Input: C hyperelliptic curve over an unramified extension K of ${\mathbb{Q}}_p$ with p a prime of good reduction Points P, Q on C.

Output: ${\displaystyle {\int }_P^Q}{\omega }_i$ .

1) Find Teichmüller points $P\mathrm{,}Q$ in the disks of $P\mathrm{,}Q$ .

2) Compute the tiny integrals ${\displaystyle {\int }_P^{P^{\prime }}{\omega }_i}$ and ${\displaystyle {\int }_Q^{Q^{\prime }}{\omega }_i}$ .

3) Calculate the action of Frobenius on each basis element ${\left({\varphi }^m\right)}^{\mathrm{<mo>\; *\; </mo>}}{\omega }_i={\displaystyle {\sum }_j^{2g-1}}{M}_{ij}{\omega }_j+d{f}_i$ .

4) Change of variables gives ${\displaystyle {\sum }_{j=0}^{2g-1}}\left(M-I\right){\displaystyle {\int }_{P^{\prime }}^{Q^{\prime }}}{\omega }_j=f_i\left(Q\right)-f_i\left(Q\right)$ and solving the linear system gives the integrals ${\displaystyle {\int }_{P^{\prime }}^{Q^{\prime }}}{\omega }_i$ .

5) Correct endpoints to recover ${\displaystyle {\int }_P^Q}{\omega }_i-{\displaystyle {\int }_P^{P^{\prime }}}{\omega }_i+{\displaystyle {\int }_{P^{\prime }}^{Q^{\prime }}}{\omega }_i+{\displaystyle {\int }_{Q^{\prime }}^Q}{\omega }_i$ .

6) Return ${\displaystyle {\int }_P^Q}{\omega }_i$ .