Differential Galois realization of double covers
Let \(G\) be one of the alternating groups \(A_4\), \(A_5\), or the symmetric group \(S_4\), and let \(2G\) be one of its double covers. Let \(k\) be a differential field of characteristic \(0\) with algebraically closed field of constants \(\mathcal C\). The authors are interested in exhibiting \(2G\) as the (polynomial or differential) Galois group of a field extension \(\widetilde{K} \supset k\). More precisely, they want to produce explicit second order differential equations over \(k\) whose differential Galois group is \(2G\). If there is an extension \(\widetilde{K}\), it contains a subextension \(K \supset k\) with Galois group \(G\). Thus they phrase their main result as follows: with \(k\), \(G\), \(2G\) as above, suppose that \(K \supset k\) is the splitting field of a polynomial \(P(x) \in k[X]\) with Galois group \(G\). Then there is a quadratic form \(Q\) in three variables over \(k\) such that the Galois embedding problem \( 2G \to G = \text{Gal}(K/k)\) is solvable if and only if \(Q\) represents \(0\) (nontrivially) over \(k\). Then for each such representation there are explicitly computable elements \(A,B,C\) of \(k\) such that \(Y^{\prime \prime \prime}+AY^{\prime \prime}+BY^\prime+CY=0\) has \(K\) as a differential Galois extension, while \(Y^{\prime \prime}+(A/3)Y'+(1/4)(B-(2/9)A^2-(1/3)A')Y=0\) has differential Galois group \(2G\). For the case that \(k=\overline{\mathbb Q}(T)\), they exhibit the constants \(A,B,C\) (and hence the corresponding differential equations) starting from explicit choices for \(P(X)\).
scientific article; zbMATH DE number 1794821