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:
WehlerK3Surface_ring
- class sage.dynamics.arithmetic_dynamics.wehlerK3.WehlerK3Surface_finite_field(polys)#
Bases:
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:
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)#
Return 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 1k
- 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)#
Return 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 1i
- Integer: 0, 1 or 2j
- 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 spaceOUTPUT: 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 spaceOUTPUT: 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
withN
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 pointN
– 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
withN
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 pointN
– 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
withN
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 pointN
– positive integer. Number of terms of the series to usebadprimes
– (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
- ringOUTPUT: 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()#
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
orQQ
.INPUT:
check
– (default: True) boolean, whether the primes are verifiedALGORITHM:
\(p\) 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)#
Return the fibers [y (component = 1) or x (Component = 0)] of a point on a K3 Surface.
This 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
forP
withN
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 pointN
– positive integer. number of terms of the series to usev
– non-negative integer. a place, use v = 0 for the Archimedean placem,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
forP
withN
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 pointN
– positive integer. number of terms of the series to usev
– non-negative integer. a place, use v = 0 for the Archimedean placem,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 surfacenormalize
– (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 (ifn
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 surfacenormalize
– (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 (ifn
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)#
Return the orbit of the \(\phi\) function defined by \(\phi = \sigma_y \circ \sigma_x\).
This function is defined in [CS1996].
INPUT:
P
- Point on the K3 surfaceN
- 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 surfacenormalize
– (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)#
Return the orbit of the \(\psi\) function defined by \(\psi = \sigma_x \circ \sigma_y\).
This function is defined in [CS1996].
INPUT:
P
- a point on the K3 surfaceN
- 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 surfacenormalize
– (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 surfacenormalize
– (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 surfacenormalize
– (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 surfacenormalize
– (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 surfacenormalize
– (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: w = random_WehlerK3Surface(PP) sage: type(w) <class 'sage.dynamics.arithmetic_dynamics.wehlerK3.WehlerK3Surface_finite_field_with_category'>