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:

The real part of the new Riemann matrix has entries that are less than or equal to $\left|\frac{1}{2}\right|$.

The length of the shortest element of L has a lower bound of $\sqrt{\frac{\sqrt{3}}{2}}$,

$\max \left(\left|N_i\right|\right)$ : {${\left|\mathrm{TN}\right|}^2\le R^2$, $N$ 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 $\left[\mathrm{s1}\,\mathrm{s2}\right]$ where $\mathrm{s1}$ is the new Riemann matrix, and $\mathrm{s2}$ is the symplectic transformation matrix on the canonical basis of the homology such that the Riemann matrix in the new basis is $\mathrm{s1}$. If B is a $g$ by $g$ matrix, then $\mathrm{s2}$ is a $2g$ by $2g$ matrix. If $\mathrm{s2}=\mathrm{Matrix}\left(\left[\left[a\,b\right]\,\left[c\,d\right]\right]\right)$, where $a,b,c$, and $d$ are $g$ by $g$ matrices, the new Riemann matrix is $\mathrm{s1}=\frac{aB+b}{cB+d}$.

$\mathrm{with}\left(\mathrm{algcurves}\right)\:$

$f≔y^3-x^9-2x^{3}y\:$

$b≔\mathrm{periodmatrix}\left(f\,x\,y\,\mathrm{Riemann}\right)$

$b≔\left[\begin{array}{ccc}0.500000060072942+0.959847052703856\mathrm{I}& \mathrm{-0.500000035130138}-0.181985139050623\mathrm{I}& 0.641559304403873+0.570916115123785\mathrm{I}\\ \mathrm{-0.500000001775655}-0.181985124346498\mathrm{I}& 0.499999991527753+0.866025410504281\mathrm{I}& \mathrm{-0.907603727073024}-0.524005265810045\mathrm{I}\\ 0.641559342066582+0.570916143568672\mathrm{I}& \mathrm{-0.907603764292705}-0.524005276538734\mathrm{I}& 0.549163001208095+1.23866447641924\mathrm{I}\end{array}\right]$

$s≔\mathrm{Siegel}\left(b\right)\:$

$s\left[1\right]$

$\left[\begin{array}{ccc}0.499999991527753+0.866025410504281\mathrm{I}& 0.499999981547104-0.181985131698561\mathrm{I}& \mathrm{-0.407603727229968}-0.342020139475828\mathrm{I}\\ 0.499999981547104-0.181985131698561\mathrm{I}& \mathrm{-0.499999939927058}+0.959847052703856\mathrm{I}& 0.141559263162286-0.388930923357628\mathrm{I}\\ \mathrm{-0.407603727229968}-0.342020139475828\mathrm{I}& 0.141559263162286-0.388930923357628\mathrm{I}& \mathrm{-0.233955585189418}+1.05667927043064\mathrm{I}\end{array}\right]$

$s\left[2\right]$

$\left[\begin{array}{cccccc}0& 1& 0& 1& 0& 1\\ 1& 0& 0& \mathrm{-1}& 1& \mathrm{-1}\\ \mathrm{-1}& 0& 1& 0& 0& 0\\ 0& 0& 0& 0& 1& 0\\ 0& 0& 0& 1& 0& 1\\ 0& 0& 0& 0& 0& 1\end{array}\right]$

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.