Semistable models of superelliptic curves of degree \(p\)¶

Let \(K\) be a field of characteritic zero and \(v_K\) a discrete valuation on \(K\) whose residue field is finite of characteristic \(p>0\).

Let \(f\in K[x]\) be a polynomial over \(K\) which is not a \(p\)-th power and whose radical has degree at least three. We consider the smooth projective curve \(Y\) over \(K\) defined generically by the equation

\[Y: y^p = f(x).\]

So \(Y\) is a superelliptic curve of degree \(p\).

In this module we define a class SuperpModel which represents a semistable model of a superelliptic curve \(Y\) of degree \(p\), with respect to a \(p\)-adic valuation on the base field \(K\) of \(Y\).

The method to define and compute a semistable model in this particular case is taken from

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

The key notion is the etale locus.

The superelliptic curve \(Y\) is, by definition, a cyclic cover

\[\phi: Y \to X\]

of degree \(p\) of the projective line \(X\) over the base field \(K\). We consider \(X\) and \(Y\) as analytic spaces over the completion of \(K\) at the base valuation \(v_K\). Let

\[\bar{\phi}: \bar{Y} \to \bar{X}\]

denote the semistable reduction of the cover \(Y\to X\). The etale locus is an affinoid subdomain \(X^{et}\) of \(X\) consisting of those points which specialize to a point on \(\bar{X}\) above which the map \(\bar{\phi}\) is etale.

While the affinoid \(X^{et}\) is determined by the semistable reduction of the cover \(\phi\), conversely \(X^{et}\) contains a lot of information on the semistable reduction. The main result of [We17] gives an explicit description of the affinoid \(X^{et}\) as a union of rational domains defined by rational functions which can be easily computed in terms of the polynomial \(f\) defining \(Y\).

Once the etale locus \(X^{et}\) is computed, we can define an inertial model \(\mathcal{X}_0\) of \(X\). A semistable model \(\mathcal{Y}\) of \(Y\) can then be obtained as the normalization of \(\mathcal{X}_0\) inside \(Y_L\), for a sufficiently large finite extension \(L/K\).

The class SuperpModel is a subclass of the class SemistableModel and can be instantiated via its parent. All methods to extract information about the semistable reduction of \(Y\) are simply inherited from SemistableModel. The subclass itself only defines the methods to compute the etale locus and to create the corresponding inertail model.

AUTHORS:

Stefan Wewers (2017-07-29): initial version

EXAMPLES:

sage: from mclf import *
sage: R.<x> = QQ[]
sage: f = x^4 + x^2 + 1
sage: Y = SuperellipticCurve(f, 3)
sage: Y
superelliptic curve y^3 = x^4 + x^2 + 1 over Rational Field
sage: v_3 = QQ.valuation(3)
sage: YY = SuperpModel(Y, v_3)
sage: YY
semistable model of superelliptic curve Y: y^3 = x^4 + x^2 + 1 over Rational Field, with respect to 3-adic valuation
sage: YY.etale_locus()
Affinoid with 3 components:
Elementary affinoid defined by
v(x) >= 3/4
Elementary affinoid defined by
v(x - 2) >= 5/4
Elementary affinoid defined by
v(x + 2) >= 5/4

sage: YY.is_semistable()
True
sage: YY.components()
[the smooth projective curve with Function field in u1 defined by u1^3 + 2*x^4 + 2*x^2 + 2,
 the smooth projective curve with Function field in u2 defined by u2^3 + u2 + 2*x^2,
 the smooth projective curve with Function field in u2 defined by u2^3 + u2 + 2*x^2,
 the smooth projective curve with Function field in u2 defined by u2^3 + u2 + 2*x^2]
sage: YY.conductor_exponent()
12

We check that issues #39 and #40 have been fixed:

sage: v_2 = QQ.valuation(2)
sage: f =  x^5 - 5*x^4 + 3*x^3 - 3*x^2 + 4*x - 1
sage: Y = SuperellipticCurve(f, 2)
sage: Y2 = SemistableModel(Y, v_2)
sage: Y2.etale_locus()
Affinoid with 2 components:
Elementary affinoid defined by
v(x + 1) >= 2/3
Elementary affinoid defined by
v(x^4 +  4*x + 4) >= 8/3
sage: Y2.is_semistable()
True

TO DO:

more doctests

class mclf.semistable_reduction.superp_models.SuperpModel(Y, vK)¶

Bases: mclf.semistable_reduction.semistable_models.SemistableModel

Return a semistable model of a superelliptic curve of degree \(p\).

