Semistable reduction of a smooth projective curve over a local field

Let \(K\) be a field and \(v_K\) a discrete valuation on \(K\). We let \(\mathcal{O}_K\) denote the valuation ring of \(v_K\) and \(\mathbb{F}_K\) the residue field.

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.

Let us define what we mean by ‘semistable reduction’ and by ‘computing’. By the famous result of Deligne and Mumford there exists a finite, separable field extension \(L/K\), an extension \(v_L\) of \(v_K\) to \(L\) (whose valuation ring we call \(\mathcal{O}_L\)) and an \(\mathcal{O}_L\)-model \(\mathcal{Y}\) of \(Y_L\) whose special fiber \(\bar{Y}:=\mathcal{Y}_s\) is reduced and has at most ordinary double points as singularities. We call \(\mathcal{Y}\) a semistabel model and \(\bar{Y}\) a semistable reduction of \(Y\).

Let us assume, for simplicity, that \(K\) is complete with respect to \(v_K\). Then the extension \(v_L\) to \(L\) is unique and \(L\) is complete with respect to \(v_L\). Then we we may moreover assume that \(L/K\) is a Galois extension and that the tautological action of the Galois group of \(L/K\) extends to the semistable model \(\mathcal{Y}\). By restriction we obtain an action of \({\rm Gal}(L/K)\) on \(\bar{Y}\). (In practise we mostly work with fields \(K\) which are not complete. To make sense of the above definitions, one simply has to replace \(K\) by its completion.)

When we say the semistable reduction of \(Y\) we actually mean the extension \(L/K\), the \(\mathbb{F}_L\)-curve \(\bar{Y}\) and the action of the former on the latter.

Note that neither \(L/K\) nor \(\bar{Y}\) are unique, but their nonuniqueness is in a sense inessential. For instance, one may replace \(L\) by a larger Galois extension \(L'/K\); as a consequence the curve \(\bar{Y}\) gets replaced by its base extension to the residue field of \(L'\). Also, certain blowups of the semistable model \(\mathcal{Y}\) may result in a new semistable model \(\mathcal{Y}\) with special fiber \(\bar{Y}'\). The only difference between \(\bar{Y}\) and \(\bar{Y}'\) are some new irreducible components, which are ‘contractible’, i.e. they are smooth of genus \(0\) and meet the rest of \(\bar{Y}'\) in at most two points.

At the moment, we do not have an effective method at our disposal to compute the semistable reduction of an arbitrary curve \(Y\), but only a set of methods which can be applied for certain classes of curves. We always assume that the curve \(Y\) is given as a finite separable cover

\[\phi: Y \to X,\]

where \(X=\mathbb{P}^1_K\) is the projective line over \(K\). There are two main cases that we can handle:

  • the order of the monodromy group of \(\phi\) (i.e. the Galois group of its Galois closure) is prime to the residue characteristic of the valuation \(v_K\).
  • \(\phi\) is a Kummer cover of degree \(p\), where \(p\) is the (positive) residue characteristic of \(v_K\)

In the first case, the method of admissible reduction is available. In the second case, the results of [We17] tell us what to do. In both cases, there exists a normal \(\mathcal{O}_K\)-model \(\mathcal{X}_0\) of \(X=\mathbb{P}^1_K\) (the inertial model) whose normalization in the function field of \(Y_L\) is a semistable model, for a sufficiently large finite extension \(L/K\). Once the right inertial model is defined, the method for computing the semistable model \(\mathcal{Y}\) and its special fiber \(\bar{Y}\) are independent of the particular case (these computations are done within the Sage class ReductionTree).

In this module we define a base class SemistableModel. An object in this class is initialized by a pair \((Y,v_K)\) , where \(Y\) is a smooth projective curve over a field \(K\) and \(v_K\) is a discrete valuation on \(K\). The class provides access to functions which compute and extract information from the semistable reduction of \(Y\) with respect to \(v_K\).

[We17]S. Wewers, Semistable reduction of superelliptic curves of degree p, preprint, 2017.

Note

For the time being, we have to assume that \(K\) is a number field. Then \(v_K\) is the valuation associated to a prime ideal of \(K\) (i.e. a maximal ideal of its ring of integers).

