Marius van der Put and Felix Ulmer
The {\em 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 {\em constructive} solution to the Riemann-Hilbert problem for finite groups. More precisely:
Let a finite group $G\subset {\rm GL}(n,\overline{\bf Q})$ be given with generators $g_0,g_1,g_{\infty}$ satisfying $g_0g_1g_{\infty}=1$. Then the aim is to produce a scalar (or matrix) Fuchsian differential equation $L$ with singular set $0,1,\infty$ and of order $n$, such that $g_0,g_1,g_{\infty}$ are the monodromy matrices for loops around $0,1,\infty$.
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 $C\rightarrow {\bf P}^1$, with group $G$ and unramified outside $0,1,\infty$. 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 $G$.