Regions in fundamental domains of period lattices¶
This module is used to represent sub-regions of a fundamental parallelogram of the period lattice of an elliptic curve, used in computing minimum height bounds.
In particular, these are the approximating sets S^{(v)} in section 3.2 of
Thotsaphon Thongjunthug’s Ph.D. Thesis and paper [Tho2010].
AUTHORS:
Robert Bradshaw (2010): initial version
John Cremona (2014): added some docstrings and doctests
- class sage.schemes.elliptic_curves.period_lattice_region.PeriodicRegion[source]¶
Bases:
objectEXAMPLES:
sage: import numpy as np sage: from sage.schemes.elliptic_curves.period_lattice_region import PeriodicRegion sage: S = PeriodicRegion(CDF(2), CDF(2*I), np.zeros((4, 4))) sage: S.plot() # needs sage.plot Graphics object consisting of 1 graphics primitive sage: data = np.zeros((4, 4)) sage: data[1,1] = True sage: S = PeriodicRegion(CDF(2), CDF(2*I+1), data) sage: S.plot() # needs sage.plot Graphics object consisting of 5 graphics primitives
>>> from sage.all import * >>> import numpy as np >>> from sage.schemes.elliptic_curves.period_lattice_region import PeriodicRegion >>> S = PeriodicRegion(CDF(Integer(2)), CDF(Integer(2)*I), np.zeros((Integer(4), Integer(4)))) >>> S.plot() # needs sage.plot Graphics object consisting of 1 graphics primitive >>> data = np.zeros((Integer(4), Integer(4))) >>> data[Integer(1),Integer(1)] = True >>> S = PeriodicRegion(CDF(Integer(2)), CDF(Integer(2)*I+Integer(1)), data) >>> S.plot() # needs sage.plot Graphics object consisting of 5 graphics primitives
import numpy as np from sage.schemes.elliptic_curves.period_lattice_region import PeriodicRegion S = PeriodicRegion(CDF(2), CDF(2*I), np.zeros((4, 4))) S.plot() # needs sage.plot data = np.zeros((4, 4)) data[1,1] = True S = PeriodicRegion(CDF(2), CDF(2*I+1), data) S.plot() # needs sage.plot
- border(raw=True)[source]¶
Return the boundary of this region as set of tile boundaries.
If raw is true, returns a list with respect to the internal bitmap, otherwise returns complex intervals covering the border.
EXAMPLES:
sage: import numpy as np sage: from sage.schemes.elliptic_curves.period_lattice_region import PeriodicRegion sage: data = np.zeros((4, 4)) sage: data[1, 1] = True sage: PeriodicRegion(CDF(1), CDF(I), data).border() [(1, 1, 0), (2, 1, 0), (1, 1, 1), (1, 2, 1)] sage: PeriodicRegion(CDF(2), CDF(I-1/2), data).border() [(1, 1, 0), (2, 1, 0), (1, 1, 1), (1, 2, 1)] sage: PeriodicRegion(CDF(1), CDF(I), data).border(raw=False) [0.25000000000000000? + 1.?*I, 0.50000000000000000? + 1.?*I, 1.? + 0.25000000000000000?*I, 1.? + 0.50000000000000000?*I] sage: PeriodicRegion(CDF(2), CDF(I-1/2), data).border(raw=False) [0.3? + 1.?*I, 0.8? + 1.?*I, 1.? + 0.25000000000000000?*I, 1.? + 0.50000000000000000?*I] sage: data[1:3, 2] = True sage: PeriodicRegion(CDF(1), CDF(I), data).border() [(1, 1, 0), (2, 1, 0), (1, 1, 1), (1, 2, 0), (1, 3, 1), (3, 2, 0), (2, 2, 1), (2, 3, 1)]
