rH^1(\Gamma_k, \Aut(E_{k^s})). $$

For example, if $E$ is a non-CM curve over a field $k$ of characteristic 0, then $\Aut(E_{k^s})=\{\pm 1\}$ with trivial Galois action, and the $\rH^1$ is just $k^{\times}/(k^{\times})^2$, and so twisted forms of $E$ correspond to square classes in $k$. Explicitly, if $E$ is given by $y^2=f(x)$ and $d \in k^{\times}$, then the twist of $E$ corresponding to $d$ is given by $dy^2=f(x)$.

Full level 3 structure

The above discussion shows that the existence of automorpisms of elliptic curves prevents the functor $F_{\Gamma(1)}$ from being representable. This suggests that we might have better luck if we look at more rigid objects.

Let $N \ge 2$ be an integer. Let $E \to S$ be an elliptic curve, with $N$ invertible on $S$. A $\Gamma(N)$-structure on $E$ is a pair of sections $(P, Q) \in  E(S)[N]$ such that the map $(P, Q) : (\bZ/N\bZ)_S^2 \to E[N]$ is an isomorphism of group schemes over $S$; this is the same as asking that $(P, Q)$ give a basis of the $N$-torsion of $E_s$ for each geometric point $s$ of $S$. One can prove the following result:

Proposition. If $N \ge 3$ then any automorphism of $E$ fixing a $\Gamma(N)$-structure is the identity.

In other words, elliptic cures equipped with $\Gamma(N)$-structures have no non-trivial automorphisms for $N \ge 3$. Let $F_{\Gamma(N)}(S)$ denote the set of isomorphism classes of data $(E, (P, Q))$, where $E$ is an elliptic curve over $S$ and $(P, Q)$ is a $\Gamma(N)$-structure on $E$. It follows from the above proposition, and general principles, that $F_{\Gamma(N)}$ is a sheaf for $N \ge 3$. In the rest of this section, we investigate the $N=3$ case.

Suppose $E/S$ is an elliptic curve and $(P, Q)$ is a $\Gamma(3)$-structure on $E$. By Riemann--Roch, locally on $S$ there is a function $x$ having a double pole at 0; it is unique up to $x \mapsto ax+b$. There is also a function $y$ having a triple pole at 0, and no other poles: it is unique up to $y \mapsto ay+bx+c$. After possibly scaling $x$ and $y$, they satisfy an equation of the form $y^2+a_1xy+a_3y=h(x)$, for some cubic $h$.

Since $P$ is 3-torsion, the divisor $3[P]-3[0]$ is principal, and so there is a unique function (up to scaling) with this divisor. Since $1$, $x$, and $y$ span the space of functions having a triple pole at 0, there is a unique such function of the form $y+ax+b$. In other words, we can choose our original function $y$ uniquely (up to scaling) so that $y$ has a triple zero at $P$. The function $y^2+a_1xy+a_3y$ has valuation 3 at $P$, and so $h(x)$ does as well. Since $h$ is cubic and $x$ is a degree 2, this implies $h(x)=(x-x(P))^3$ and $x$ vanishes to order 1 at $P$. We now replace $x$ with $x-x(P)$.

We have thus shown that $x$ and $y$ can be chosen uniquely, up to scaling, such that the equation becomes $y^2+a_1xy+a_3y=x^3$, and $P$ is the point $(0, 0)$.

Now, $3[Q]-3[0]$ is also a principal divisor, and so is the divisor of a function of the form $y-Ax-B$. We claim $A$ is invertible on $S$. To prove this it suffices to treat the case where $S$ is a field, and show $A \ne 0$. Suppose $A=0$, so that $y-B$ has a triple zero at $Q$. Since $y$ is a degree 3 function, this means $Q$ is the only point at which $y-B$ vanishes, and so plugging $y=B$ into the defining equation gives a polynomial in $x$ with a triple root. Thus $x^3-(B^2+a_1Bx+a_3B)=(x-x(Q))^3$. Comparing the coefficients of $x^2$, we see that $x(Q)=0$. (Here we use that we are not in characteristic 3.) But then $y(Q)^2+a_3y(Q)=0$, and so $y(Q)$ is either equal to 0 or $-a_3$. Thus $Q$ is given in coordinates by $(0, 0)$ or $(0, -a_3)$. But these two points are $P$ and $-P$, and $Q$ is not equal to either of them. Thus $A \ne 0$.

