Wehler K3 Surfaces

AUTHORS:

  • Ben Hutz (11-2012)
  • Joao Alberto de Faria (10-2013)

Todo

Hasse-Weil Zeta Function

Picard Number

Number Fields

REFERENCES: [FH2015], [CS1996], [Weh1998], [Hutz2007]

sage.dynamics.arithmetic_dynamics.wehlerK3.WehlerK3Surface(polys)

Defines a K3 Surface over \(\mathbb{P}^2 \times \mathbb{P}^2\) defined as the intersection of a bilinear and biquadratic form. [Weh1998]

INPUT: Bilinear and biquadratic polynomials as a tuple or list

OUTPUT: WehlerK3Surface_ring

EXAMPLES:

sage: PP.<x0,x1, x2, y0, y1, y2> = ProductProjectiveSpaces([2, 2],QQ)
sage: L = x0*y0 + x1*y1 - x2*y2
sage: Q = x0*x1*y1^2 + x2^2*y0*y2
sage: WehlerK3Surface([L, Q])
Closed subscheme of Product of projective spaces P^2 x P^2 over Rational
Field defined by:
 x0*y0 + x1*y1 - x2*y2,
 x0*x1*y1^2 + x2^2*y0*y2
class sage.dynamics.arithmetic_dynamics.wehlerK3.WehlerK3Surface_field(polys)

Bases: sage.dynamics.arithmetic_dynamics.wehlerK3.WehlerK3Surface_ring

class sage.dynamics.arithmetic_dynamics.wehlerK3.WehlerK3Surface_finite_field(polys)

Bases: sage.dynamics.arithmetic_dynamics.wehlerK3.WehlerK3Surface_field

cardinality()

Counts the total number of points on the K3 surface.

ALGORITHM:

Enumerate points over \(\mathbb{P}^2\), and then count the points on the fiber of each of those points.

OUTPUT: Integer - total number of points on the surface

EXAMPLES:

sage: PP.<x0,x1,x2,y0,y1,y2> = ProductProjectiveSpaces([2, 2], GF(7))
sage: Z = x0^2*y0^2 + 3*x0*x1*y0^2 + x1^2*y0^2 + 4*x0^2*y0*y1 + \
3*x0*x1*y0*y1 -2*x2^2*y0*y1 - x0^2*y1^2 + 2*x1^2*y1^2 - x0*x2*y1^2 \
- 4*x1*x2*y1^2 + 5*x0*x2*y0*y2 -4*x1*x2*y0*y2 + 7*x0^2*y1*y2 + 4*x1^2*y1*y2 \
+ x0*x1*y2^2 + 3*x2^2*y2^2
sage: Y = x0*y0 + x1*y1 + x2*y2
sage: X = WehlerK3Surface([Z, Y])
sage: X.cardinality()
55
class sage.dynamics.arithmetic_dynamics.wehlerK3.WehlerK3Surface_ring(polys)

Bases: sage.schemes.product_projective.subscheme.AlgebraicScheme_subscheme_product_projective

A K3 surface in \(\mathbb{P}^2 \times \mathbb{P}^2\) defined as the intersection of a bilinear and biquadratic form. [Weh1998]

EXAMPLES:

sage: R.<x,y,z,u,v,w> = PolynomialRing(QQ, 6)
sage: L = x*u - y*v
sage: Q = x*y*v^2 + z^2*u*w
sage: WehlerK3Surface([L, Q])
Closed subscheme of Product of projective spaces P^2 x P^2 over Rational
Field defined by:
  x*u - y*v,
  x*y*v^2 + z^2*u*w
Gpoly(component, k)

Returns the G polynomials \(G^*_k\).

They are defined as: \(G^*_k = \left(L^*_j\right)^2Q^*_{ii}-L^*_iL^*_jQ^*_{ij}+\left(L^*_i\right)^2Q^*_{jj}\)where {i, j, k} is some permutation of (0, 1, 2) and * is either x (Component = 1) or y (Component = 0).

INPUT:

  • component - Integer: 0 or 1
  • k - Integer: 0, 1 or 2

OUTPUT: polynomial in terms of either y (Component = 0) or x (Component = 1)

EXAMPLES:

sage: R.<x0,x1,x2,y0,y1,y2> = PolynomialRing(ZZ, 6)
sage: Y = x0*y0 + x1*y1 - x2*y2
sage: Z = x0^2*y0*y1 + x0^2*y2^2 - x0*x1*y1*y2 + x1^2*y2*y1 \
+ x2^2*y2^2 + x2^2*y1^2 + x1^2*y2^2
sage: X = WehlerK3Surface([Z, Y])
sage: X.Gpoly(1, 0)
x0^2*x1^2 + x1^4 - x0*x1^2*x2 + x1^3*x2 + x1^2*x2^2 + x2^4
Hpoly(component, i, j)

Returns the H polynomials defined as \(H^*_{ij}\).

This polynomial is defined by:

\(H^*_{ij} = 2L^*_iL^*_jQ^*_{kk}-L^*_iL^*_kQ^*_{jk} - L^*_jL^*_kQ^*_{ik}+\left(L^*_k\right)^2Q^*_{ij}\) where {i, j, k} is some permutation of (0, 1, 2) and * is either y (Component = 0) or x (Component = 1).

INPUT:

  • component - Integer: 0 or 1
  • i - Integer: 0, 1 or 2
  • j - Integer: 0, 1 or 2

OUTPUT: polynomial in terms of either y (Component = 0) or x (Component = 1)

EXAMPLES:

sage: R.<x0,x1,x2,y0,y1,y2> = PolynomialRing(ZZ, 6)
sage: Y = x0*y0 + x1*y1 - x2*y2
sage: Z = x0^2*y0*y1 + x0^2*y2^2 - x0*x1*y1*y2 + x1^2*y2*y1 \
+ x2^2*y2^2 + x2^2*y1^2 + x1^2*y2^2
sage: X = WehlerK3Surface([Z, Y])
sage: X.Hpoly(0, 1, 0)
 2*y0*y1^3 + 2*y0*y1*y2^2 - y1*y2^3
Lxa(a)

Function will return the L polynomial defining the fiber, given by \(L^{x}_{a}\).

This polynomial is defined as:

\(L^{x}_{a} = \{(a, y) \in \mathbb{P}^{2} \times \mathbb{P}^{2} \colon L(a, y) = 0\}\).

Notation and definition from: [CS1996]

INPUT: a - Point in \(\mathbb{P}^2\)

OUTPUT: A polynomial representing the fiber

EXAMPLES:

sage: PP.<x0,x1,x2,y0,y1,y2> = ProductProjectiveSpaces([2, 2], QQ)
sage: Z = x0^2*y0^2 + 3*x0*x1*y0^2 + x1^2*y0^2 + 4*x0^2*y0*y1 \
+ 3*x0*x1*y0*y1 - 2*x2^2*y0*y1 - \
x0^2*y1^2 + 2*x1^2*y1^2 - x0*x2*y1^2 - 4*x1*x2*y1^2 \
+ 5*x0*x2*y0*y2 - 4*x1*x2*y0*y2 + 7*x0^2*y1*y2 \
+ 4*x1^2*y1*y2 + x0*x1*y2^2 + 3*x2^2*y2^2
sage: Y = x0*y0 + x1*y1 + x2*y2
sage: X = WehlerK3Surface([Z, Y])
sage: T = PP(1, 1, 0, 1, 0, 0)
sage: X.Lxa(T[0])
y0 + y1
Lyb(b)

Function will return a fiber by \(L^{y}_{b}\).

This polynomial is defined as:

\(L^{y}_{b} = \{(x,b) \in \mathbb{P}^{2} \times \mathbb{P}^{2} \colon L(x,b) = 0\}\).

