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