>>> from sage.all import * >>> import numpy as np >>> from sage.schemes.elliptic_curves.period_lattice_region import PeriodicRegion >>> data = np.zeros((Integer(4), Integer(4))) >>> data[Integer(1), Integer(1)] = True >>> PeriodicRegion(CDF(Integer(1)), CDF(I), data).border() [(1, 1, 0), (2, 1, 0), (1, 1, 1), (1, 2, 1)] >>> PeriodicRegion(CDF(Integer(2)), CDF(I-Integer(1)/Integer(2)), data).border() [(1, 1, 0), (2, 1, 0), (1, 1, 1), (1, 2, 1)] >>> PeriodicRegion(CDF(Integer(1)), CDF(I), data).border(raw=False) [0.25000000000000000? + 1.?*I, 0.50000000000000000? + 1.?*I, 1.? + 0.25000000000000000?*I, 1.? + 0.50000000000000000?*I] >>> PeriodicRegion(CDF(Integer(2)), CDF(I-Integer(1)/Integer(2)), data).border(raw=False) [0.3? + 1.?*I, 0.8? + 1.?*I, 1.? + 0.25000000000000000?*I, 1.? + 0.50000000000000000?*I] >>> data[Integer(1):Integer(3), Integer(2)] = True >>> PeriodicRegion(CDF(Integer(1)), CDF(I), data).border() [(1, 1, 0), (2, 1, 0), (1, 1, 1), (1, 2, 0), (1, 3, 1), (3, 2, 0), (2, 2, 1), (2, 3, 1)]
import numpy as np from sage.schemes.elliptic_curves.period_lattice_region import PeriodicRegion data = np.zeros((4, 4)) data[1, 1] = True PeriodicRegion(CDF(1), CDF(I), data).border() PeriodicRegion(CDF(2), CDF(I-1/2), data).border() PeriodicRegion(CDF(1), CDF(I), data).border(raw=False) PeriodicRegion(CDF(2), CDF(I-1/2), data).border(raw=False) data[1:3, 2] = True PeriodicRegion(CDF(1), CDF(I), data).border()
- contract(corners=True)[source]¶
Opposite (but not inverse) of expand; removes neighbors of complement.
EXAMPLES:
sage: import numpy as np sage: if int(np.version.short_version[0]) > 1: ....: np.set_printoptions(legacy="1.25") sage: from sage.schemes.elliptic_curves.period_lattice_region import PeriodicRegion sage: data = np.zeros((10, 10)) sage: data[1:4,1:4] = True sage: S = PeriodicRegion(CDF(1), CDF(I + 1/2), data) sage: S.plot() # needs sage.plot Graphics object consisting of 13 graphics primitives sage: S.contract().plot() # needs sage.plot Graphics object consisting of 5 graphics primitives sage: S.contract().data.sum() 1 sage: S.contract().contract().is_empty() True
>>> from sage.all import * >>> import numpy as np >>> if int(np.version.short_version[Integer(0)]) > Integer(1): ... np.set_printoptions(legacy="1.25") >>> from sage.schemes.elliptic_curves.period_lattice_region import PeriodicRegion >>> data = np.zeros((Integer(10), Integer(10))) >>> data[Integer(1):Integer(4),Integer(1):Integer(4)] = True >>> S = PeriodicRegion(CDF(Integer(1)), CDF(I + Integer(1)/Integer(2)), data) >>> S.plot() # needs sage.plot Graphics object consisting of 13 graphics primitives >>> S.contract().plot() # needs sage.plot Graphics object consisting of 5 graphics primitives >>> S.contract().data.sum() 1 >>> S.contract().contract().is_empty() True
import numpy as np if int(np.version.short_version[0]) > 1: np.set_printoptions(legacy="1.25") from sage.schemes.elliptic_curves.period_lattice_region import PeriodicRegion data = np.zeros((10, 10)) data[1:4,1:4] = True S = PeriodicRegion(CDF(1), CDF(I + 1/2), data) S.plot() # needs sage.plot S.contract().plot() # needs sage.plot S.contract().data.sum() S.contract().contract().is_empty()
- ds()[source]¶
Return the sides of each parallelogram tile.