Notation and definition from: [CS1996]

INPUT: b - Point in projective space

OUTPUT: A polynomial representing the fiber

EXAMPLES:

sage: PP.<x0,x1,x2,y0,y1,y2> = ProductProjectiveSpaces([2, 2], QQ)
sage: Z =x0^2*y0^2 + 3*x0*x1*y0^2 + x1^2*y0^2 + 4*x0^2*y0*y1 \
+ 3*x0*x1*y0*y1 \
- 2*x2^2*y0*y1 - x0^2*y1^2 + 2*x1^2*y1^2 - x0*x2*y1^2 - 4*x1*x2*y1^2 \
+ 5*x0*x2*y0*y2 \
- 4*x1*x2*y0*y2 + 7*x0^2*y1*y2 + 4*x1^2*y1*y2 + x0*x1*y2^2 + 3*x2^2*y2^2
sage: Y = x0*y0 + x1*y1 + x2*y2
sage: Y = x0*y0 + x1*y1 + x2*y2
sage: X = WehlerK3Surface([Z, Y])
sage: T = PP(1, 1, 0, 1, 0, 0)
sage: X.Lyb(T[1])
  x0
Qxa(a)

Function will return the Q polynomial defining a fiber given by \(Q^{x}_{a}\).

This polynomial is defined as:

\(Q^{x}_{a} = \{(a,y) \in \mathbb{P}^{2} \times \mathbb{P}^{2} \colon Q(a,y) = 0\}\).

Notation and definition from: [CS1996]

INPUT: a - Point in \(\mathbb{P}^2\)

OUTPUT: A polynomial representing the fiber

EXAMPLES:

sage: PP.<x0,x1,x2,y0,y1,y2> = ProductProjectiveSpaces([2, 2], QQ)
sage: Z = x0^2*y0^2 + 3*x0*x1*y0^2 + x1^2*y0^2 + 4*x0^2*y0*y1 + 3*x0*x1*y0*y1 \
- 2*x2^2*y0*y1 - x0^2*y1^2 + 2*x1^2*y1^2 - x0*x2*y1^2 - 4*x1*x2*y1^2 \
+ 5*x0*x2*y0*y2 \
- 4*x1*x2*y0*y2 + 7*x0^2*y1*y2 + 4*x1^2*y1*y2 + x0*x1*y2^2 + 3*x2^2*y2^2 \
sage: Y = x0*y0 + x1*y1 + x2*y2
sage: X = WehlerK3Surface([Z, Y])
sage: T = PP(1, 1, 0, 1, 0, 0)
sage: X.Qxa(T[0])
5*y0^2 + 7*y0*y1 + y1^2 + 11*y1*y2 + y2^2
Qyb(b)

Function will return a fiber by \(Q^{y}_{b}\).

This polynomial is defined as:

\(Q^{y}_{b} = \{(x,b) \in \mathbb{P}^{2} \times \mathbb{P}^{2} \colon Q(x,b) = 0\}\).

Notation and definition from: [CS1996]

INPUT: b - Point in projective space

OUTPUT: A polynomial representing the fiber

EXAMPLES:

sage: PP.<x0,x1,x2,y0,y1,y2> = ProductProjectiveSpaces([2, 2], QQ)
sage: Z = x0^2*y0^2 + 3*x0*x1*y0^2 + x1^2*y0^2 + 4*x0^2*y0*y1 \
+ 3*x0*x1*y0*y1 - 2*x2^2*y0*y1 - x0^2*y1^2 + 2*x1^2*y1^2 - x0*x2*y1^2 \
- 4*x1*x2*y1^2 + 5*x0*x2*y0*y2 - 4*x1*x2*y0*y2 + 7*x0^2*y1*y2 \
+ 4*x1^2*y1*y2 + x0*x1*y2^2 + 3*x2^2*y2^2
sage: Y = x0*y0 + x1*y1 + x2*y2
sage: X = WehlerK3Surface([Z, Y])
sage: T = PP(1, 1, 0, 1, 0, 0)
sage: X.Qyb(T[1])
x0^2 + 3*x0*x1 + x1^2
Ramification_poly(i)

Function will return the Ramification polynomial \(g^*\).

This polynomial is defined by:

\(g^* = \frac{\left(H^*_{ij}\right)^2 - 4G^*_iG^*_j}{\left(L^*_k\right)^2}\).

The roots of this polynomial will either be degenerate fibers or fixed points of the involutions \(\sigma_x\) or \(\sigma_y\) for more information, see [CS1996].

INPUT: i - Integer, either 0 (polynomial in y) or 1 (polynomial in x)

OUTPUT: Polynomial in the coordinate ring of the ambient space

EXAMPLES:

sage: PP.<x0,x1,x2,y0,y1,y2> = ProductProjectiveSpaces([2, 2], QQ)
sage: Z = x0^2*y0^2 + 3*x0*x1*y0^2 + x1^2*y0^2 + 4*x0^2*y0*y1 + 3*x0*x1*y0*y1\
- 2*x2^2*y0*y1 - x0^2*y1^2 + 2*x1^2*y1^2 - x0*x2*y1^2 - 4*x1*x2*y1^2 + 5*x0*x2*y0*y2\
- 4*x1*x2*y0*y2 + 7*x0^2*y1*y2 + 4*x1^2*y1*y2 + x0*x1*y2^2 + 3*x2^2*y2^2
sage: Y = x0*y0 + x1*y1 + x2*y2
sage: X = WehlerK3Surface([Z, Y])
sage: X.Ramification_poly(0)
8*y0^5*y1 - 24*y0^4*y1^2 + 48*y0^2*y1^4 - 16*y0*y1^5 + y1^6 + 84*y0^3*y1^2*y2
+ 46*y0^2*y1^3*y2 - 20*y0*y1^4*y2 + 16*y1^5*y2 + 53*y0^4*y2^2 + 56*y0^3*y1*y2^2
- 32*y0^2*y1^2*y2^2 - 80*y0*y1^3*y2^2 - 92*y1^4*y2^2 - 12*y0^2*y1*y2^3
- 168*y0*y1^2*y2^3 - 122*y1^3*y2^3 + 14*y0^2*y2^4 + 8*y0*y1*y2^4 - 112*y1^2*y2^4 + y2^6
Sxa(a)

Function will return fiber by \(S^{x}_{a}\).

This function is defined as:

\(S^{x}_{a} = L^{x}_{a} \cap Q^{x}_{a}\).

Notation and definition from: [CS1996]

INPUT: a - Point in \(\mathbb{P}^2\)

OUTPUT: A subscheme representing the fiber

EXAMPLES:

sage: PP.<x0,x1,x2,y0,y1,y2> = ProductProjectiveSpaces([2, 2], QQ)
sage: Z = x0^2*y0^2 + 3*x0*x1*y0^2 + x1^2*y0^2 + 4*x0^2*y0*y1 \
+ 3*x0*x1*y0*y1 \
- 2*x2^2*y0*y1 - x0^2*y1^2 + 2*x1^2*y1^2 - x0*x2*y1^2 - 4*x1*x2*y1^2 \
+ 5*x0*x2*y0*y2 \
- 4*x1*x2*y0*y2 + 7*x0^2*y1*y2 + 4*x1^2*y1*y2 + x0*x1*y2^2 + 3*x2^2*y2^2
sage: Y = x0*y0 + x1*y1 + x2*y2
sage: Y = x0*y0 + x1*y1 + x2*y2
sage: X = WehlerK3Surface([Z, Y])
sage: T = PP(1, 1, 0, 1, 0, 0)
sage: X.Sxa(T[0])
  Closed subscheme of Projective Space of dimension 2 over Rational Field defined by:
   y0 + y1,
   5*y0^2 + 7*y0*y1 + y1^2 + 11*y1*y2 + y2^2
Syb(b)

Function will return fiber by \(S^{y}_{b}\).

This function is defined by:

\(S^{y}_{b} = L^{y}_{b} \cap Q^{y}_{b}\).

Notation and definition from: [CS1996]

INPUT: b - Point in \(\mathbb{P}^2\)

OUTPUT: A subscheme representing the fiber

EXAMPLES:

sage: PP.<x0,x1,x2,y0,y1,y2> = ProductProjectiveSpaces([2, 2], QQ)
sage: Z = x0^2*y0^2 + 3*x0*x1*y0^2 + x1^2*y0^2 + 4*x0^2*y0*y1 + \
3*x0*x1*y0*y1 - 2*x2^2*y0*y1 - x0^2*y1^2 + 2*x1^2*y1^2 - x0*x2*y1^2 \
- 4*x1*x2*y1^2 + 5*x0*x2*y0*y2 - 4*x1*x2*y0*y2 + 7*x0^2*y1*y2 \
+ 4*x1^2*y1*y2 + x0*x1*y2^2 + 3*x2^2*y2^2
sage: Y = x0 * y0 + x1 * y1 + x2 * y2
sage: X = WehlerK3Surface([Z, Y])
sage: T = PP(1, 1, 0, 1, 0, 0)
sage: X.Syb(T[1])
 Closed subscheme of Projective Space of dimension 2 over Rational Field defined by:
   x0,
   x0^2 + 3*x0*x1 + x1^2
canonical_height(P, N, badprimes=None, prec=100)

Evaluates the canonical height for P with N terms of the series of the local heights.

