Global sections of $p$-torsion of elliptic curve

algebraic-geometryelliptic-curvesschemes

Suppose that $R$ is a ring and $E$ is an elliptic curve over $R$. (That is, we have an elliptic curve $E$ which is a group scheme over the affine scheme $\text{Spec }R$.) Then for any prime $p$, we have the multiplication by $p$-map $E \to E$, which has group-scheme theoretic kernel $E[p]$:
$$0 \to E[p] \to E \to E.$$
I'd like to emphasize that I'm looking at $E[p]$ as a group scheme over $R$, as opposed to just an abelian group.

My question is: what are the global sections of the scheme $E[p]$? That is, if $\mathcal{O}$ denotes the structure sheaf of the scheme $E[p]$, then what is $H^0(E[p], \mathcal{O})$?

What I've tried: $E[p]$ is a closed subscheme of $\mathbf{P}^2_{R}$, and so it is of the form $\text{Proj}(R[x,y,z]/I)$ for some homogenous ideal $I$. Here is where I'm stuck: if $R$ was an algebraically closed field, then the global sections would be the constants. But I'm not sure what the global sections would be for general rings $R$. Any clarification / help would be appreciated. Thanks!

Best Answer

Note that $E[p]$ is an affine scheme, so $E[p] \cong \operatorname{Spec} A$ for some commutative ring $A$, and then $H^0(E[p], \mathcal{O}_{E[p]}) \cong A$. Moreover, $E[p]$ is zero-dimensional, and, since $K$ has characteristic zero, $E[p]$ is also a reduced scheme. Zero-dimensional reduced schemes of finite type over a field $K$ have a very simple description: they're the spectra of étale $K$-algebras, that is, finite products of finite separable extensions of the base field $K$.

Here's what that looks like in this case: $E[p](\bar{K}) \cong (\mathbb{Z}/p\mathbb{Z})^2$ consists of $p^2$ points, and these points are partitioned into Galois orbits by the action of the absolute Galois group $\operatorname{Gal}(\bar{K}/K)$. The points of the scheme $E[p]$ correspond to these Galois orbits, and the residue field of each point is the field of definition of the coordinates of one of the $\bar{K}$-points in the corresponding orbit. So, $E[p](\bar{K}) \cong \operatorname{Spec}(L_1 \times \ldots \times L_n)$, where $L_1, \ldots, L_n$ are finite separable extensions of $K$. (Actually, "separable" is redundant since all algebraic extensions are separable in characteristic zero.)

It may help to illustrate with a few examples:

  • Elliptic curve 24.a3, given by the equation $y^2 = x^3-x^2-24x-36 = (x - 6) (x + 2) (x + 3)$, has all four of its $2$-torsion points defined over $\mathbb{Q}$, so $H^0(E[2], \mathcal{O}) \cong \mathbb{Q}^4$.
  • Elliptic curve 20.a1, given by the equation $y^2=x^3+x^2-41x-116 = (x + 4) (x^2 - 3x - 29)$, has two $2$-torsion points defined over $\mathbb{Q}$: the point at infinity and $(-4, 0)$. The other two $2$-torsion points are a conjugate pair defined over $\mathbb{Q}(\sqrt{5})$, over which $x^2 - 3x - 29$ factors. Thus, the ring of global sections of $E[2]$ is $\mathbb{Q}^2 \times \mathbb{Q}(\sqrt{5})$.
  • Elliptic curve 88.a1, given by the equation $y^2=x^3-4x+4$, has no $2$-torsion points defined over $\mathbb{Q}$ except the point at infinity. The other three $2$-torsion points are defined over number field 3.1.44.1, the cubic field $K$ generated by adjoining a root of $x^3 - 4x + 4$. So here the ring of global sections of $E[2]$ is isomorphic to $\mathbb{Q} \times K$. (Note that we take the field $K$ itself here, not its sextic splitting field over which all four $2$-torsion points can be defined simultaneously.)

In positive characteristic $\ell \neq p$, the same is true—the scheme $E[p]$ is a reduced zero-dimensional scheme of length $p^2$—except that now we really do need to say "separable". In characteristic $p$, the situation is more complicated: $E[p]$ is no longer a reduced scheme, so its ring of global sections isn't a reduced $K$-algebra, and we have two very different cases depending on whether $E$ is an ordinary or supersingular elliptic curve.

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