
Calling Sequence


AbelMap(F, x, y, P, P_0, t, accuracy)


Parameters


F



irreducible polynomial in x and y specifying a Riemann surface by F(x,y) = 0

x



variable

y



variable

P



Puiseux representation, in a parameter t of a point on the Riemann surface specified by F(x,y)=0

P_0



same as P

accuracy



number of desired accurate decimal digits





Description


•

The AbelMap command computes the Abel map between two points P and P_0 on a Riemann surface R of genus g, that is a gtuple of complex numbers. The jth element of the Abel map is the integral of the jth normalized holomorphic differential integrated along a path from P to P_0.

•

The Riemann surface is entered as F; an irreducible, squarefree polynomial in x and y. Floating point numbers are not allowed as coefficients of F. Algebraic numbers are allowed. Curves of arbitrary finite genus with arbitrary singularities are allowed.

•

The points P and P_0 are entered as $\[x=a+b{t}^{r},y=\left(\mathrm{Laurent\; series}\mathrm{in}t\right)\]$, where a and b are constants, and r is an integer. If r < 0, that is, if entering one of the points for $x=\mathrm{\infty}$, then a = 0.


Note: The Abel map will almost always be computed along with other objects associated with some polynomial F, such as the Riemann matrix. It is imperative that the order of the differential be the same for each of the objects, and at each stage of the calculation. As no order is imposed by algcurves[differentials], make sure to compute AbelMap and, for instance algcurves[periodmatrix], without a restart (or quit) in between.



Notes


•

This command is based on code written by Bernard Deconinck, Michael A. Nivala, and Matthew S. Patterson.



Examples


>

$\mathrm{with}\left(\mathrm{algcurves}\,\mathrm{AbelMap}\,\mathrm{genus}\,\mathrm{puiseux}\right)$

$\left[{\mathrm{AbelMap}}{\,}{\mathrm{genus}}{\,}{\mathrm{puiseux}}\right]$
 (1) 
>

$f\u2254{y}^{2}\left({x}^{2}1\right)\left({x}^{2}4\right)\left({x}^{2}9\right)\left({x}^{2}16\right)$

${f}{\u2254}{{y}}^{{2}}{}\left({{x}}^{{2}}{}{1}\right){}\left({{x}}^{{2}}{}{4}\right){}\left({{x}}^{{2}}{}{9}\right){}\left({{x}}^{{2}}{}{16}\right)$
 (2) 
Give a look first at the genus
>

$\mathrm{genus}\left(f\,x\,y\right)$

>

$\mathrm{puiseux}\left(f\,x=1\,y\,0\,t\right)$

$\left\{\left[{x}{=}{}{720}{}{{t}}^{{2}}{+}{1}{\,}{y}{=}{}{720}{}{t}\right]\right\}$
 (4) 
>

$\mathrm{puiseux}\left(f\,x=4\,y\,0\,t\right)$

$\left\{\left[{x}{=}{10080}{}{{t}}^{{2}}{+}{4}{\,}{y}{=}{10080}{}{t}\right]\right\}$
 (5) 
Compute the Abel map for this curve
>

$\mathrm{P\_0},P\u2254\mathrm{op}\left(\right),\mathrm{op}\left(\right)$

${\mathrm{P\_0}}{,}{P}{\u2254}\left[{x}{=}{}{720}{}{{t}}^{{2}}{+}{1}{\,}{y}{=}{}{720}{}{t}\right]{,}\left[{x}{=}{10080}{}{{t}}^{{2}}{+}{4}{\,}{y}{=}{10080}{}{t}\right]$
 (6) 
>

$A\u2254\mathrm{AbelMap}\left(f\,x\,y\,P\,\mathrm{P\_0}\,t\,7\right)$

${A}{\u2254}\left[{\mathrm{0.5086732390}}{}{1.395818333}{}{\mathrm{I}}{\,}{0.5158465233}{+}{0.3733240360}{}{\mathrm{I}}{\,}{0.00716997551}{}{0.3585270829}{}{\mathrm{I}}\right]$
 (7) 