ALGORITHM:

The sum of the canonical height minus and canonical height plus, for more info see section 4 of [CS1996].

INPUT:

  • P – a surface point
  • N – positive integer (number of terms of the series to use)
  • badprimes – (optional) list of integer primes (where the surface is degenerate)
  • prec – (default: 100) float point or p-adic precision

OUTPUT: A real number

EXAMPLES:

sage: set_verbose(None)
sage: R.<x0,x1,x2,y0,y1,y2> = PolynomialRing(QQ, 6)
sage: L =  (-y0 - y1)*x0 + (-y0*x1 - y2*x2)
sage: Q = (-y2*y0 - y1^2)*x0^2 + ((-y0^2 - y2*y0 + (-y2*y1 - y2^2))*x1 + \
(-y0^2 - y2*y1)*x2)*x0 + ((-y0^2 - y2*y0 - y2^2)*x1^2 + (-y2*y0 - y1^2)*x2*x1 \
+ (-y0^2 + (-y1 - y2)*y0)*x2^2)
sage: X = WehlerK3Surface([L, Q])
sage: P = X([1, 0, -1, 1,- 1, 0]) #order 16
sage: X.canonical_height(P, 5)  # long time
0.00000000000000000000000000000

Call-Silverman example:

sage: set_verbose(None)
sage: PP.<x0,x1,x2,y0,y1,y2> = ProductProjectiveSpaces([2, 2], QQ)
sage: Z = x0^2*y0^2 + 3*x0*x1*y0^2 + x1^2*y0^2 + 4*x0^2*y0*y1 + 3*x0*x1*y0*y1 - \
2*x2^2*y0*y1 - x0^2*y1^2 + 2*x1^2*y1^2 - x0*x2*y1^2 - 4*x1*x2*y1^2 + 5*x0*x2*y0*y2 \
-4*x1*x2*y0*y2 + 7*x0^2*y1*y2 + 4*x1^2*y1*y2 + x0*x1*y2^2 + 3*x2^2*y2^2
sage: Y = x0*y0 + x1*y1 + x2*y2
sage: X = WehlerK3Surface([Z, Y])
sage: P = X(0, 1, 0, 0, 0, 1)
sage: X.canonical_height(P, 4)
0.69826458668659859569990618895
canonical_height_minus(P, N, badprimes=None, prec=100)

Evaluates the canonical height minus function of Call-Silverman for P with N terms of the series of the local heights.

Must be over \(\ZZ\) or \(\QQ\).

ALGORITHM:

Sum over the lambda minus heights (local heights) in a convergent series, for more detail see section 7 of [CS1996].

INPUT:

  • P – a surface point
  • N – positive integer (number of terms of the series to use)
  • badprimes – (optional) list of integer primes (where the surface is degenerate)
  • prec – (default: 100) float point or p-adic precision

OUTPUT: A real number

EXAMPLES:

sage: set_verbose(None)
sage: R.<x0,x1,x2,y0,y1,y2> = PolynomialRing(QQ, 6)
sage: L =  (-y0 - y1)*x0 + (-y0*x1 - y2*x2)
sage: Q = (-y2*y0 - y1^2)*x0^2 + ((-y0^2 - y2*y0 + (-y2*y1 - y2^2))*x1\
 + (-y0^2 - y2*y1)*x2)*x0 + ((-y0^2 - y2*y0 - y2^2)*x1^2 + (-y2*y0 - y1^2)*x2*x1\
  + (-y0^2 + (-y1 - y2)*y0)*x2^2)
sage: X = WehlerK3Surface([L, Q])
sage: P = X([1, 0, -1, 1, -1, 0]) #order 16
sage: X.canonical_height_minus(P, 5)  # long time
0.00000000000000000000000000000

Call-Silverman example:

sage: set_verbose(None)
sage: PP.<x0,x1,x2,y0,y1,y2> = ProductProjectiveSpaces([2, 2], QQ)
sage: Z = x0^2*y0^2 + 3*x0*x1*y0^2 + x1^2*y0^2 + 4*x0^2*y0*y1 +\
 3*x0*x1*y0*y1 - 2*x2^2*y0*y1 - x0^2*y1^2 + 2*x1^2*y1^2 - x0*x2*y1^2 - \
 4*x1*x2*y1^2 + 5*x0*x2*y0*y2 - 4*x1*x2*y0*y2 + 7*x0^2*y1*y2 + 4*x1^2*y1*y2 + \
 x0*x1*y2^2 + 3*x2^2*y2^2
sage: Y = x0*y0 + x1*y1 + x2*y2
sage: X = WehlerK3Surface([Z, Y])
sage: P = X([0, 1, 0, 0, 0, 1])
sage: X.canonical_height_minus(P, 4) # long time
0.55073705369676788175590206734
canonical_height_plus(P, N, badprimes=None, prec=100)

Evaluates the canonical height plus function of Call-Silverman for P with N terms of the series of the local heights.

Must be over \(\ZZ\) or \(\QQ\).

ALGORITHM:

Sum over the lambda plus heights (local heights) in a convergent series, for more detail see section 7 of [CS1996].

INPUT:

  • P – a surface point
  • N – positive integer. Number of terms of the series to use
  • badprimes – (optional) list of integer primes (where the surface is degenerate)
  • prec – (default: 100) float point or p-adic precision

OUTPUT: A real number

EXAMPLES:

sage: set_verbose(None)
sage: R.<x0,x1,x2,y0,y1,y2> = PolynomialRing(QQ, 6)
sage: L =  (-y0 - y1)*x0 + (-y0*x1 - y2*x2)
sage: Q = (-y2*y0 - y1^2)*x0^2 + ((-y0^2 - y2*y0 + (-y2*y1 - y2^2))*x1 + \
(-y0^2 - y2*y1)*x2)*x0 + ((-y0^2 - y2*y0 - y2^2)*x1^2 + (-y2*y0 - y1^2)*x2*x1 \
+ (-y0^2 + (-y1 - y2)*y0)*x2^2)
sage: X = WehlerK3Surface([L, Q])
sage: P = X([1, 0, -1, 1, -1, 0]) #order 16
sage: X.canonical_height_plus(P, 5)  # long time
0.00000000000000000000000000000

Call-Silverman Example:

sage: set_verbose(None)
sage: PP.<x0,x1,x2,y0,y1,y2> = ProductProjectiveSpaces([2, 2], QQ)
sage: Z = x0^2*y0^2 + 3*x0*x1*y0^2 + x1^2*y0^2 + 4*x0^2*y0*y1 + \
3*x0*x1*y0*y1 - 2*x2^2*y0*y1 - x0^2*y1^2 + 2*x1^2*y1^2 - x0*x2*y1^2 \
- 4*x1*x2*y1^2 + 5*x0*x2*y0*y2 -4*x1*x2*y0*y2 + 7*x0^2*y1*y2 + 4*x1^2*y1*y2 \
+ x0*x1*y2^2 + 3*x2^2*y2^2
sage: Y = x0*y0 + x1*y1 + x2*y2
sage: X = WehlerK3Surface([Z, Y])
sage: P = X([0, 1, 0, 0, 0, 1])
sage: X.canonical_height_plus(P, 4) # long time
0.14752753298983071394400412161
change_ring(R)

Changes the base ring on which the Wehler K3 Surface is defined.

