DIFFERENTIAL EQUATIONS AND FINITE GROUPS

MARIUS VAN DER PUT FELIX ULMER

Abstract. The constructive inverse problem of differential Galois theory for connected linear algebraic groups has been solved by M.F. Singer and C. Mitschi. The theme of the present paper is “perpendicular” to this, namely to give a constructive solution to the Riemann-Hilbert problem for finite groups. More precisely:

Let a finite group G ⊂ GL(n, \overline{Q}) be given with generators g_{0}, g_{1}, g_{∞} satisfying g_{0}g_{1}g_{∞} = 1. Then the aim is to produce a scalar (or matrix) Fuchsian differential equation L with singular set 0, 1, ∞ and of order n, such that g_{0}, g_{1}, g_{∞} are the monodromy matrices for loops around 0, 1, ∞.

The theory developed in the paper produces an algorithm which is effective at least for n = 2 and 3, as is shown by the many new examples found.

The given data determine a Galois covering C → P^{1}, with group G and unramified outside 0, 1, ∞. The algorithm starts by determining whether the given (say irreducible) representation of G is present in the space of the holomorphic differential forms on the curve C. From this information one deduces proposals for the exponents of L. The accessory parameters in L are determined by computing “invariants” for the group G and the equation L. Finally one needs some techniques for verifying that the proposed equation L actually has differential Galois group G.