Communications in Mathematical Physics

, Volume 355, Issue 2, pp 741–766 | Cite as

An Elliptic Garnier System



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 of a set of kernels at the singular points. We regard the system of discrete isomonodromic deformations as an elliptic analogue of the Garnier system. We identify the special case in which m = 1 with the elliptic Painlevé equation; hence, this work provides an explicit form and Lax pair for the elliptic Painlevé equation.


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© Springer-Verlag GmbH Germany 2017

Authors and Affiliations

  1. 1.Department of Mathematics and StatisticsUniversity of MaineOronoUSA
  2. 2.CaltechPasadenaUSA

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