INPUT: R - ring

OUTPUT: K3 Surface defined over input ring

EXAMPLES:

sage: PP.<x0,x1,x2,y0,y1,y2> = ProductProjectiveSpaces([2, 2], GF(3))
sage: L = x0*y0 + x1*y1 - x2*y2
sage: Q = x0*x1*y1^2 + x2^2*y0*y2
sage: W = WehlerK3Surface([L, Q])
sage: W.base_ring()
Finite Field of size 3
sage: T = W.change_ring(GF(7))
sage: T.base_ring()
Finite Field of size 7
degenerate_fibers()

Function will return the (rational) degenerate fibers of the surface defined over the base ring, or the fraction field of the base ring if it is not a field.

ALGORITHM:

The criteria for degeneracy by the common vanishing of the polynomials self.Gpoly(1, 0), self.Gpoly(1, 1), self.Gpoly(1, 2), self.Hpoly(1, 0, 1),``self.Hpoly(1, 0, 2)``, self.Hpoly(1, 1, 2) (for the first component), is from Proposition 1.4 in the following article: [CS1996].

This function finds the common solution through elimination via Groebner bases by using the .variety() function on the three affine charts in each component.

OUTPUT: The output is a list of lists where the elements of lists are points in the appropriate projective space. The first list is the points whose pullback by the projection to the first component (projective space) is dimension greater than 0. The second list is points in the second component

EXAMPLES:

sage: R.<x0,x1,x2,y0,y1,y2> = PolynomialRing(ZZ, 6)
sage: Y = x0*y0 + x1*y1 - x2*y2
sage: Z = x0^2*y0*y1 + x0^2*y2^2 - x0*x1*y1*y2 + x1^2*y2*y1 + x2^2*y2^2\
+ x2^2*y1^2 + x1^2*y2^2
sage: X = WehlerK3Surface([Z, Y])
sage: X.degenerate_fibers()
[[], [(1 : 0 : 0)]]
sage: PP.<x0,x1,x2,y0,y1,y2> = ProductProjectiveSpaces([2, 2], QQ)
sage: Z = x0^2*y0^2 + 3*x0*x1*y0^2 + x1^2*y0^2 + 4*x0^2*y0*y1 + 3*x0*x1*y0*y1\
- 2*x2^2*y0*y1 - x0^2*y1^2 + 2*x1^2*y1^2 - x0*x2*y1^2 - 4*x1*x2*y1^2 + 5*x0*x2*y0*y2\
- 4*x1*x2*y0*y2 + 7*x0^2*y1*y2 + 4*x1^2*y1*y2 + x0*x1*y2^2 + 3*x2^2*y2^2
sage: Y = x0*y0 + x1*y1 + x2*y2
sage: X = WehlerK3Surface([Z, Y])
sage: X.degenerate_fibers()
[[], []]
sage: PP.<x0,x1,x2,y0,y1,y2> = ProductProjectiveSpaces([2, 2], QQ)
sage: R = PP.coordinate_ring()
sage: l = y0*x0 + y1*x1 + (y0 - y1)*x2
sage: q = (y1*y0 + y2^2)*x0^2 + ((y0^2 - y2*y1)*x1 + (y0^2 + (y1^2 - y2^2))*x2)*x0 \
+ (y2*y0 + y1^2)*x1^2 + (y0^2 + (-y1^2 + y2^2))*x2*x1
sage: X = WehlerK3Surface([l,q])
sage: X.degenerate_fibers()
[[(-1 : 1 : 1), (0 : 0 : 1)], [(-1 : -1 : 1), (0 : 0 : 1)]]
degenerate_primes(check=True)

Determine which primes \(p\) self has degenerate fibers over \(GF(p)\).

If check is False, then may return primes that do not have degenerate fibers. Raises an error if the surface is degenerate. Works only for ZZ or QQ.

INPUT: check – (default: True) boolean, whether the primes are verified

ALGORITHM:

\(p\) is a prime of bad reduction if and only if the defining polynomials of self plus the G and H polynomials have a common zero. Or stated another way, \(p\) is a prime of bad reduction if and only if the radical of the ideal defined by the defining polynomials of self plus the G and H polynomials is not \((x_0,x_1,\ldots,x_N)\). This happens if and only if some power of each \(x_i\) is not in the ideal defined by the defining polynomials of self (with G and H). This last condition is what is checked. The lcm of the coefficients of the monomials \(x_i\) in a groebner basis is computed. This may return extra primes.

OUTPUT: List of primes.

EXAMPLES:

sage: R.<x0,x1,x2,y0,y1,y2> = PolynomialRing(QQ, 6)
sage: L =  y0*x0 + (y1*x1 + y2*x2)
sage: Q = (2*y0^2 + y2*y0 + (2*y1^2 + y2^2))*x0^2 + ((y0^2 + y1*y0 + \
(y1^2 + 2*y2*y1 + y2^2))*x1 + (2*y1^2 + y2*y1 + y2^2)*x2)*x0 + ((2*y0^2\
+ (y1 + 2*y2)*y0 + (2*y1^2 + y2*y1))*x1^2 + ((2*y1 + 2*y2)*y0 + (y1^2 + \
y2*y1 + 2*y2^2))*x2*x1 + (2*y0^2 + y1*y0 + (2*y1^2 + y2^2))*x2^2)
sage: X = WehlerK3Surface([L, Q])
sage: X.degenerate_primes()
[2, 3, 5, 11, 23, 47, 48747691, 111301831]
fiber(p, component)

Returns the fibers [y (component = 1) or x (Component = 0)] of a point on a K3 Surface, will work for nondegenerate fibers only.

For algorithm, see [Hutz2007].

INPUT:

-p - a point in \(\mathbb{P}^2\)

OUTPUT: The corresponding fiber (as a list)

EXAMPLES:

sage: R.<x0,x1,x2,y0,y1,y2> = PolynomialRing(ZZ, 6)
sage: Y = x0*y0 + x1*y1 - x2*y2
sage: Z = y0^2*x0*x1 + y0^2*x2^2 - y0*y1*x1*x2 + y1^2*x2*x1 + y2^2*x2^2 +\
y2^2*x1^2 + y1^2*x2^2
sage: X = WehlerK3Surface([Z, Y])
sage: Proj = ProjectiveSpace(QQ, 2)
sage: P = Proj([1, 0, 0])
sage: X.fiber(P, 1)
Traceback (most recent call last):
...
TypeError: fiber is degenerate
sage: P.<x0,x1,x2,y0,y1,y2> = ProductProjectiveSpaces([2, 2], QQ)
sage: Z = x0^2*y0^2 + 3*x0*x1*y0^2 + x1^2*y0^2 + 4*x0^2*y0*y1 + 3*x0*x1*y0*y1 - \
2*x2^2*y0*y1 - x0^2*y1^2 + 2*x1^2*y1^2 - x0*x2*y1^2 -4*x1*x2*y1^2 + 5*x0*x2*y0*y2 - \
4*x1*x2*y0*y2 + 7*x0^2*y1*y2 + 4*x1^2*y1*y2 + x0*x1*y2^2 + 3*x2^2*y2^2
sage: Y = x0*y0 + x1*y1 + x2*y2
sage: X = WehlerK3Surface([Z, Y])
sage: Proj = P[0]
sage: T = Proj([0, 0, 1])
sage: X.fiber(T, 1)
[(0 : 0 : 1 , 0 : 1 : 0), (0 : 0 : 1 , 2 : 0 : 0)]
sage: PP.<x0,x1,x2,y0,y1,y2> = ProductProjectiveSpaces([2, 2], GF(7))
sage: L = x0*y0 + x1*y1 - 1*x2*y2
sage: Q=(2*x0^2 + x2*x0 + (2*x1^2 + x2^2))*y0^2 + ((x0^2 + x1*x0 +(x1^2 + 2*x2*x1 + x2^2))*y1 + \
(2*x1^2 + x2*x1 + x2^2)*y2)*y0 + ((2*x0^2+ (x1 + 2*x2)*x0 + (2*x1^2 + x2*x1))*y1^2 + ((2*x1 + 2*x2)*x0 + \
(x1^2 +x2*x1 + 2*x2^2))*y2*y1 + (2*x0^2 + x1*x0 + (2*x1^2 + x2^2))*y2^2)
sage: W = WehlerK3Surface([L, Q])
sage: W.fiber([4, 0, 1], 0)
[(0 : 1 : 0 , 4 : 0 : 1), (4 : 0 : 2 , 4 : 0 : 1)]
is_degenerate()

Function will return True if there is a fiber (over the algebraic closure of the base ring) of dimension greater than 0 and False otherwise.

