Primitive unimodular groups of degree 2 as differential Galois groups
In this paper a method is given for the construction of irreducible linear differential equations of order 2 with Galois group a primitive unimodular group of degree 2, i.e., one of the groups \(2A_4\), \(2S_4\), or \(2A_5\), which are double covers of \(G=A_4\), \(S_4\), or \(A_5\), respectively. Let \(k\) be a differential field of characteristic 0 with algebraically closed field of constants. The construction starts with a polynomial \(Q(X)\in k[X]\) with Galois group \(G\) over \(k\) such that the Galois embedding problem associated to the double cover \(2G\) over \(k\) is solvable. Using previous work of the first author, one can compute an element \(\gamma\in K\), \(K\) the splitting field of \(Q(X)\), such that \(K(\sqrt\gamma)/k\) has Galois group \(2G\). The authors determine explicitly in each case a basis of the invariant space of dimension 2 corresponding to an irreducible differential equation of order 2 of the group \(2G\) and they reduce the problem to the resolution of a system of linear equations. The presentation of explicit examples is omitted because, as the authors explain, the coefficients of differential equations obtained are too big. On the other hand, using other methods, \textit{M. van der Put} and \textit{F. Ulmer} in [J. Algebra 226, 920-966 (2000; Zbl 0979.12003)] have constructed differential equations with Galois groups \(2A_4\), \(2S_4\) and \(2A_5\), contained in the list of Schwarz for hypergeometric differential equations.
scientific article; zbMATH DE number 1502171
scientific article