EXAMPLES:
sage: import numpy as np sage: from sage.schemes.elliptic_curves.period_lattice_region import PeriodicRegion sage: data = np.zeros((4, 4)) sage: S = PeriodicRegion(CDF(2), CDF(2*I), data, full=False) sage: S.ds() (0.5, 0.25*I) sage: _ = S._ensure_full() sage: S.ds() (0.5, 0.25*I) sage: data = np.zeros((8, 8)) sage: S = PeriodicRegion(CDF(1), CDF(I + 1/2), data) sage: S.ds() (0.125, 0.0625 + 0.125*I)
>>> from sage.all import * >>> import numpy as np >>> from sage.schemes.elliptic_curves.period_lattice_region import PeriodicRegion >>> data = np.zeros((Integer(4), Integer(4))) >>> S = PeriodicRegion(CDF(Integer(2)), CDF(Integer(2)*I), data, full=False) >>> S.ds() (0.5, 0.25*I) >>> _ = S._ensure_full() >>> S.ds() (0.5, 0.25*I) >>> data = np.zeros((Integer(8), Integer(8))) >>> S = PeriodicRegion(CDF(Integer(1)), CDF(I + Integer(1)/Integer(2)), data) >>> S.ds() (0.125, 0.0625 + 0.125*I)
import numpy as np from sage.schemes.elliptic_curves.period_lattice_region import PeriodicRegion data = np.zeros((4, 4)) S = PeriodicRegion(CDF(2), CDF(2*I), data, full=False) S.ds() _ = S._ensure_full() S.ds() data = np.zeros((8, 8)) S = PeriodicRegion(CDF(1), CDF(I + 1/2), data) S.ds()
- expand(corners=True)[source]¶
Return a region containing this region by adding all neighbors of internal tiles.
EXAMPLES:
sage: import numpy as np sage: from sage.schemes.elliptic_curves.period_lattice_region import PeriodicRegion sage: data = np.zeros((4, 4)) sage: data[1,1] = True sage: S = PeriodicRegion(CDF(1), CDF(I + 1/2), data) sage: S.plot() # needs sage.plot Graphics object consisting of 5 graphics primitives sage: S.expand().plot() # needs sage.plot Graphics object consisting of 13 graphics primitives sage: S.expand().data array([[1, 1, 1, 0], [1, 1, 1, 0], [1, 1, 1, 0], [0, 0, 0, 0]], dtype=int8) sage: S.expand(corners=False).plot() # needs sage.plot Graphics object consisting of 13 graphics primitives sage: S.expand(corners=False).data array([[0, 1, 0, 0], [1, 1, 1, 0], [0, 1, 0, 0], [0, 0, 0, 0]], dtype=int8)
>>> from sage.all import * >>> import numpy as np >>> from sage.schemes.elliptic_curves.period_lattice_region import PeriodicRegion >>> data = np.zeros((Integer(4), Integer(4))) >>> data[Integer(1),Integer(1)] = True >>> S = PeriodicRegion(CDF(Integer(1)), CDF(I + Integer(1)/Integer(2)), data) >>> S.plot() # needs sage.plot Graphics object consisting of 5 graphics primitives >>> S.expand().plot() # needs sage.plot Graphics object consisting of 13 graphics primitives >>> S.expand().data array([[1, 1, 1, 0], [1, 1, 1, 0], [1, 1, 1, 0], [0, 0, 0, 0]], dtype=int8) >>> S.expand(corners=False).plot() # needs sage.plot Graphics object consisting of 13 graphics primitives >>> S.expand(corners=False).data array([[0, 1, 0, 0], [1, 1, 1, 0], [0, 1, 0, 0], [0, 0, 0, 0]], dtype=int8)
import numpy as np from sage.schemes.elliptic_curves.period_lattice_region import PeriodicRegion data = np.zeros((4, 4)) data[1,1] = True S = PeriodicRegion(CDF(1), CDF(I + 1/2), data) S.plot() # needs sage.plot S.expand().plot() # needs sage.plot S.expand().data S.expand(corners=False).plot() # needs sage.plot S.expand(corners=False).data
- innermost_point()[source]¶
Return a point well inside the region, specifically the center of (one of) the last tile(s) to be removed on contraction.
