Specialization of sections in an elliptic fibration

Question

Let $\pi: X \rightarrow S$ be the Neron model of an elliptic curve over a dedekind domain (but probably any minimal elliptic fibration will suffice).

Let $\eta$ be the generic point of $S$, $K = S(\eta)$ and $ 0_\eta \in X(K)$ denote the zero element of the generic fiber. Suppose $\alpha$ is a section of $\pi$ and specifies a torsion element of the generic fiber $E(K)$. Suppose that $X$ has a fiber of Kodaira type $I_n$ with $n> 1$ at $p \in S$. There is a (nameless?) natural specialization homomorphism $$ E(K)_{tors} \xrightarrow{\phi} G_p $$ to the group of irreducible components of the fiber over $p$: elements of $E(K)$ correspond to sections of $\pi$, and such a section meets exactly one irreducible component of an $I_n$ fiber. Label the components $0, 1, ..., n-1$ starting with the component the $0$-element meets and then going around the $I_n$ fiber consecutively (either direction).

$\textbf{Question:}$ For my own sake, I would like to know how to calculate the image of a torsion point under this map - the completely naive approach (represent torsion points as sections, and calculate intersections in the Neron Severi group) is not very computable - either by hand or computer.

Dispite it's simplicity, I can't find any resources that might explain how to do this. In particular, I didn't find anything useful in BLR's Neron Models book. For generalities on this map where $S$ is a smooth proper curve over $\mathbb{C}$,see p.75 of Miranda's Book http://www.math.colostate.edu/~miranda/BTES-Miranda.pdf

Answer 1

Your question is local, so we may take $S$ to be the ring of integers in a local field $K$. If the point $P\in E(K)$ is torsion or not does not matter in my answer here. I think here of an ellpitic curve over a number field.

I think the best theoretical tool to study fibres of type $I_n$ is by using Tate's uniformisation. This works over $K$ if the reduction is split multiplicative. If it is non-split then the question is rather easier as the group of components is small.

Tate's parametrisation gives an isomorphism of $E(K)$ to $K^{\times}/q^{\mathbb{Z}}$ for some $q$ in $K$ of valuation $n$. The components can now simply be labeled by the valuation map $E(K) \to K^{\times}/q^{\mathbb{Z}} \to \mathbb{Z}/n\mathbb{Z}$. See Silverman 2.

Alternatively, for an explicit approach one can use Tate's algorithm. More precisely, Tate's algorithm is a simplification of the concrete blow-ups needed to build the minimal regular model. One can now follow the point $P\in E(K)$ along these blow-ups and find the corresponding label of the component. The function component_of_a_point(P,v) in https://www.maths.nottingham.ac.uk/personal/cw/download/class_group_pairing.py does exactly what you want.

Finally, not that the index is not exactly well-defined. It is really in $\mathbb{Z}/n\mathbb{Z}$ modulo $\pm 1$ as we can label in one direction or the other.