Admissible reduction of curves¶
Let \(K\) be a field equipped with a discrete valuation \(v_K\). For the time being, we assume that \(K\) is a number field. Then \(v_K\) is the \(\mathfrak{p}\)-adic valuation corresponding to a prime ideal \(\mathfrak{p}\) of \(K\).
We consider a smooth projective curve \(Y\) over \(K\). Our goal is to compute the semistable reduction of \(Y\) at \(v_K\) and to extract nontrivial arithmetic information on \(Y\) from this.
In this module we realize a class AdmissibleModel
which computes the semistable reduction of a given curve \(Y\) at \(v_K\) provided
that it has admissible reduction at \(v_K\). This is always the case if the
residue characteristic of \(v_K\) is zero or strictly larger than the degree
of \(Y\) (as a cover of the projective line).
AUTHORS:
- Stefan Wewers (2018-7-03): initial version
EXAMPLES:
<Lots and lots of examples>
TO DO:
- more doctests
-
class
mclf.semistable_reduction.admissible_reduction.
AdmissibleModel
(Y, vK)¶ Bases:
mclf.semistable_reduction.semistable_models.SemistableModel
A class representing a curve \(Y\) over a field \(K\) with a discrete valuation \(v_K\). Assuming that \(Y\) has (potentially) admissible reduction at \(v_K\), we are able to compute the semistable reduction of \(Y\) at \(v_K\).
INPUT:
Y
– a smooth projective curve over a field \(K\)vK
– a discrete valuation on \(K\)
OUTPUT: the object representing the curve \(Y\) and the valuation \(v_K\). This object has various functionalities to compute the semistable reduction of \(Y\) relative to \(v_K\), and some arithmetic invariants associated to it (for instance the “exponent of conductor” of \(Y\) with respect to \(v_K\)).
EXAMPLES:
-
original_model_of_curve
()¶ Return the original model of the curves.