de.jtem.riemann.theta

Class ModularPropertySupport

java.lang.Object

de.jtem.riemann.theta.ModularPropertySupport

All Implemented Interfaces:

Serializable, Cloneable

public class ModularPropertySupport
extends Object
implements Serializable, Cloneable

Supports the use of the modular property of Riemann theta functions. Unlike the transformation property is the modular transformation property a deep result in the theory of Riemann theta functions. For all elements

σ =	/ a b \ \ c d /

of the ModularGroup of degree 2g holds:

θ(z|B) = k exp( fσ(z|B) ) θ( σ(z|B) | σ(B) ) ,

with

fσ(z|B)	=	Rσ(B)tz - ztAσ(B)z + δσ(B)
σ(z|B)	=	Hσ(B)z + Sσ(B)
σ(B)	=	2πi(aB+2πib)(cB+2πid)-1

and further with

Rσ(B)	=	πi (cB + 2πid)-1Diag(cdt)
Aσ(B)	=	½ ((cB + 2πid)-1)c
δσ(B)	=	½ ( ¼ Diag(cdt)tσ(B)Diag(cdt) - g log(2πi) - 2log( (det(cB +2πid) )½ )
Hσ(B)	=	2πi( (cB + 2πid)-1)t
Sσ(B)	=	πi Diag(abt) + ½ σ(B)Diag(cdt)

Finally k is a nasty 8^th-root of unity, which we do not control. This is usually neglectable because one only deals with ratios of theta functions.

To use this class you have to provide a period Matrix B (setPeriodMatrix(de.jtem.blas.ComplexMatrix)), an element of the ModularGroup (#setModular) and, the argument vector z (setZ(de.jtem.blas.ComplexVector)). ModularPropertySupport than determines the transformed argument σ(z|B), the exponential transformation factor fσ(z|B) and, the transformed period matrix σ(B) which can be accessed by calling getTransformedZ(), getFactor() and, getTransformedPeriodMatrix(). Further are Rσ(B), Aσ(B), δσ(B), Hσ(B) and, Sσ(B) attributes of this class.

Author:

Markus Schmies

See Also:

SiegelReduction, ModularTransformation, TransformPropertySupport, Theta, Serialized Form

Constructor Summary

Constructors

Constructor and Description
ModularPropertySupport(de.jtem.blas.ComplexMatrix periodMatrix) Creates instance with presrcibed period matrix.
ModularPropertySupport(de.jtem.blas.ComplexMatrix periodMatrix, ModularTransformation sigma) Creates instance with presrcibed period matrix and modular transformation σ..

Method Summary

Modifier and Type	Method and Description
de.jtem.blas.ComplexMatrix	getA() Returns the quadratic component of the transformation factor.
de.jtem.mfc.field.Complex	getDelta() Returns the constant component of the transformation factor.
de.jtem.mfc.field.Complex	getFactor() Returns the exponential transformation factor.
de.jtem.blas.ComplexMatrix	getH() Returns the linear component of the argument transformation.
ModularTransformation	getModularTransformation() Returns the element of the modular group that determines the transformation.
de.jtem.blas.ComplexMatrix	getPeriodMatrix() Returns period matrix which was transformed.
de.jtem.blas.ComplexVector	getR() Returns the linear component of the transformation factor.
de.jtem.blas.ComplexVector	getS() Returns the constant component of the argument transformation.
de.jtem.blas.ComplexMatrix	getTransformedPeriodMatrix() Returns transformed period matrix.
de.jtem.blas.ComplexVector	getTransformedZ() Returns the transformed argument vector.
de.jtem.blas.ComplexVector	getZ() Returns the argument vector which was transform.
void	setModularTransformation(ModularTransformation modular) Sets the element of the modular group that determines the transformation.
void	setPeriodMatrix(de.jtem.blas.ComplexMatrix aPeriodMatrix) Sets the period matrix which will be transformed.
void	setZ(de.jtem.blas.ComplexVector aZ) Sets the argument vector which will be transformed.
static de.jtem.blas.ComplexMatrix	transformPeriodMatrix(de.jtem.blas.ComplexMatrix B, ModularTransformation sigma) Transforms the period matrix B using the element of the modular group σ.

