An elliptic Garnier system

Understand

We present a linear system of difference equations whose entries are expressed in terms of theta functions.

• This linear system is singular at $4m+12$ points for $m \geq 1$, which appear in pairs due to a symmetry condition.
• We parameterize this linear system in terms a set of kernels at the singular points.
• We regard the system of discrete isomonodromic deformations as an elliptic analogue of the Garnier system.