Points of type V on the Berkovich line

Let \(X^{an}\) be a Berkovich line over a discretely valued field \(K\). A “point” \(\eta\) of type V on \(X^{an}\) corresponds to a pair \((v,\bar{v})\), where \(v\) is a type-II-valuation and \(\bar{v}\) is a function field valuation on the residue field of \(v\). We call \(v\) the “major valuation” and \(\bar{v}\) the “minor valuation” associated to \(\eta\).

Note that \(\eta\) is not, properly speaking, a point on the analytic space \(X^{an}\), but rather a point on the adic space \(X^{ad}\).

Equivalent ways to describe \(\eta\) are:

  • the rank-2-valuation given as the composition of \(v\) and \(\bar{v}\)

  • a “residue class” on \(X^{an}\); more precisely, \(\eta\) corresponds to a connected component of \(X^{an}-\{\xi\}\), where \(\xi\) is the type-II-point corresponding to \(v\) (and then \(\xi\) is the unique boundary point of the residue class)

  • an “open discoid”: more precise, a pair \((\phi,s)\), where \(\phi\) is a rational function such that the open discoid

    \[D = \{ v \mid v(\phi) > s \}\]

    is the residue class corresponding to \(\eta\). Moreover, either \(\phi\) of \(1/\phi\) is a monic, integral and irreducible polynomial in \(x\) or in \(1/x\).

  • a “tangent vector” on \(X^{an}\); more precisely a group homomorphism

    \[\partial: K(x)^* \to \mathbb{Z}\]

with the following properties: let \((\phi,s)\) be the discoid representation of \(\eta\). We define, for \(t\geq s\), the valuation \(v_t\) as the valuation corresponding to the boundary point of the open discoid \(v(\phi)>t\). Then \(\partial(f)\) is the right derivative at \(t=s\) of the function

\[t \mapsto v_t(f).\]

The most convenient way to determine a point of type V is as follows. Let \(\xi_1\) be a point of type II and \(\xi_2\) be of type I or II, distinct from \(\xi_1\). Then

\[\eta = \eta(\xi_1,\xi_2)\]

is the point of type V corresponding to the connected component of \(X-\{\xi_1\}\) containing \(\xi_2\). We call \(\eta\) the direction from \(\xi_1\) towards \(\xi_2\).

class mclf.berkovich.type_V_points.TypeVPointOnBerkovichLine(xi0, xi1)

Bases: SageObject

A point of type V on the Berkovich line.

Let \(\xi_1\) be a point of type II, and \(\xi_2\) a point of type I or II. Then we can define the point of type V \(\eta:=\eta(\xi_1,\xi_2)\) as the unique residue class with boundary point \(\xi_1\) containing \(\xi_2\).

INPUT:

  • xi0 – point of type II
  • xi1 – arbitrary point of X, distinct from xi0

OUTPUT:

The type-V-point corresponding to the connected component of \(X^{an}-{\xi0}\) which contains \(\xi1\).

EXAMPLES:

sage: from mclf import *
sage: K = QQ
sage: vK = K.valuation(2)
sage: F.<x> = FunctionField(K)
sage: X = BerkovichLine(F, vK)
sage: xi1 = X.point_from_discoid(x,1)
sage: xi2 = X.point_from_discoid(x^2+4,3)
sage: eta = TypeVPointOnBerkovichLine(xi1, xi2)

We see that eta represents an open disk inside the closed unit disk.

sage: eta
Point of type V given by residue class v(x + 2) > 1

Here is an example of a type-V-point representing an open disk in the complement of the closed unit disk:

sage: xi0 = X.gauss_point()
sage: xi3 = X.point_from_discoid(2*x+1, 2)
sage: eta = TypeVPointOnBerkovichLine(xi0, xi3)
sage: eta
Point of type V given by residue class v(1/x) > 0

We check that xi0 lies outside the open disk and xi3 inside:

sage: eta.is_in_residue_class(xi0)
False
sage: eta.is_in_residue_class(xi3)
True

sage: xi4 = X.point_from_discoid(2*x+1, 4)
sage: TypeVPointOnBerkovichLine(xi3, xi4)
Point of type V given by residue class v((2*x + 1)/x) > 3
sage: TypeVPointOnBerkovichLine(xi4, xi3)
Point of type V given by residue class v(1/2*x/(x + 1/2)) > -5

The following example shows that the minor valuation is computed correctly

sage: xi5 = X.point_from_discoid(1/x,1)
sage: eta = TypeVPointOnBerkovichLine(xi0,xi5)
sage: eta
Point of type V given by residue class v(1/x) > 0
sage: eta.minor_valuation()
Valuation at the infinite place
berkovich_line()

Return the Berkovich line underlying the point.

boundary_point()

Return the boundary point of the type-V-point.

minor_valuation()

Return the minor valuation of this type V point.

open_discoid()

Return the representation of self as an open discoid.

INPUT:

  • self: a point of type V on a Berkovich line

OUTPUT:

a pair \((\phi, s)\), where \(\phi\) is a rational function and \(s\) a rational number is such that

\[D = \{ v\in X \mid v(\phi) > s \}\]

is the open discoid representing the type-V-point self.

Either \(\phi\) of \(1/\phi\) is a monic, integral and strongly irreducible polynomial in \(x\) or in \(1/x\).

point_inside_discoid(t)

Return the point inside the residue class at the value \(t\).

The type-V-point corresponds to an open discoid defined by

\[v(\phi) > s.\]

For for a rational number \(t>s\) we can define the type-II-point \(\xi_t\) corresponding to the closed discoid defined by

\[v(\phi) >= t.\]

If \(t=\infty\) we obtain the type-I-point corresponding to \(\phi=0\).

INPUT:

  • t – a rational number or Infinity

OUTPUT:

The point \(\xi_t\) inside the residue class corresponding to the closed discoid defined by \(v(\phi) >= t\).

If \(t <= s\) then an error is raised.