use Siegel's algorithm for reducing a Riemann matrix - Maple Help

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algcurves[Siegel] - use Siegel's algorithm for reducing a Riemann matrix

Calling Sequence

Siegel(B)

Parameters

B

-

Riemann matrix

Description

• 

A Riemann matrix is a symmetric matrix whose imaginary part is strictly positive definite. In the context of algebraic curves, such a matrix is obtained as a normalized periodmatrix of the algebraic curve.

• 

A Siegel transformation is a transformation from the canonical basis of the homology of a Riemann surface to a new canonical basis of the homology on the Riemann surface such that:

1. 

The real part of the new Riemann matrix has entries that are less than or equal to 12.

The imaginary part of B is strictly positive definite. Then it can be decomposed as B=transposeTT. The columns of T generate a lattice L. Then

2. 

The length of the shortest element of L has a lower bound of 123,

and

3. 

maxNi : {TN2R2, N an integer vector} has an upper bound depending only on R and g (=dimension of B) (thus not on B).

• 

The Siegel(B) command returns a list s1,s2 where s1 is the new Riemann matrix, and s2 is the symplectic transformation matrix on the canonical basis of the homology such that the Riemann matrix in the new basis is s1. If B is a g by g matrix, then s2 is a 2g by 2g matrix. If s2=Matrixa,b,c,d, where a,b,c, and d are g by g matrices, the new Riemann matrix is s1=Ba+bBc+d.

Examples

withalgcurves:

f:=y3x92x3y:

b:=periodmatrixf,x,y,Riemann

b:=0.500000055125149+0.959847047127724I0.5000000301918730.181985134690607I0.641559302373033+0.570916110133834I0.5000000019703520.181985123303614I0.499999991378705+0.866025409210244I0.9076037269559980.524005264964518I0.641559334375792+0.570916144445057I0.9076037573826220.524005277968408I0.549162995371542+1.23866447525944I

(1)

s:=Siegelb:

s1

0.499999991378705+0.866025409210244I0.4999999839188880.181985128997111I0.4076037260881980.342020142469353I0.4999999839188880.181985128997111I0.499999944874851+0.959847047127724I0.1415592632492630.388930919838278I0.4076037260881980.342020142469353I0.1415592632492630.388930919838278I0.233955586252134+1.05667926780828I

(2)

s2

010101100111101000000010000101000001

(3)

See Also

algcurves[homology], algcurves[periodmatrix], RiemannTheta

References

  

Deconinck, B., and van Hoeij, M. "Computing Riemann Matrices of Algebraic Curves." Physica D Vol 152-153, (2001): 28-46.

  

Siegel, C. L. Topics in Complex Function Theory. Vol. 3. Now York: Wiley, 1973.


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