Since $A$ is a unit, we can replace $y$ by $y/A^3$ and $x$ by $x/A^2$, and assume $A=1$. As $y-x-B$ vanishes only at $Q$, when we substitute $y=x+B$ into the defining equation the resulting polynomial has a triple root. Thus $x^3-((x+B)^2+a_1x(x+B)+a_3(x+B))=(x-C)^3$, where $C=x(Q)$. Equating like powers, we find

$$ \begin{cases} 3C=a_1+1 \\ -3C^2=2B+a_1B+a_3 \\ C^3=B^2+a_3B \end{cases} $$

The first two define $a_1$ and $a_3$ in terms of $B$ and $C$. Subtracting the third from $B$ times the second gives $C^3+3C^2B+(a_1+1)B^2=0$. But $a_1+1=3C$, and so this equation is equivalent to $(B+C)^3=B^3$. We have thus proved the following result:

Proposition. Let $E/S$ be an elliptic curve and let $(P, Q)$ be a $\Gamma(3)$-structure on $E$. Then there exist unique functions $x$ and $y$ on $E$ such that the following conditions hold:

  • $x$ and $y$ have poles of order 2 and 3 at 0, and no other poles, and $y^2/x^3=1$ at 0.

  • $y$ vanishes to order 3 at $P$, and $x$ vanishes at $P$ (to order 1).

  • $y-x-B$ vanishes to order 3 at $Q$, for some function $B$ on $S$.

Furthermore, if $C=x(Q)$ then $(B+C)^3=B^3$, and $E$ is defined by the equation

$$ y^2+a_1xy+a_3y=x^3 $$

with $a_1=3C-1$ and $a_3=-3C^2-B-3BC$.

Remark. In coordinates, $P=(0,0)$ and $Q=(C,B+C)$.

The converse to this proposition holds as well. Namely, given functions $B$ and $C$ on $S$, let $E$ be the curve defined by the above equations, let $P$ be the point $(0, 0)$ and let $Q$ be the point $(C,B+C)$. Then as long as $E$ is an elliptic curve (i.e., the discriminant of the equation is a unit), the above statements hold. These statements imply $\div(y)=3[P]-3[0]$ and $\div(y-x-B)=3[Q]-3[0]$, and so $P$ and $Q$ are 3-torsion. Furthermore, since the discriminant is a unit, $C$ is a unit, which implies $Q \ne \pm P$. Thus $(P, Q)$ is a $\Gamma(3)$-structure on $E$. This proves the following theorem:

Theorem. Let $R=\bZ[1/3,B,C][1/\Delta]/(B^3=(B+C)^3)$. Then $Y(3)=\Spec(R)$ represents the functor $F_{\Gamma(3)}$.

Full level N structure

We would now like to prove that $F_{\Gamma(N)}$ is representable for all $N \ge 3$. However, the argument we gave in the $N=3$ is too complicated if $N$ is much larger than 3. We therefore need an alternate approach. Fortunately, we can use the work we did for $N=3$ to (almost) prove the result in general. We begin with the following.

Proposition. Let $E/S$ be an elliptic curve. Let $F$ be the functor on schemes over $S$ which attaches to $S' \to S$ the set $F(S')$ of $\Gamma(N)$-structures on $E_{S'}$. Then $F$ is represented by a finite étale scheme $T \to S$.

Proof. Let $T_0=E[N] \times E[N]$, a finite étale group scheme over $S$. Let $T$ be the kernel of the Weil pairing $T_0 \to (\mu_N)_S$. Then $T$ is a closed subscheme of $T_0$, and therefore finite étale over $S$. Furthermore, giving a map $S' \to T$ is the same as giving $P, Q \in E(S')[N]$ such that in each fiber the Weil pairing of $P$ and $Q$ is non-zero, i.e., $(P, Q)$ is a $\Gamma(N)$-structure.

Theorem. Suppose $N$ is prime to 3. Then $F_{\Gamma(3N)}$ is represented by a smooth affine scheme $Y(3N)$ over $\bZ[1/3N]$.