OUTPUT: boolean

EXAMPLES:

sage: R.<x0,x1,x2,y0,y1,y2> = PolynomialRing(ZZ, 6)
sage: Y = x0*y0 + x1*y1 - x2*y2
sage: Z = x0^2*y0*y1 + x0^2*y2^2 - x0*x1*y1*y2 + x1^2*y2*y1 + x2^2*y2^2 + \
x2^2*y1^2 + x1^2*y2^2
sage: X = WehlerK3Surface([Z, Y])
sage: X.is_degenerate()
True
sage: PP.<x0,x1,x2,y0,y1,y2> = ProductProjectiveSpaces([2, 2], QQ)
sage: Z = x0^2*y0^2 + 3*x0*x1*y0^2 + x1^2*y0^2 + 4*x0^2*y0*y1 + 3*x0*x1*y0*y1 - \
2*x2^2*y0*y1 - x0^2*y1^2 + 2*x1^2*y1^2 - x0*x2*y1^2 -4*x1*x2*y1^2 + 5*x0*x2*y0*y2 - \
4*x1*x2*y0*y2 + 7*x0^2*y1*y2 + 4*x1^2*y1*y2 + x0*x1*y2^2 + 3*x2^2*y2^2
sage: Y = x0*y0 + x1*y1 + x2*y2
sage: X = WehlerK3Surface([Z, Y])
sage: X.is_degenerate()
False
sage: PP.<x0,x1,x2,y0,y1,y2> = ProductProjectiveSpaces([2, 2], GF(3))
sage: Z = x0^2*y0^2 + 3*x0*x1*y0^2 + x1^2*y0^2 + 4*x0^2*y0*y1 + 3*x0*x1*y0*y1 - \
2*x2^2*y0*y1 - x0^2*y1^2 + 2*x1^2*y1^2 - x0*x2*y1^2 -4*x1*x2*y1^2 + 5*x0*x2*y0*y2 - \
4*x1*x2*y0*y2 + 7*x0^2*y1*y2 + 4*x1^2*y1*y2 + x0*x1*y2^2 + 3*x2^2*y2^2
sage: Y = x0*y0 + x1*y1 + x2*y2
sage: X = WehlerK3Surface([Z, Y])
sage: X.is_degenerate()
True
is_isomorphic(right)

Checks to see if two K3 surfaces have the same defining ideal.

INPUT:

  • right - the K3 surface to compare to the original

OUTPUT: Boolean

EXAMPLES:

sage: PP.<x0,x1,x2,y0,y1,y2> = ProductProjectiveSpaces([2, 2], QQ)
sage: Z = x0^2*y0^2 + 3*x0*x1*y0^2 + x1^2*y0^2 + 4*x0^2*y0*y1 + \
3*x0*x1*y0*y1 -2*x2^2*y0*y1 - x0^2*y1^2 + 2*x1^2*y1^2 - x0*x2*y1^2 \
-4*x1*x2*y1^2 + 5*x0*x2*y0*y2 - 4*x1*x2*y0*y2 + 7*x0^2*y1*y2 + 4*x1^2*y1*y2 \
+ x0*x1*y2^2 + 3*x2^2*y2^2
sage: Y = x0*y0 + x1*y1 + x2*y2
sage: X = WehlerK3Surface([Z, Y])
sage: W = WehlerK3Surface([Z + Y^2, Y])
sage: X.is_isomorphic(W)
True
sage: R.<x,y,z,u,v,w> = PolynomialRing(QQ, 6)
sage: L = x*u-y*v
sage: Q = x*y*v^2 + z^2*u*w
sage: W1 = WehlerK3Surface([L, Q])
sage: PP.<x0,x1,x2,y0,y1,y2> = ProductProjectiveSpaces([2, 2], QQ)
sage: L = x0*y0 + x1*y1 + x2*y2
sage: Q = x1^2*y0^2 + 2*x2^2*y0*y1 + x0^2*y1^2 -x0*x1*y2^2
sage: W2 = WehlerK3Surface([L, Q])
sage: W1.is_isomorphic(W2)
False
is_smooth()

Function will return the status of the smoothness of the surface.

ALGORITHM:

Checks to confirm that all of the 2x2 minors of the Jacobian generated from the biquadratic and bilinear forms have no common vanishing points.

OUTPUT: Boolean

EXAMPLES:

sage: R.<x0,x1,x2,y0,y1,y2> = PolynomialRing(ZZ, 6)
sage: Y = x0*y0 + x1*y1 - x2*y2
sage: Z = x0^2*y0*y1 + x0^2*y2^2 - x0*x1*y1*y2 + x1^2*y2*y1 +\
 x2^2*y2^2 + x2^2*y1^2 + x1^2*y2^2
sage: X = WehlerK3Surface([Z, Y])
sage: X.is_smooth()
False
sage: PP.<x0,x1,x2,y0,y1,y2> = ProductProjectiveSpaces([2, 2], QQ)
sage: Z = x0^2*y0^2 + 3*x0*x1*y0^2 + x1^2*y0^2 + 4*x0^2*y0*y1 + \
3*x0*x1*y0*y1 - 2*x2^2*y0*y1 - x0^2*y1^2 + 2*x1^2*y1^2 - x0*x2*y1^2 \
- 4*x1*x2*y1^2 + 5*x0*x2*y0*y2 - 4*x1*x2*y0*y2 + 7*x0^2*y1*y2 + 4*x1^2*y1*y2 \
+ x0*x1*y2^2 + 3*x2^2*y2^2
sage: Y = x0*y0 + x1*y1 + x2*y2
sage: X = WehlerK3Surface([Z, Y])
sage: X.is_smooth()
True
is_symmetric_orbit(orbit)

Checks to see if the orbit is symmetric (i.e. if one of the points on the orbit is fixed by ‘sigma_x’ or ‘sigma_y’).

INPUT:

  • orbit- a periodic cycle of either psi or phi

OUTPUT: Boolean

EXAMPLES:

sage: PP.<x0,x1,x2,y0,y1,y2> = ProductProjectiveSpaces([2, 2], GF(7))
sage: Z = x0^2*y0^2 + 3*x0*x1*y0^2 + x1^2*y0^2 + 4*x0^2*y0*y1 + 3*x0*x1*y0*y1 \
-2*x2^2*y0*y1 - x0^2*y1^2 + 2*x1^2*y1^2 - x0*x2*y1^2 -4*x1*x2*y1^2 + 5*x0*x2*y0*y2 \
-4*x1*x2*y0*y2 + 7*x0^2*y1*y2 + 4*x1^2*y1*y2 + x0*x1*y2^2 + 3*x2^2*y2^2
sage: Y = x0*y0 + x1*y1 + x2*y2
sage: X = WehlerK3Surface([Z, Y])
sage: T = PP([0, 0, 1, 1, 0, 0])
sage: orbit = X.orbit_psi(T, 4)
sage: X.is_symmetric_orbit(orbit)
True
sage: PP.<x0,x1,x2,y0,y1,y2> = ProductProjectiveSpaces([2, 2], QQ)
sage: L = x0*y0 + x1*y1 + x2*y2
sage: Q = x1^2*y0^2 + 2*x2^2*y0*y1 + x0^2*y1^2 - x0*x1*y2^2
sage: W = WehlerK3Surface([L, Q])
sage: T = W([-1, -1, 1, 1, 0, 1])
sage: Orb = W.orbit_phi(T, 7)
sage: W.is_symmetric_orbit(Orb)
False
lambda_minus(P, v, N, m, n, prec=100)

Evaluates the local canonical height minus function of Call-Silverman at the place v for P with N terms of the series.

Use v = 0 for the Archimedean place. Must be over \(\ZZ\) or \(\QQ\).

ALGORITHM:

Sum over local heights using convergent series, for more details, see section 4 of [CS1996].

