Studies on monodromy preserving deformation of linear differential equations on elliptic curves

Abstract

We study a monodromy preserving deformation (MPD) of linear differential equations on elliptic curves. As the first of our results, we describe asymptotic behaviors of solutions to the MPD system when the elliptic curve degenerates to a rational curve. As the second, we find explicit solutions for special values of parameters where the MPD system is linearizable. Our solutions are written in terms of integrals of theta functions. We also show that they converge to the hypergeometric functions applying the above asymptotic formula when the elliptic curve degenerates to a rational curve.