Proof. Let $Y(3N) \to Y(3)$ be the finite étale scheme constructed in the previous proposition using the universal family over $Y(3)$. Giving a map $S \to Y(3N)$ is the same as giving an elliptic curve $E/S$ with both $\Gamma(3)$ and $\Gamma(N)$ structures. But giving $\Gamma(3)$ and $\Gamma(N)$ structures is the same as giving a $\Gamma(3N)$ structure by the Chinese remainder theorem. Thus $Y(3N)$ represents $F_{\Gamma(3N)}$. Since $Y(3N)$ is finite étale over $Y(3)$, and $Y(3)$ is smooth and affine, the same is true for $Y(3N)$.

Proposition. Suppose $N \ge 4$ is prime to 3. Then $\GL_2(\bZ/3\bZ)$ acts freely on $F_{\Gamma(3N)}$ and the quotient sheaf is $F_{\Gamma(N)}$.

Proof. The action is by moving around the $\Gamma(3)$-structure. Suppose $(E, (P, Q), (P', Q'))$ were fixed by the action of $g \in \GL_2(\bZ/3\bZ)$, where $(P, Q)$ is a $\Gamma(3)$-structure and $(P', Q')$ is a $\Gamma(N)$-structure. Thus there is some automorphism $f \colon E \to E$ taking $(P, Q)$ to $g(P, Q)$ and $(P', Q')$ to $(P', Q')$ since $N \ge 3$, this implies $f=1$, and so $g=1$. This proves the action is free. Finally, suppose $E/S$ is an elliptic curve with $\Gamma(N)$-structure. Let $T \to S$ be the finite étale cover parametrizing $\Gamma(3)$-structures on $E$. Then $E_T$ canonically has a $\Gamma(3N)$ structure. This shows that the map $F_{\Gamma(3N)}(T) \to F_{\Gamma(N)}(S)$ is surjective, and so the map of sheaves $F_{\Gamma(3N)} \to F_{\Gamma(N)}$ is surjective. It is clear that $F_{\Gamma(3N)}/\GL_2(\bZ/3\bZ)$ is the maximal quotient it factors through.

Theorem. Suppose $N \ge 4$ is prime to 3. Then $F_{\Gamma(N)}$ is represented by a smooth affine scheme $Y(N)$ over $\bZ[1/3N]$.

Proof. By the above proposition, the group $\GL_2(\bZ/3\bZ)$ acts freely on $Y(3N)$. The quotient $Y(N)$ therefore exists as a smooth affine scheme over $\bZ[1/3N]$, and represents the quotient sheaf $F_{\Gamma(N)}$.

Obviously, this is still less than what we want. The full result is:

Theorem. Suppose $N \ge 3$. Then $F_{\Gamma(N)}$ is represented by a smooth affine scheme $Y(N)$ over $\bZ[1/N]$.

We just give the idea of the proof. One first proves that a suitable replacement of $F_{\Gamma(2)}$ is representable. This done explicitly, similar to what we did for $F_{\Gamma(3)}$. Then one reasons as we did above to show that if $N \ge 3$ is prime to 2 then $F_{\Gamma(N)}$ is representable by a smooth affine scheme over $\bZ[1/2N]$. A patching argument now shows that for any $N$ prime to 6, $F_{\Gamma(N)}$ is representable by a smooth affine scheme over $\bZ[1/N]$. A small amount of extra work is needed to remove the "$N$ prime to 6" condition.


Let us re-exaimine why $F_{\Gamma(1)}$ fails to be a sheaf. If we are given an elliptic curve $E$ over $k^s$ such that $\sigma^*(E)$ is isomorphic to $E$ for all Galois automorphisms $\sigma$, we cannot necessarily descend $E$ to $k$, because these isomorphisms may not be compatible. This suggests that we shouldn't simply take isomorphism classes of elliptic curves, but consider the whole category (or at least groupoid) of elliptic curves.

A stack (on some topological space or site) is a rule $\cF$ that assigns to each open set $U$ a groupoid $\cF(U)$ and to each inclusion $U' \subset U$ a restriction functor $\cF(U) \to \cF(U')$ such that appropriate analogs of the sheaf axioms hold. The glueing axiom states that if $U=\bigcup U_i$ is an open cover and $X_i$ are objects of $\cF(U_i)$ equipped with isomorphisms $\varphi_{ij} \colon X_i \vert_{U_{ij}} \to X_j \vert_{U_{ij}}$ satisfying the cocycle codition, then the $X_i$ glue to an object $X$ over $U$.