INPUT:

  • P – a projective point
  • N – positive integer. number of terms of the series to use
  • v – non-negative integer. a place, use v = 0 for the Archimedean place
  • m,n – positive integers, We compute the local height for the divisor \(E_{mn}^{+}\).
    These must be indices of non-zero coordinates of the point P.
  • prec – (default: 100) float point or p-adic precision

OUTPUT: A real number

EXAMPLES:

sage: PP.<x0,x1,x2,y0,y1,y2> = ProductProjectiveSpaces([2, 2], QQ)
sage: Z = x0^2*y0^2 + 3*x0*x1*y0^2 + x1^2*y0^2 + 4*x0^2*y0*y1 + 3*x0*x1*y0*y1 \
- 2*x2^2*y0*y1 - x0^2*y1^2 + 2*x1^2*y1^2 - x0*x2*y1^2 -4*x1*x2*y1^2 + 5*x0*x2*y0*y2\
- 4*x1*x2*y0*y2 + 7*x0^2*y1*y2 + 4*x1^2*y1*y2 + x0*x1*y2^2 + 3*x2^2*y2^2
sage: Y = x0*y0 + x1*y1 + x2*y2
sage: X = WehlerK3Surface([Z, Y])
sage: P = X([0, 0, 1, 1, 0, 0])
sage: X.lambda_minus(P, 2, 20, 2, 0, 200)
-0.18573351672047135037172805779671791488351056677474271893705
lambda_plus(P, v, N, m, n, prec=100)

Evaluates the local canonical height plus function of Call-Silverman at the place v for P with N terms of the series.

Use v = 0 for the archimedean place. Must be over \(\ZZ\) or \(\QQ\).

ALGORITHM:

Sum over local heights using convergent series, for more details, see section 4 of [CS1996].

INPUT:

  • P – a surface point
  • N – positive integer. number of terms of the series to use
  • v – non-negative integer. a place, use v = 0 for the Archimedean place
  • m,n – positive integers, We compute the local height for the divisor \(E_{mn}^{+}\).
    These must be indices of non-zero coordinates of the point P.
  • prec – (default: 100) float point or p-adic precision

OUTPUT: A real number

EXAMPLES:

sage: PP.<x0,x1,x2,y0,y1,y2> = ProductProjectiveSpaces([2, 2], QQ)
sage: Z = x0^2*y0^2 + 3*x0*x1*y0^2 + x1^2*y0^2 + 4*x0^2*y0*y1 + 3*x0*x1*y0*y1\
- 2*x2^2*y0*y1 - x0^2*y1^2 + 2*x1^2*y1^2 - x0*x2*y1^2 -4*x1*x2*y1^2 + 5*x0*x2*y0*y2\
- 4*x1*x2*y0*y2 + 7*x0^2*y1*y2 + 4*x1^2*y1*y2 + x0*x1*y2^2 + 3*x2^2*y2^2
sage: Y = x0*y0 + x1*y1 + x2*y2
sage: X = WehlerK3Surface([Z, Y])
sage: P = X([0, 0, 1, 1, 0, 0])
sage: X.lambda_plus(P, 0, 10, 2, 0)
0.89230705169161608922595928129
nth_iterate_phi(P, n, **kwds)

Computes the nth iterate for the phi function.

INPUT:

  • P – - a point in \(\mathbb{P}^2 \times \mathbb{P}^2\)
  • n – an integer

kwds:

  • check - (default: True) boolean checks to see if point is on the surface
  • normalize – (default: False) boolean normalizes the point

OUTPUT: The nth iterate of the point given the phi function (if n is positive), or the psi function (if n is negative)

EXAMPLES:

sage: PP.<x0,x1,x2,y0,y1,y2> = ProductProjectiveSpaces([2, 2], QQ)
sage: L = x0*y0 + x1*y1 + x2*y2
sage: Q = x1^2*y0^2 + 2*x2^2*y0*y1 + x0^2*y1^2 - x0*x1*y2^2
sage: W = WehlerK3Surface([L ,Q])
sage: T = W([-1, -1, 1, 1, 0, 1])
sage: W.nth_iterate_phi(T, 7)
(-1 : 0 : 1 , 1 : -2 : 1)
sage: PP.<x0,x1,x2,y0,y1,y2> = ProductProjectiveSpaces([2, 2], QQ)
sage: L = x0*y0 + x1*y1 + x2*y2
sage: Q = x1^2*y0^2 + 2*x2^2*y0*y1 + x0^2*y1^2 - x0*x1*y2^2
sage: W = WehlerK3Surface([L, Q])
sage: T = W([-1, -1, 1, 1, 0, 1])
sage: W.nth_iterate_phi(T, -7)
(1 : 0 : 1 , -1 : 2 : 1)
sage: R.<x0,x1,x2,y0,y1,y2>=PolynomialRing(QQ, 6)
sage: L = (-y0 - y1)*x0 + (-y0*x1 - y2*x2)
sage: Q = (-y2*y0 - y1^2)*x0^2 + ((-y0^2 - y2*y0 + (-y2*y1 - y2^2))*x1 + (-y0^2 - y2*y1)*x2)*x0 \
+ ((-y0^2 - y2*y0 - y2^2)*x1^2 + (-y2*y0 - y1^2)*x2*x1 + (-y0^2 + (-y1 - y2)*y0)*x2^2)
sage: X = WehlerK3Surface([L, Q])
sage: P = X([1, 0, -1, 1, -1, 0])
sage: X.nth_iterate_phi(P, 8) == X.nth_iterate_psi(P, 8)
True
nth_iterate_psi(P, n, **kwds)

Computes the nth iterate for the psi function.

INPUT:

  • P – - a point in \(\mathbb{P}^2 \times \mathbb{P}^2\)
  • n – an integer

kwds:

  • check – (default: True) boolean, checks to see if point is on the surface
  • normalize – (default: False) boolean, normalizes the point

OUTPUT: The nth iterate of the point given the psi function (if n is positive), or the phi function (if n is negative)

EXAMPLES:

sage: PP.<x0,x1,x2,y0,y1,y2> = ProductProjectiveSpaces([2, 2], QQ)
sage: L = x0*y0 + x1*y1 + x2*y2
sage: Q = x1^2*y0^2 + 2*x2^2*y0*y1 + x0^2*y1^2 - x0*x1*y2^2
sage: W = WehlerK3Surface([L, Q])
sage: T = W([-1, -1, 1, 1, 0, 1])
sage: W.nth_iterate_psi(T, -7)
(-1 : 0 : 1 , 1 : -2 : 1)
sage: PP.<x0,x1,x2,y0,y1,y2> = ProductProjectiveSpaces([2, 2], QQ)
sage: L = x0*y0 + x1*y1 + x2*y2
sage: Q = x1^2*y0^2 + 2*x2^2*y0*y1 + x0^2*y1^2 - x0*x1*y2^2
sage: W = WehlerK3Surface([L, Q])
sage: T = W([-1, -1, 1, 1, 0, 1])
sage: W.nth_iterate_psi(T, 7)
(1 : 0 : 1 , -1 : 2 : 1)
orbit_phi(P, N, **kwds)

Returns the orbit of the \(\phi\) function defined by \(\phi = \sigma_y \circ \sigma_x\) Function is defined in [CS1996].

INPUT:

  • P - Point on the K3 surface
  • N - a non-negative integer or list or tuple of two non-negative integers

kwds:

  • check – (default: True) boolean, checks to see if point is on the surface
  • normalize – (default: False) boolean, normalizes the point

OUTPUT: List of points in the orbit

EXAMPLES:

sage: PP.<x0,x1,x2,y0,y1,y2> = ProductProjectiveSpaces([2, 2], QQ)
sage: Z = x0^2*y0^2 + 3*x0*x1*y0^2 + x1^2*y0^2 + 4*x0^2*y0*y1 + \
3*x0*x1*y0*y1 -2*x2^2*y0*y1 - x0^2*y1^2 + 2*x1^2*y1^2 - x0*x2*y1^2 - \
4*x1*x2*y1^2 + 5*x0*x2*y0*y2 - 4*x1*x2*y0*y2 + 7*x0^2*y1*y2 + 4*x1^2*y1*y2 + \
x0*x1*y2^2 + 3*x2^2*y2^2
sage: Y = x0*y0 + x1*y1 + x2*y2
sage: X = WehlerK3Surface([Z, Y])
sage: T = PP(0, 0, 1, 1, 0, 0)
sage: X.orbit_phi(T,2, normalize = True)
[(0 : 0 : 1 , 1 : 0 : 0), (-1 : 0 : 1 , 0 : 1 : 0), (-12816/6659 : 55413/6659 : 1 , 1 : 1/9 : 1)]
sage: X.orbit_phi(T,[2,3], normalize = True)
[(-12816/6659 : 55413/6659 : 1 , 1 : 1/9 : 1),
(7481279673854775690938629732119966552954626693713001783595660989241/18550615454277582153932951051931712107449915856862264913424670784695
: 3992260691327218828582255586014718568398539828275296031491644987908/18550615454277582153932951051931712107449915856862264913424670784695 :
1 , -117756062505511/54767410965117 : -23134047983794359/37466994368025041 : 1)]
orbit_psi(P, N, **kwds)

Returns the orbit of the \(\psi\) function defined by \(\psi = \sigma_x \circ \sigma_y\).