EXAMPLES:
sage: import numpy as np sage: from sage.schemes.elliptic_curves.period_lattice_region import PeriodicRegion sage: data = np.zeros((10, 10)) sage: data[1:4, 1:4] = True sage: data[1, 0:8] = True sage: S = PeriodicRegion(CDF(1), CDF(I+1/2), data) sage: S.innermost_point() 0.375 + 0.25*I sage: S.plot() + point(S.innermost_point()) # needs sage.plot Graphics object consisting of 24 graphics primitives
>>> from sage.all import * >>> import numpy as np >>> from sage.schemes.elliptic_curves.period_lattice_region import PeriodicRegion >>> data = np.zeros((Integer(10), Integer(10))) >>> data[Integer(1):Integer(4), Integer(1):Integer(4)] = True >>> data[Integer(1), Integer(0):Integer(8)] = True >>> S = PeriodicRegion(CDF(Integer(1)), CDF(I+Integer(1)/Integer(2)), data) >>> S.innermost_point() 0.375 + 0.25*I >>> S.plot() + point(S.innermost_point()) # needs sage.plot Graphics object consisting of 24 graphics primitives
import numpy as np from sage.schemes.elliptic_curves.period_lattice_region import PeriodicRegion data = np.zeros((10, 10)) data[1:4, 1:4] = True data[1, 0:8] = True S = PeriodicRegion(CDF(1), CDF(I+1/2), data) S.innermost_point() S.plot() + point(S.innermost_point()) # needs sage.plot
- is_empty()[source]¶
Return whether this region is empty.
EXAMPLES:
sage: import numpy as np sage: if int(np.version.short_version[0]) > 1: ....: np.set_printoptions(legacy="1.25") sage: from sage.schemes.elliptic_curves.period_lattice_region import PeriodicRegion sage: data = np.zeros((4, 4)) sage: PeriodicRegion(CDF(2), CDF(2*I), data).is_empty() True sage: data[1,1] = True sage: PeriodicRegion(CDF(2), CDF(2*I), data).is_empty() False
>>> from sage.all import * >>> import numpy as np >>> if int(np.version.short_version[Integer(0)]) > Integer(1): ... np.set_printoptions(legacy="1.25") >>> from sage.schemes.elliptic_curves.period_lattice_region import PeriodicRegion >>> data = np.zeros((Integer(4), Integer(4))) >>> PeriodicRegion(CDF(Integer(2)), CDF(Integer(2)*I), data).is_empty() True >>> data[Integer(1),Integer(1)] = True >>> PeriodicRegion(CDF(Integer(2)), CDF(Integer(2)*I), data).is_empty() False
import numpy as np if int(np.version.short_version[0]) > 1: np.set_printoptions(legacy="1.25") from sage.schemes.elliptic_curves.period_lattice_region import PeriodicRegion data = np.zeros((4, 4)) PeriodicRegion(CDF(2), CDF(2*I), data).is_empty() data[1,1] = True PeriodicRegion(CDF(2), CDF(2*I), data).is_empty()
- plot(**kwds)[source]¶
Plot this region in the fundamental lattice. If
fullisFalse, plots only the lower half. Note that the true nature of this region is periodic.EXAMPLES:
sage: import numpy as np sage: from sage.schemes.elliptic_curves.period_lattice_region import PeriodicRegion sage: data = np.zeros((10, 10)) sage: data[2, 2:8] = True sage: data[2:5, 2] = True sage: data[3, 3] = True sage: S = PeriodicRegion(CDF(1), CDF(I + 1/2), data) sage: plot(S) + plot(S.expand(), rgbcolor=(1, 0, 1), thickness=2) # needs sage.plot Graphics object consisting of 46 graphics primitives
>>> from sage.all import * >>> import numpy as np >>> from sage.schemes.elliptic_curves.period_lattice_region import PeriodicRegion >>> data = np.zeros((Integer(10), Integer(10))) >>> data[Integer(2), Integer(2):Integer(8)] = True >>> data[Integer(2):Integer(5), Integer(2)] = True >>> data[Integer(3), Integer(3)] = True >>> S = PeriodicRegion(CDF(Integer(1)), CDF(I + Integer(1)/Integer(2)), data) >>> plot(S) + plot(S.expand(), rgbcolor=(Integer(1), Integer(0), Integer(1)), thickness=Integer(2)) # needs sage.plot Graphics object consisting of 46 graphics primitives
import numpy as np from sage.schemes.elliptic_curves.period_lattice_region import PeriodicRegion data = np.zeros((10, 10)) data[2, 2:8] = True data[2:5, 2] = True data[3, 3] = True S = PeriodicRegion(CDF(1), CDF(I + 1/2), data) plot(S) + plot(S.expand(), rgbcolor=(1, 0, 1), thickness=2) # needs sage.plot
- refine(condition=None, times=1)[source]¶
Recursive function to refine the current tiling.