Suppose $G$ is a finite group acting on a variety $X$. The usual quotient $X/G$ is not well-behaved when $G$ has fixed points: for instance, the fibers of $X \to X/G$ do not all have cardinality $G$. However, the is always a well-behaved stack quotient $[X/G]$. It has the following property: if $Y \to [X/G]$ is any map from a scheme $Y$, then the fiber product $X \times_{[X/G]} Y$ is a scheme, and the fibers of the projection map $X \times_{[X/G]} Y \to Y$ are permuted simply transitively by $G$; in other words, $X \times_{[X/G]} Y \to Y$ is a $G$-torsor. In fact, the functor of points of $[X/G]$ takes a scheme $Y$ to the groupoid of objects $(T, f, g)$ where $T$ is a scheme with a $G$-action, $f \colon T \to Y$ gives $T$ the structure of a $G$-torsor over $Y$, and $g \colon T \to X$ is a $G$-equivariant map.

The stack $[X/G]$ above is not a scheme. However, it is not very far off from a scheme. A Deligne--Mumford stack is stack having similar properties to $[X/G]$. More precisely, it is a stack $X$ for which there exists a map $\wt{X} \to X$ (with $\wt{X}$ a scheme) with the following property: if $T \to X$ is any map with $T$ a scheme, then $T \times_X \wt{X}$ is a scheme and the projection map to $T$ is surjective and étale. We say that $\wt{X}$ is an étale cover of $X$.

Geometric properties of Deligne--Mumford stacks are defined by appealing to analogies with schemes. For example, if $\wt{X} \to X$ is an étale cover of schemes then $X$ is smooth (over whatever base) if and only if $\wt{X}$ is. Thus one says that a Deligne--Mumford stack is smooth if it admits an étale cover by a smooth scheme.

Back to level 1

For a scheme $S$, let $\cF_{\Gamma(1)}(S)$ denote the groupoid of elliptic curves over $S$. One can then show that $\cF_{\Gamma(1)}$ is a stack for the fppf site of schemes. Furthermore, $Y_{\Gamma(N)} \to \cF_{\Gamma(1)}$ is relatively representable and étale over $\bZ[1/N]$: indeed, giving a map $S \to \cF_{\Gamma(1)}$ is the same as giving an elliptic curve $E/S$, and the fiber product is then the scheme of $\Gamma(N)$-structures on $E$, which we know to be finite étale over $S$. From this, we see that $Y(1)=\cF_{\Gamma(1)}$ is a Deligne--Mumford stack.

Other moduli problems

Let $N \ge 2$ be an integer. We only consider schemes over $\bZ[1/N]$. Let $E/S$ be an elliptic curve. A $\Gamma_1(N)$-structure on $E$ is a section $P \in E(S)[N]$ of order $N$. Let $\cF_{\Gamma_1(N)}$ be the stack associating to $S$ the groupoid of $(E, P)$ with $E$ an elliptic curve over $S$ and $P$ a $\Gamma_1(N)$-structure. For $N \ge 3$, this problem is rigid, and $\cF_{\Gamma_1(N)}$ is equivalent to a sheaf $F_{\Gamma_1(N)}$.

Theorem. The stack $Y_1(N)=\cF_{\Gamma_1(N)}$ is a smooth Deligne--Mumford stack over $\bZ[1/N]$. For $N \ge 3$, it is a smooth affine scheme.

A $\Gamma_0(N)$-structure on $E$ is a closed étale subgroup $G \subset E$ which is cyclic of cardinality $N$ in each geometric fiber. Let $\cF_{\Gamma_0(N)}$ be the associated stack.

Theorem. The stack $Y_0(N)=\cF_{\Gamma_0(N)}$ is a smooth Deligne--Mumford stack over $\bZ[1/N]$.

Unlike previous cases, this example is never a scheme, since multiplication by $-1$ preserves and $\Gamma_0(N)$-structure. In fact, the automorphism groups can even be larger than order 2 in certain cases. For example, suppose $N$ is a prime congruent to 1 modulo 4, and let $E$ be an elliptic curve with $\End(E)=\bZ[i]$. Under a suitable basis $(P, Q)$ of $E[N]$, we have $iP=aP$ and $iQ=-aQ$, where $a$ is a square root of $-1$ in $\bZ/N\bZ$. Thus if $G$ is the subgroup of $E[N]$ generated by $P$ or $Q$ then $(E, G)$ has $i$ as an automorphism.