Calculation of Kleinian hyperelliptic functions for curves of genus $2$

This talk will present an approach to the computer calculation of Kleinian functions associated with a complex curve of genus $2$, similar to the well-known Landen method. The analogue of the Landen transform for genus $2$ is the Richelot transform, which associates a genus $2$ curve with another curve whose period lattice is obtained by doubling all periods from a certain Lagrangian subgroup. For curves with isogenous Jacobians, relations between Kleinian functions can be obtained from an explicit calculation of the coordinate representation of the corresponding isogeny of Kummer surfaces. Ultimately, the calculation of Kleinian functions for a given curve can be reduced to another curve that is isogenous to the original one. Iterations of the Richelot transform reduce the problem to a special curve for which the Kleinian functions are expressed in terms of elementary functions. Like the classical Landen method, the described procedure has a quadratic rate of convergence and therefore represents an effective approach to calculating Kleinian functions.