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.