INPUT:

Y – a superelliptic curve over a field \(K\)
vK – a discrete valuation on \(K\)

The field \(K\) must be of characteristic \(0\) and the residue characteristic of vK must be a prime \(p\) which is equal to the covering degree of \(Y\).

OUTPUT: the object representing a semistable model of \(Y\).

Note

For the time being, we need to make the following additional assumptions on the curve \(Y\):

the polynomial \(f\) which is part of the defining equation \(y^p = f(x)\) is of degree prime to \(p\).

This restrictions is preliminary and will be removed in a future version. Note that a superelliptic curve of degree \(p\) can be written in the required form if and only if the map \(Y\to X\) has a \(K\)-rational branch point.

EXAMPLES:

compute_semistable_reduction()¶: Compute the semistable reduction of this curve, and report on the computation and the result.

etale_locus()¶

Return the etale locus of the cover \(Y\to X\).

OUTPUT: the etal locus, an affinoid subdomain of the Berkovich line \(X\) (the analytic space associated to the projektive line over the completion of the base field \(K\) with respect to the valuation \(v_K\)).

EXAMPLES:
sage: from mclf import *
sage: v_2 = QQ.valuation(2)
sage: R.<x> = QQ[]
sage: f = x^3 + x^2 + 1
sage: Y = SuperellipticCurve(f, 2)
sage: Y
superelliptic curve y^2 = x^3 + x^2 + 1 over Rational Field
sage: YY = SuperpModel(Y, v_2)
sage: YY.etale_locus()
Elementary affinoid defined by
v(x^4 + 4*x + 4) >= 8/3
We check Example 4.14 from [BouWe16]. The original equation is \(y^2 = f(x) = 2x^3 + x^2 + 32\), and \(f\) is not monic, as required. To fix this, we substitute \(x/2\) and multiply with \(4\). Then the new equation is \(y^2 = x^3 + x^2 + 128\):
sage: f = x^3 + x^2 + 128
sage: Y = SuperellipticCurve(f, 2)
sage: YY = SuperpModel(Y, v_2)
sage: YY.etale_locus()
Elementary affinoid defined by
v(x) >= 2
v(1/x) >= -5/2
<BLANKLINE>

Note

At the moment, the construction of the superelliptic curve \(Y\) requires that the polynomial \(f\) defining \(Y\) is monic, integral with respect to \(v_K\) and of degree prime to \(p\). The motivation for this restriction, and its result is that the etale locus is contained in the closed unit disk.

reduction_tree()¶: Return the reduction tree which determines the semistabel model.

mclf.semistable_reduction.superp_models.p_approximation(f, p)¶

Return the \(p\)-approximation of f.

INPUT:

f – a polynomial of degree \(n\) over a field \(K\), with nonvanishing

constant coefficient
p – a prime number

OUTPUT:

Two polynomials \(h\) and \(g\) in \(K[x]\), such that

\(f=a_0(h^p+g)\), where \(a_0\) is the constant coefficient of \(f\)
\(r:=deg(h)\leq n/p\), and
\(x^{(r+1)}\) divides \(g\)

Note that \(h, g\) are uniquely determined by these conditions.

mclf.semistable_reduction.superp_models.p_approximation_generic(f, p)¶

Return the generic \(p\)-approximation of f.

INPUT:

f – a polynomial of degree \(n\) over a field \(K\), with nonvanishing constant coefficient
p – a prime number

OUTPUT:

Two polynomials \(H\) and \(G\) in \(K(x)[t]\) which are the \(p\)-approximation of the polynomial \(F:=f(x+t)\), considered as polynomial in \(t\).