INPUT:
condition– function (default:None); if notNone, only keep tiles in the refinement which satisfy the conditiontimes– integer (default: 1); the number of times to refine. Each refinement step halves the mesh size.
OUTPUT: the refined PeriodicRegion
EXAMPLES:
sage: import numpy as np sage: from sage.schemes.elliptic_curves.period_lattice_region import PeriodicRegion sage: data = np.zeros((4, 4)) sage: S = PeriodicRegion(CDF(2), CDF(2*I), data, full=False) sage: S.ds() (0.5, 0.25*I) sage: S = S.refine() sage: S.ds() (0.25, 0.125*I) sage: S = S.refine(2) sage: S.ds() (0.125, 0.0625*I)
>>> from sage.all import * >>> import numpy as np >>> from sage.schemes.elliptic_curves.period_lattice_region import PeriodicRegion >>> data = np.zeros((Integer(4), Integer(4))) >>> S = PeriodicRegion(CDF(Integer(2)), CDF(Integer(2)*I), data, full=False) >>> S.ds() (0.5, 0.25*I) >>> S = S.refine() >>> S.ds() (0.25, 0.125*I) >>> S = S.refine(Integer(2)) >>> S.ds() (0.125, 0.0625*I)
import numpy as np from sage.schemes.elliptic_curves.period_lattice_region import PeriodicRegion data = np.zeros((4, 4)) S = PeriodicRegion(CDF(2), CDF(2*I), data, full=False) S.ds() S = S.refine() S.ds() S = S.refine(2) S.ds()
- verify(condition)[source]¶
Given a condition that should hold for every line segment on the boundary, verify that it actually does so.
INPUT:
condition– boolean-valued function on \(\CC\)
OUTPUT:
boolean according to whether the condition holds for all lines on the boundary.
EXAMPLES:
sage: import numpy as np sage: from sage.schemes.elliptic_curves.period_lattice_region import PeriodicRegion sage: data = np.zeros((4, 4)) sage: data[1, 1] = True sage: S = PeriodicRegion(CDF(1), CDF(I), data) sage: S.border() [(1, 1, 0), (2, 1, 0), (1, 1, 1), (1, 2, 1)] sage: condition = lambda z: z.real().abs()<1/2 sage: S.verify(condition) False sage: condition = lambda z: z.real().abs()<1 sage: S.verify(condition) True
>>> from sage.all import * >>> import numpy as np >>> from sage.schemes.elliptic_curves.period_lattice_region import PeriodicRegion >>> data = np.zeros((Integer(4), Integer(4))) >>> data[Integer(1), Integer(1)] = True >>> S = PeriodicRegion(CDF(Integer(1)), CDF(I), data) >>> S.border() [(1, 1, 0), (2, 1, 0), (1, 1, 1), (1, 2, 1)] >>> condition = lambda z: z.real().abs()<Integer(1)/Integer(2) >>> S.verify(condition) False >>> condition = lambda z: z.real().abs()<Integer(1) >>> S.verify(condition) True
import numpy as np from sage.schemes.elliptic_curves.period_lattice_region import PeriodicRegion data = np.zeros((4, 4)) data[1, 1] = True S = PeriodicRegion(CDF(1), CDF(I), data) S.border() condition = lambda z: z.real().abs()<1/2 S.verify(condition) condition = lambda z: z.real().abs()<1 S.verify(condition)