Function is defined in [CS1996].

INPUT:

  • P - a point on the K3 surface
  • N - a non-negative integer or list or tuple of two non-negative integers

kwds:

  • check - (default: True) boolean, checks to see if point is on the surface
  • normalize – (default: False) boolean, normalizes the point

OUTPUT: a list of points in the orbit

EXAMPLES:

sage: PP.<x0,x1,x2,y0,y1,y2> = ProductProjectiveSpaces([2, 2], QQ)
sage: Z = x0^2*y0^2 + 3*x0*x1*y0^2 + x1^2*y0^2 + 4*x0^2*y0*y1 + \
 3*x0*x1*y0*y1 -2*x2^2*y0*y1 - x0^2*y1^2 + 2*x1^2*y1^2 - x0*x2*y1^2 - \
 4*x1*x2*y1^2 + 5*x0*x2*y0*y2 -4*x1*x2*y0*y2 + 7*x0^2*y1*y2 + 4*x1^2*y1*y2 + \
  x0*x1*y2^2 + 3*x2^2*y2^2
sage: Y = x0*y0 + x1*y1 + x2*y2
sage: X = WehlerK3Surface([Z, Y])
sage: T = X(0, 0, 1, 1, 0, 0)
sage: X.orbit_psi(T, 2, normalize = True)
[(0 : 0 : 1 , 1 : 0 : 0), (0 : 0 : 1 , 0 : 1 : 0), (-1 : 0 : 1 , 1 : 1/9 : 1)]
sage: X.orbit_psi(T,[2,3], normalize = True)
[(-1 : 0 : 1 , 1 : 1/9 : 1),
(-12816/6659 : 55413/6659 : 1 , -117756062505511/54767410965117 : -23134047983794359/37466994368025041 : 1)]
phi(a, **kwds)

Evaluates the function \(\phi = \sigma_y \circ \sigma_x\).

ALGORITHM:

Refer to Section 6: “An algorithm to compute \(\sigma_x\), \(\sigma_y\), \(\phi\), and \(\psi\)” in [CS1996].

For the degenerate case refer to [FH2015].

INPUT:

  • a - Point in \(\mathbb{P}^2 \times \mathbb{P}^2\)

kwds:

  • check - (default: True) boolean checks to see if point is on the surface
  • normalize – (default: True) boolean normalizes the point

OUTPUT: A point on this surface

EXAMPLES:

sage: PP.<x0,x1,x2,y0,y1,y2> = ProductProjectiveSpaces([2, 2], QQ)
sage: Z = x0^2*y0^2 + 3*x0*x1*y0^2 + x1^2*y0^2 + 4*x0^2*y0*y1 + \
3*x0*x1*y0*y1 -2*x2^2*y0*y1 - x0^2*y1^2 + 2*x1^2*y1^2 - x0*x2*y1^2 \
- 4*x1*x2*y1^2 + 5*x0*x2*y0*y2 -4*x1*x2*y0*y2 + 7*x0^2*y1*y2 + 4*x1^2*y1*y2 \
+ x0*x1*y2^2 + 3*x2^2*y2^2
sage: Y = x0*y0 + x1*y1 + x2*y2
sage: X = WehlerK3Surface([Z, Y])
sage: T = PP([0, 0, 1, 1 ,0, 0])
sage: X.phi(T)
(-1 : 0 : 1 , 0 : 1 : 0)
psi(a, **kwds)

Evaluates the function \(\psi = \sigma_x \circ \sigma_y\).

ALGORITHM:

Refer to Section 6: “An algorithm to compute \(\sigma_x\), \(\sigma_y\), \(\phi\), and \(\psi\)” in [CS1996].

For the degenerate case refer to [FH2015].

INPUT:

  • a - Point in \(\mathbb{P}^2 \times \mathbb{P}^2\)

kwds:

  • check - (default: True) boolean checks to see if point is on the surface
  • normalize – (default: True) boolean normalizes the point

OUTPUT: A point on this surface

EXAMPLES:

sage: PP.<x0,x1,x2,y0,y1,y2> = ProductProjectiveSpaces([2, 2], QQ)
sage: Z = x0^2*y0^2 + 3*x0*x1*y0^2 + x1^2*y0^2 + 4*x0^2*y0*y1 + \
3*x0*x1*y0*y1 -2*x2^2*y0*y1 - x0^2*y1^2 + 2*x1^2*y1^2 - x0*x2*y1^2 \
- 4*x1*x2*y1^2 + 5*x0*x2*y0*y2 - 4*x1*x2*y0*y2 + 7*x0^2*y1*y2 + 4*x1^2*y1*y2 \
+ x0*x1*y2^2 + 3*x2^2*y2^2
sage: Y = x0*y0 + x1*y1 + x2*y2
sage: X = WehlerK3Surface([Z, Y])
sage: T = PP([0, 0, 1, 1, 0, 0])
sage: X.psi(T)
(0 : 0 : 1 , 0 : 1 : 0)
sigmaX(P, **kwds)

Function returns the involution on the Wehler K3 surface induced by the double covers.

In particular, it fixes the projection to the first coordinate and swaps the two points in the fiber, i.e. \((x, y) \to (x, y')\). Note that in the degenerate case, while we can split fiber into pairs of points, it is not always possibleto distinguish them, using this algorithm.

ALGORITHM:

Refer to Section 6: “An algorithm to compute \(\sigma_x\), \(\sigma_y\), \(\phi\), and \(\psi\)” in [CS1996FH2015. For the degenerate case refer to [FH2015].

INPUT:

  • P - a point in \(\mathbb{P}^2 \times \mathbb{P}^2\)

kwds:

  • check - (default: True) boolean checks to see if point is on the surface
  • normalize – (default: True) boolean normalizes the point

OUTPUT: A point on the K3 surface

EXAMPLES:

sage: PP.<x0,x1,x2,y0,y1,y2> = ProductProjectiveSpaces([2, 2], QQ)
sage: Z = x0^2*y0^2 + 3*x0*x1*y0^2 + x1^2*y0^2 + 4*x0^2*y0*y1 +\
3*x0*x1*y0*y1 -2*x2^2*y0*y1 - x0^2*y1^2 + 2*x1^2*y1^2 - x0*x2*y1^2 -\
4*x1*x2*y1^2 + 5*x0*x2*y0*y2 -4*x1*x2*y0*y2 + 7*x0^2*y1*y2 +\
4*x1^2*y1*y2 + x0*x1*y2^2 + 3*x2^2*y2^2
sage: Y = x0*y0 + x1*y1 + x2*y2
sage: X = WehlerK3Surface([Z, Y])
sage: T = PP(0, 0, 1, 1, 0, 0)
sage: X.sigmaX(T)
(0 : 0 : 1 , 0 : 1 : 0)

degenerate examples:

sage: PP.<x0,x1,x2,y0,y1,y2> = ProductProjectiveSpaces([2, 2], QQ)
sage: l = y0*x0 + y1*x1 + (y0 - y1)*x2
sage: q = (y1*y0)*x0^2 + ((y0^2)*x1 + (y0^2 + (y1^2 - y2^2))*x2)*x0\
+ (y2*y0 + y1^2)*x1^2 + (y0^2 + (-y1^2 + y2^2))*x2*x1
sage: X = WehlerK3Surface([l, q])
sage: X.sigmaX(X([1, 0, 0, 0, 1, -2]))
(1 : 0 : 0 , 0 : 1/2 : 1)
sage: X.sigmaX(X([1, 0, 0, 0, 0, 1]))
(1 : 0 : 0 , 0 : 0 : 1)
sage: X.sigmaX(X([-1, 1, 1, -1, -1, 1]))
(-1 : 1 : 1 , 2 : 2 : 1)
sage: X.sigmaX(X([0, 0, 1, 1, 1, 0]))
(0 : 0 : 1 , 1 : 1 : 0)
sage: X.sigmaX(X([0, 0, 1, 1, 1, 1]))
(0 : 0 : 1 , -1 : -1 : 1)

