4
$\begingroup$

Let $K$ be a local field that is complete with respect to a discrete valuation. When an elliptic curve, $E/K$, has reduction type represented by the Kodaira symbol $I_{2n}^{*}$, its component group can be ${\mathbb Z}/2{\mathbb Z} \times {\mathbb Z}/2{\mathbb Z}$.

In this case, is there a criterion to determine which component a point is on? In particular, to tell if it is on the component $(1,0)$ (or $(0,1)$) or on the component $(1,1)$?

In Lemma 5.1 of Silverman's ``Computing Heights on Elliptic Curves'', Math Comp. v. 51 (1988), 339--358, he shows how to determine the component when we have split multiplicative reduction. Ideally something like this is the sort of thing I am looking for.

Improve this question
$\endgroup$
1
  • $\begingroup$ I don't think that is all that easy. Of course you can blow up the singularity and follow the points to see where they land. However, I doubt that you will often need on which components the points really lie. It is easy to test if two points lie on the same component (the difference has good reduction). Operations and comparisons on a group of the form $\mathbb{Z}/2 \times\mathbb{Z}/2$ does not need any thing else. In particular the monodromy pairing etc can be terermined completely using just this test. $\endgroup$
    – Chris Wuthrich
    Commented Sep 13, 2017 at 19:55

0

Reset to default

You must log in to answer this question.