Regions in fundamental domains of period lattices

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 [TT].

AUTHORS:

  • Robert Bradshaw (2010): initial version
  • John Cremona (2014): added some docstrings and doctests

REFERENCES:

[T]T. Thongjunthug, Computing a lower bound for the canonical height on elliptic curves over number fields, Math. Comp. 79 (2010), pages 2431-2449.
class sage.schemes.elliptic_curves.period_lattice_region.PeriodicRegion

Bases: object

EXAMPLE:

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()
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()
Graphics object consisting of 5 graphics primitives
border(raw=True)

Returns 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)]
contract(corners=True)

Opposite (but not inverse) of expand; removes neighbors of complement.

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: S = PeriodicRegion(CDF(1), CDF(I + 1/2), data)
sage: S.plot()
Graphics object consisting of 13 graphics primitives
sage: S.contract().plot()
Graphics object consisting of 5 graphics primitives
sage: S.contract().data.sum()
1
sage: S.contract().contract().is_empty()
True
data
ds()

Returns 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)
expand(corners=True)

Returns 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()
Graphics object consisting of 5 graphics primitives
sage: S.expand().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()
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)
full
innermost_point()

Returns 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())
Graphics object consisting of 24 graphics primitives
is_empty()

Returns whether this region is empty.

EXAMPLES:

sage: import numpy as np
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
plot(**kwds)

Plots this region in the fundamental lattice. If full is False 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)
Graphics object consisting of 46 graphics primitives
refine(condition=None, times=1)

Recursive function to refine the current tiling.

INPUT:

  • condition (function, default None) - if not None, only keep tiles in the refinement which satisfy the condition.
  • times (int, 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)
verify(condition)

Given a condition that should hold for every line segment on the boundary, verify that it actually does so.

INPUT:

  • condition (function) - a boolean-valued function on \(\CC\).

OUTPUT:

True or False 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()<0.5
sage: S.verify(condition)
False
sage: condition = lambda z: z.real().abs()<1
sage: S.verify(condition)
True
w1
w2

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