Case where we cannot distinguish the two points:

sage: PP.<y0,y1,y2,x0,x1,x2>=ProductProjectiveSpaces([2, 2], GF(3))
sage: l = x0*y0 + x1*y1 + x2*y2
sage: q=-3*x0^2*y0^2 + 4*x0*x1*y0^2 - 3*x0*x2*y0^2 - 5*x0^2*y0*y1 - \
190*x0*x1*y0*y1- 5*x1^2*y0*y1 + 5*x0*x2*y0*y1 + 14*x1*x2*y0*y1 + \
5*x2^2*y0*y1 - x0^2*y1^2 - 6*x0*x1*y1^2- 2*x1^2*y1^2 + 2*x0*x2*y1^2 - \
4*x2^2*y1^2 + 4*x0^2*y0*y2 - x1^2*y0*y2 + 3*x0*x2*y0*y2+ 6*x1*x2*y0*y2 - \
6*x0^2*y1*y2 - 4*x0*x1*y1*y2 - x1^2*y1*y2 + 51*x0*x2*y1*y2 - 7*x1*x2*y1*y2 - \
9*x2^2*y1*y2 - x0^2*y2^2 - 4*x0*x1*y2^2 + 4*x1^2*y2^2 - x0*x2*y2^2 + 13*x1*x2*y2^2 - x2^2*y2^2
sage: X = WehlerK3Surface([l, q])
sage: P = X([1, 0, 0, 0, 1, 1])
sage: X.sigmaX(X.sigmaX(P))
Traceback (most recent call last):
...
ValueError: cannot distinguish points in the degenerate fiber
sigmaY(P, **kwds)

Function returns the involution on the Wehler K3 surfaces induced by the double covers.

In particular,it fixes the projection to the second coordinate and swaps the two points in the fiber, i.e. \((x,y) \to (x',y)\). Note that in the degenerate case, while we can split the fiber into two points, it is not always possibleto distinguish them, using this algorithm.

ALGORITHM:

Refer to Section 6: “An algorithm to compute \(\sigma_x\), \(\sigma_y\), \(\phi\), and \(\psi\)” in [CS1996]. For the degenerate case refer to [FH2015].

INPUT:

  • P - a point in \(\mathbb{P}^2 \times \mathbb{P}^2\)

kwds:

  • check - (default: True) boolean checks to see if point is on the surface
  • normalize – (default: True) boolean normalizes the point

OUTPUT: A point on the K3 surface

EXAMPLES:

sage: PP.<x0,x1,x2,y0,y1,y2> = ProductProjectiveSpaces([2, 2], QQ)
sage: Z = x0^2*y0^2 + 3*x0*x1*y0^2 + x1^2*y0^2 + 4*x0^2*y0*y1 + \
3*x0*x1*y0*y1 -2*x2^2*y0*y1 - x0^2*y1^2 + 2*x1^2*y1^2 - x0*x2*y1^2 \
- 4*x1*x2*y1^2 + 5*x0*x2*y0*y2 - 4*x1*x2*y0*y2 + 7*x0^2*y1*y2 + 4*x1^2*y1*y2 \
+ x0*x1*y2^2 + 3*x2^2*y2^2
sage: Y = x0*y0 + x1*y1 + x2*y2
sage: X = WehlerK3Surface([Z, Y])
sage: T = PP(0, 0, 1, 1, 0, 0)
sage: X.sigmaY(T)
(0 : 0 : 1 , 1 : 0 : 0)

degenerate examples:

sage: PP.<x0,x1,x2,y0,y1,y2> = ProductProjectiveSpaces([2, 2], QQ)
sage: l = y0*x0 + y1*x1 + (y0 - y1)*x2
sage: q = (y1*y0)*x0^2 + ((y0^2)*x1 + (y0^2 + (y1^2 - y2^2))*x2)*x0 +\
 (y2*y0 + y1^2)*x1^2 + (y0^2 + (-y1^2 + y2^2))*x2*x1
sage: X = WehlerK3Surface([l, q])
sage: X.sigmaY(X([1, -1, 0 ,-1, -1, 1]))
(1/10 : -1/10 : 1 , -1 : -1 : 1)
sage: X.sigmaY(X([0, 0, 1, -1, -1, 1]))
(-4 : 4 : 1 , -1 : -1 : 1)
sage: X.sigmaY(X([1, 2, 0, 0, 0, 1]))
(-3 : -3 : 1 , 0 : 0 : 1)
sage: X.sigmaY(X([1, 1, 1, 0, 0, 1]))
(1 : 0 : 0 , 0 : 0 : 1)

Case where we cannot distinguish the two points:

sage: PP.<x0,x1,x2,y0,y1,y2>=ProductProjectiveSpaces([2, 2], GF(3))
sage: l = x0*y0 + x1*y1 + x2*y2
sage: q=-3*x0^2*y0^2 + 4*x0*x1*y0^2 - 3*x0*x2*y0^2 - 5*x0^2*y0*y1 - 190*x0*x1*y0*y1 \
- 5*x1^2*y0*y1 + 5*x0*x2*y0*y1 + 14*x1*x2*y0*y1 + 5*x2^2*y0*y1 - x0^2*y1^2 - 6*x0*x1*y1^2 \
- 2*x1^2*y1^2 + 2*x0*x2*y1^2 - 4*x2^2*y1^2 + 4*x0^2*y0*y2 - x1^2*y0*y2 + 3*x0*x2*y0*y2 \
+ 6*x1*x2*y0*y2 - 6*x0^2*y1*y2 - 4*x0*x1*y1*y2 - x1^2*y1*y2 + 51*x0*x2*y1*y2 - 7*x1*x2*y1*y2 \
- 9*x2^2*y1*y2 - x0^2*y2^2 - 4*x0*x1*y2^2 + 4*x1^2*y2^2 - x0*x2*y2^2 + 13*x1*x2*y2^2 - x2^2*y2^2
sage: X = WehlerK3Surface([l ,q])
sage: P = X([0, 1, 1, 1, 0, 0])
sage: X.sigmaY(X.sigmaY(P))
Traceback (most recent call last):
...
ValueError: cannot distinguish points in the degenerate fiber
sage.dynamics.arithmetic_dynamics.wehlerK3.random_WehlerK3Surface(PP)

Produces a random K3 surface in \(\mathbb{P}^2 \times \mathbb{P}^2\) defined as the intersection of a bilinear and biquadratic form. [Weh1998]

INPUT: Projective space cartesian product

OUTPUT: WehlerK3Surface_ring

EXAMPLES:

sage: PP.<x0, x1, x2, y0, y1, y2> = ProductProjectiveSpaces([2, 2], GF(3))
sage: random_WehlerK3Surface(PP)
Closed subscheme of Product of projective spaces P^2 x P^2 over Finite Field of size 3 defined by:
x0*y0 + x1*y1 + x2*y2,
-x1^2*y0^2 - x2^2*y0^2 + x0^2*y0*y1 - x0*x1*y0*y1 - x1^2*y0*y1
+ x1*x2*y0*y1 + x0^2*y1^2 + x0*x1*y1^2 - x1^2*y1^2 + x0*x2*y1^2
- x0^2*y0*y2 - x0*x1*y0*y2 + x0*x2*y0*y2 + x1*x2*y0*y2 + x0*x1*y1*y2
- x1^2*y1*y2 - x1*x2*y1*y2 - x0^2*y2^2 + x0*x1*y2^2 - x1^2*y2^2 - x0*x2*y2^2