multiplicative reduction of a elliptic curve $E$ splits

Question

In Silverman's "The Arithmetic of Elliptic Curves" in Cap. VII.5 (Good and Bad Reduction) a multiplicative reduction of a elliptic curve $E$ , is said to be split if the slopes of the tangent lines at the node are in $k$, and otherwise it is said to be nonsplit.

Q: what is the intuition and the origin of the usage of the word "split" in this context? what "splits"? can we associate a certain s.e.s. to this reduction, which then splits or does the notation split come from another reason?

Answer 1

Here's some more evidence (and slightly alternate interpretations) for my comment that slopes in $k$ is equivalent to the equation defining the node splitting as a product of linear factors in the completed local ring. I am as yet unable to find the definitive history of the term, but I hope this sheds some light on the subject for the asker and bountier:

Vakil's Rising Sea, section 29.3, "Defining types of singularities":

Singularities are best defined in terms of completions. As an important first example, we finally define "node".

29.3.1. Definition. Suppose $X$ is a dimension $1$ variety over $\overline{k}$, and $p\in X$ is a closed point. We say that $X$ has a node at $p$ if the completion of $\mathcal{O}_{X,p}$ at $\mathfrak{m}_{X,p}$ is isomorphic (as topological rings) to $\overline{k}[[x,y]]/(xy)$.

29.3.B. Exercise. Suppose $k=\overline{k}$ and $\operatorname{char} k\neq 2$, and we have $f(x,y)\in k[x,y]$. Show that $\operatorname{Spec} k[x,y]/(f(x,y))$ has a node at the origin iff $f$ has no terms of degree $0$ or $1$, and the degree $2$ terms are not a perfect square.

The definition of node outside the case of varieties over algebraically closed fields is more problematic, and we give some possible ways forward. For varieties over a non-algebraically closed field $k$, one can always base-change to the closure $\overline{k}$. As an alternative approach, if $p$ is a $k$-valued point of a variety over $k$ (not necessarily algebraically closed), then we could take the same definition as 29.3.1; this might reasonably be called a split node, because the branches (or more precisely, the tangent directions) are distinguished. Those singularities that are not split nodes, but which become nodes after base change to $\overline{k}$ (such as the origin in $\operatorname{Spec} \Bbb R[x,y]/(x^2+y^2)$) might reasonably be called non-split nodes.

Stacks Project Tag 0C46, Nodal Curves:

We have already defined ordinary double points over algebraically closed fields as follows: if $x\in X$ is a closed point of a $1$-dimensional scheme over an algebraically closed field $k$, then $x$ is an ordinary double point if $$ \mathcal{O}_{X,x}^\wedge \cong k[[x,y]]/(xy).$$

Definition 0C47. Let $k$ be a field. Let $X$ be a $1$-dimensional locally algebraic $k$-scheme. We say a closed point $x\in X$ is a node if there exists an ordinary double point $\overline{x}\in X_{\overline{k}}$ mapping to $x$.

Stacks goes on to prove that if $x\in X$ is a node, then (under mild niceness hypotheses) the completion of the local ring at $x$ is isomorphic to $k[[x,y]]/(q(x,y))$ where $q$ is a nondegenerate quadratic form. Saying that this node is split is then equivalent to $q(x,y)$ being choosable as $xy$, which is the same as saying it splits in to distinct linear factors. There's also another characterization - to each $q$, we can associate a degree-two algebra extension of the residue field at $x$, and saying that the node $x$ is split is equivalent to this algebra extension splitting as a direct product of the residue field with itself (see 0CBT + OCBU).

Answer 2

I think one way to understand this terminology is the following (though, like the other answer, I am not really positive about the etymology).

If $C$ is a singular projective plane curve over a field $k$ then $C$ has exactly one singular point $x_0$ and, in fact, $x_0$ is a $k$-point. This can be seen by applying Bezout's theorem over $\overline{k}$ (i.e. if $p,q\in C(\overline{k})$ were both singular then if $\ell$ is a line passing through $p$ and $q$ then the multiplicity of $\ell\cap C$ at $p,q$ both have to be at least $2$ since $C$ is singular there, but then $\ell\cdot C\geqslant 4$ which contradicts Bezout's theorem).

