The Jacobi inversion problem can be used to accomplish addition on a curve.
As mentioned in the beginning of this discussion, a divisor
$D$
can be of arbitrary degree
$N\geqslant g$.
Then we compute
$\mathcal{A}(D)$ point by point, and find values of the eight
$\wp$-functions at
$u=\mathcal{A}(D)$. With these
$\wp$-functions entire rational functions
$\mathcal{R}_{8}$,
$\mathcal{R}_{9}$ are obtained.
The divisor
$\tilde{D}$ of common zeros of
$\mathcal{R}_{8}$,
$\mathcal{R}_{9}$ is of degree
$4$. This is the reduced divisor linearly equivalent to
$D$. In order to find the common divisor, we use NSolve.
Example 1c: Addition of one point
We use the points
$P_1$,
$P_2$,
$P_3$,
$P_4$ which form the divisor
$D$ from Example 1a, and add one more point
$$P_5 = \big( 7-\imath, -\sqrt{\mathcal{P}(7-\imath)}\big) \approx (7-\imath, -4099.87 - 27571.7 \imath).$$
$P_5$ is located on Sheet 2.
The Abel image of
$P_5$ can be computed along a path
$\gamma_{P_5}$ on Sheet 2, as shown in Example 1a. We choose another path
$\tilde{\gamma}_{P_5}$: from
$-\infty$ to
$e_8$ on Sheet 1, and from
$e_8$ to
$P_5$ on Sheet 2:
$$ \mathcal{A}(P_5) =
\mathcal{A}_{0,1}^{[+]}
+ \mathcal{A}_{1,2}^{[-]}
+ \mathcal{A}_{2,3}^{[-]}
+ \mathcal{A}_{3,4}^{[-]}
+ \mathcal{A}_{4,5}^{[+]}
+ \mathcal{A}_{5,6}^{[-]}
+ \mathcal{A}_{6,7}^{[-]}
+ \mathcal{A}_{7,8}^{[-]}
+ \int_{e_8}^{P_5} \mathrm{d} u^{[-]}.
$$
The two Abel images differ in a full period:
$$\omega^{-1} \big(\mathcal{A}(P_5; \tilde{\gamma}_{P_5}) - \mathcal{A}(P_5; \gamma_{P_5}) \big)
= \begin{pmatrix} 1 \\ 1 \\ 1 \\ 1 \end{pmatrix} + \tau \, \begin{pmatrix} 0 \\ 0 \\ 0 \\ 1 \end{pmatrix}.$$
Then we compute values of the eight
$\wp$-functions at
$u=\sum_{i=1}^5
\mathcal{A}(P_i)$. These eight values define a degree
$4$ divisor
$\tilde{D}$ on
$\mathcal{C}$, which is called the reduced divisor. This is the unique representative of the degree equal to the genus of a curve in a class of linearly equivalent non-special divisors. Divisor
$\tilde{D}$ serves as
a result of adding point
$P_5$ to
$D$.
Example 1d: Addition on divisors
We add two divisors
$D$ from Example 1a. Thus, the Abel image is
$2 \mathcal{A}(D)$.
Values of the eight
$\wp$-functions are computed at
$u=2\mathcal{A}(D)$, and the corresponding
$\mathcal{R}_{8}$ and
$\mathcal{R}_{9}$ are obtained.
The divisor
$\tilde{D}$ of common zeros
of
$\mathcal{R}_{8}$ and
$\mathcal{R}_{9}$ gives the divisor of degree
$4$ linearly equivalent to
$2D$.
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