AUTHORS:

  • Stefan Wewers (2018-5-16): initial version

EXAMPLES:

We compute the stable reduction and the conductor exponent of the Picard curve

\[Y:\; y^3 = x^4 - 1.\]

at the primes \(p=2,3\):

sage: from mclf import *
sage: v_2 = QQ.valuation(2)
sage: R.<x> = QQ[]
sage: Y = SuperellipticCurve(x^4-1, 3)
sage: Y
superelliptic curve y^3 = x^4 - 1 over Rational Field
sage: Y2 = SemistableModel(Y, v_2)
sage: Y2.is_semistable()
True

The stable reduction of \(Y\) at \(p=2\) has four components, one of genus \(0\) and three of genus \(1\).

sage: [Z.genus() for Z in Y2.components()]
[0, 1, 1, 1]
sage: Y2.components_of_positive_genus()
[the smooth projective curve with Function field in u2 defined by u2^3 + x^4 + x^2,
 the smooth projective curve with Function field in u2 defined by u2^3 + x^2 + x,
 the smooth projective curve with Function field in u2 defined by u2^3 + x^2 + x + 1]
sage: Y2.conductor_exponent()
6
sage: v_3 = QQ.valuation(3)
sage: Y3 = SemistableModel(Y, v_3)
sage: Y3.is_semistable()
True
sage: Y3.components_of_positive_genus()
[the smooth projective curve with Function field in u2 defined by u2^3 + u2 + 2*x^4]
sage: Y3.conductor_exponent()
6
class mclf.semistable_reduction.semistable_models.SemistableModel(Y, vK, check=True)

Bases: SageObject

This is a base class for various classes of curves and methods for computing the semistable reduction. Objects of this class are determined by a smooth projective curve \(Y\) over a field \(K\) and a discrete valuation \(v_K\) on \(K\).

INPUT:

  • Y – a smooth projective curve
  • vK – a discrete valuation on the base field \(K\) of \(Y\)
  • check – a boolean (default: True)

Instantiation of this class actually creates an instant of a suitable subclass, which represents the kind of curve for which an algorithm for computing the semistable reduction has been implemented. At the moment, there are two such subclasses:

  • If the degree of \(Y\) as a cover of the projective line is prime to the residue characteristic of \(v_K\) then we invoke the subclass AdmissibleModel. Note that this may not work: we can only guarantee that \(Y\) has admissible reduction if the order of the Galois group of the cover \(Y\to X=\mathbb{P}^1_K\) is prime to the residue characteristic.
  • If \(Y\) is a superelliptic curve of degree \(p\), where \(p\) is the residue characteristic of \(v_K\) and \(K\) has characteristic \(0\) then the subclass SuperpModel is invoked.
  • if none of the above holds, then we either raise a NotImplementedError (if check=True) or we create an AdmissibleModel (if check=False)

EXAMPLES:

sage: from mclf import *
sage: v_5 = QQ.valuation(5)
sage: FX.<x> = FunctionField(QQ)
sage: R.<y> = FX[]
sage: FY.<y> = FX.extension(y^3 - y^2 + x^4 + x + 1)
sage: Y = SmoothProjectiveCurve(FY)
sage: YY = SemistableModel(Y, v_5)
sage: YY
semistable model of the smooth projective curve with Function field in y defined by y^3 - y^2 + x^4 + x + 1, with respect to 5-adic valuation

The degree of \(Y\) as a cover of the projective line is \(4\), which is strictly less than \(p=5\). Hence \(Y\) has admissible reduction and we have created an instance of the class AdmissibleModel:

sage: isinstance(YY, AdmissibleModel)
True

Actually, \(Y\) has good reduction at \(p=5\):

sage: YY.is_semistable()
True
sage: YY.components_of_positive_genus()
[the smooth projective curve with Function field in u1 defined by u1^3 + 4*u1^2 + x^4 + x + 1]
base_valuation()

Return the valuation on the constant base field of the curve.

components()

Return the list of all components of the admissible reduction of the curve.

components_of_positive_genus()

Return the list of all components of of the admissible reduction of the curve which have positive genus.

compute_semistable_reduction(verbosity=1)

Compute the semistable reduction of this curve (and report on the ongoing computation).

INPUT:

– verbosity - a nonnegative integer (default: 1)

OUTPUT:

Calling this function initiates the creation of a ReductionTree which essentially encodes a semistable model of the curve. Depending on the verbosity level, messages will be printed which report on the ongoing computation. If verbosity is set to \(0\), no message will be printed.

This method must be implemented by the subclasses of SemistableModel (which are characterized by a particular method for computing a semistable model). At the moment, these subclasses are - AdmissibleModel - Superell - Superp

conductor_exponent()

Return the conductor exponent at p of this curve.

EXAMPLES

In this example the conductor exponent was computed wrongly in a previous version:

sage: from mclf import *
sage: R.<x> = QQ[]
sage: f = x^4+2*x^3+2*x^2+x
sage: Y = SuperellipticCurve(f, 3)
sage: Y3 = SemistableModel(Y, QQ.valuation(3))
sage: Y3.conductor_exponent()
11
constant_base_field()

Return the constant base field of this curve.

curve()

Return the curve.

is_semistable()

Check whether the model is really (potentially) semistable.

reduction_tree()

Return the reduction tree underlying this semistable model.

semistable_reduction()

Return the special fiber of this semistable model.

Note: not yet implemented

stable_reduction()

Return the special fiber of the stable model of this curve.

The stable model is obtained from this semistable model by contracting all ‘unstable’ components.

Note: not yet implemented