Thus, if $C$ is singular we see that the smooth locus $C^\text{sm}=C-\{x_0\}$ is a smooth integral affine $k$ curve. Moreover, note that if $\ell'$ is any line passing through $x_0$ then $\ell\cdot C=3$, again by Bezout's theorem, which since the multiplicity of $\ell\cap C$ at $x_0$ is $3$ implies that $\ell\cap C$ contains another point of multiplicity $1$ which then is clearly a smooth $k$-point. In particular, $C^\text{sm}(k)\ne\varnothing$.

So, fix a point $e\in C^\text{sm}(k)$. Then, the exact same chord-tangent construction for elliptic curves endows $C^\text{sm}$ with a unique group structure such that $e$ is the identity--the point is that again if one takes $p,q$ in $C^\text{sm}(L)$ for any field extension $L$ then for any line $\ell$ in $\mathbb{P}^2_L$ passing through $p,q$ we have that $\ell\cdot C_L$ is $3$ which, again by multiplicity reasoning, implies that $\ell\cap C$ intersects $C$ at a third point which is automatically smooth and $L$-rational, so the same chord-tangent construction applies.

Thus, we see that $C^\text{sm}$ is a smooth integral $1$-dimensional affine group variety over $k$! As it turns out, there are not so many of those over $k$ if $k$ is finite:

Fact: Let $G$ be a smooth integral $1$-dimensional affine group variety over $k=\mathbb{F}_q$. Then, $G$ is isomorphic to $\mathbb{G}_{a,\mathbb{F}_q}$, $\mathbb{G}_{m,\mathbb{F}_q}$ or $\mathsf{Res}^1_{\mathbb{F}_{q^2}/\mathbb{F}_q}\mathbb{G}_{m,\mathbb{F}_{q^2}}$.

For a proof you can see the following (DISCLAIMER: THIS IS MY BLOG):

• https://ayoucis.wordpress.com/2014/11/29/classifying-one-dimensional-algebraic-groups/
• https://ayoucis.wordpress.com/2019/11/19/classifying-one-dimensional-groups-ii/

The group $\mathbb{G}_{a,\mathbb{F}_q}$ is called the additive group and the groups $\mathbb{G}_{m,\mathbb{F}_q}$ and $\mathsf{Res}^1_{\mathbb{F}_{q^2}/\mathbb{F}_q}\mathbb{G}_{m,\mathbb{F}_{q^2}}$ are tori. By definition, a group variety $G$ over a field $k$ is called a torus if $G_{\overline{k}}$ is isomorphic $\mathbb{G}_{m,\overline{k}}^n$ where $n=\dim(G)$. We call $G$ split if $G\cong \mathbb{G}_{m,k}^n$ (i.e. it's isomorphic rationally to $\mathbb{G}_{m,k}^n$). Note that $\mathbb{G}_{m,k}$ is called the multiplicative group and so another name for tori is groups of multiplicative type.

Remark: This last statement is not quite standard-- with the usual terminology of 'multiplicative group' tori are the connected groups of multiplicative type, but let's not worry about this here.

In particular, we see that every smooth integral $1$-dimensional affine group variety over $\mathbb{F}_q$ is either

• The (one-dimensional) additive group.
• The (one-dimensional) split multiplicative group.
• The (one-dimensional) non-split multiplicative group.

So, if $E$ is an elliptic curve over a $p$-adic field $K$ then it has a unique minimal Weierstrass model $\mathcal{E}^\text{min}$ which is a certain cubic curve over $\mathcal{O}_K$. If $k$ is the residue field of $K$ then we see from our above discussion that the reduction $\mathcal{E}_k$ is a cubic curve and thus $\mathcal{E}_k^\text{sm}$ is either

• An elliptic curve (this is the case when $\mathcal{E}_k$ has no singular point).
• The (one-dimensional) additive group.
• The (one-dimensional) split multiplicative group.
• The (one-dimensional) non-split multiplicative group.

One can then check that the usual definitions of good, additive, split multiplicative, and non-split multiplicative reduction match up precisely with this classification.