Methods inherited from class java.lang.Object

clone, equals, finalize, getClass, hashCode, notify, notifyAll, toString, wait, wait, wait

Constructor Detail

ModularPropertySupport

public ModularPropertySupport(de.jtem.blas.ComplexMatrix periodMatrix)

Creates instance with presrcibed period matrix.

Parameters:

periodMatrix - symmetric complex matrix with negative definite real part

ModularPropertySupport

public ModularPropertySupport(de.jtem.blas.ComplexMatrix periodMatrix,
                              ModularTransformation sigma)

Creates instance with presrcibed period matrix and modular transformation σ..

Parameters:

periodMatrix - symmetric complex matrix with negative definite real part

sigma - element of the modular group

Method Detail

getA

public de.jtem.blas.ComplexMatrix getA()

Returns the quadratic component of the transformation factor.

Returns:

Aσ(B) = ½ ((cB + 2πid)-1)c

getDelta

public de.jtem.mfc.field.Complex getDelta()

Returns the constant component of the transformation factor.

Returns:

&delta = ½ ( ¼ Diag(cdt)tσ(B)Diag(cdt) - g log(2πi) - 2log( (det(cB +2πid) )½ )

getFactor

public de.jtem.mfc.field.Complex getFactor()

Returns the exponential transformation factor.

Returns:

fσ(z|B) = Rσ(B)tz - ztAσ(B)z + δσ(B)

getH

public de.jtem.blas.ComplexMatrix getH()

Returns the linear component of the argument transformation.

Returns:

Hσ(B) = 2πi( (cB + 2πid)-1)t

getModularTransformation

public ModularTransformation getModularTransformation()

Returns the element of the modular group that determines the transformation.

getPeriodMatrix

public de.jtem.blas.ComplexMatrix getPeriodMatrix()

Returns period matrix which was transformed.

getR

public de.jtem.blas.ComplexVector getR()

Returns the linear component of the transformation factor.

Returns:

Rσ(B) = πi (cB + 2πid)-1Diag(cdt)

getS

public de.jtem.blas.ComplexVector getS()

Returns the constant component of the argument transformation.

Returns:

Sσ(B) = πi Diag(abt) + ½ σ(B)Diag(cdt)

getTransformedPeriodMatrix

public de.jtem.blas.ComplexMatrix getTransformedPeriodMatrix()

Returns transformed period matrix.

Returns:

&sigma(B) = 2πi(aB+2πib)(cB+2πid)-1

getTransformedZ

public de.jtem.blas.ComplexVector getTransformedZ()

Returns the transformed argument vector.

Returns:

σ(z|B) = Hσ(B)z + Sσ(B)

getZ

public de.jtem.blas.ComplexVector getZ()

Returns the argument vector which was transform.

setModularTransformation

public void setModularTransformation(ModularTransformation modular)

Sets the element of the modular group that determines the transformation.

Parameters:

modular - element of the modular group

setPeriodMatrix

public void setPeriodMatrix(de.jtem.blas.ComplexMatrix aPeriodMatrix)

Sets the period matrix which will be transformed.

Parameters:

periodMatrix - symmetric complex matrix with negative definite real part

setZ

public void setZ(de.jtem.blas.ComplexVector aZ)

Sets the argument vector which will be transformed.

Parameters:

z - argument vector

transformPeriodMatrix

public static de.jtem.blas.ComplexMatrix transformPeriodMatrix(de.jtem.blas.ComplexMatrix B,
                                                               ModularTransformation sigma)

Transforms the period matrix B using the element of the modular group σ. The period matrix is complex a symmetric complex matrix with negative definite real part. σ is of the form:

/ A B \
\ C D /	,

Parameters:

periodMatrix - symmetric complex matrix with negative definite real part

sigma - element of the modular group

Returns:

2πi(aB+2πib)(cB+2πid)^-1