The Lamé family of connections on the projective line

Abstract

This paper deals with rank two connections on the projective line having four simple poles with prescribed local exponents 1/4 and -1/4. This Lame family of connections has been extensively studied in the literature. The differential Galois group of a Lame connection is never maximal: it is either dihedral (finite or infinite) or reducible. We provide an explicit moduli space of those connections having a free underlying vector bundle and compute the algebraic locus of those reducible connections. The irreducible Lame connections are derived from the rank 1 regular connections on the Legendre elliptic curve w^2=z(z-1)(z-t); those connections having a finite Galois group are known to be related to points of finite order on the elliptic curve. In the paper, we provide a very efficient algorithm to compute the locus of those Lame connections having a finite Galois group of a given order. We also give an efficient algorithm to compute the minimal polynomial for the corresponding field extension. We do this computation for low order and recover this way known algebraic solutions of the Painleve VI equation and of the classical Lame equation. In the final section we compare our moduli space with the classical one due to Okamoto.