Mathieu-group coverings of the affine line
Let \(L_ k\) be the affine line over an algebraically closed field \(k\), with \(\text{char }k=p\neq 0\). It has been conjectured by the first author in a previous paper that the algebraic fundamental group \(\pi_ A(L_ k)=Q(p)\), the set of all quasi \(p\)-groups. In this paper, several Galois groups of coverings of \(L_ k\) are computed, in support of this conjecture. Recall that \(\pi_ A(L_ k)\) is the set of all finite Galois groups of unramified coverings of \(L_ k\), while a quasi \(p\)-group is a finite group which is generated by all its \(p\)-Sylow subgroups. Namely, the Galois group of the coverings given by equations of the following types are computed: \[ y^{11}-ax^ sy^ 2+1=0;\;y^{22}-ax^ sy^ 4+b=0;\;y^{12}-ay+x^ s=0;\;y^{11}+y^ 8+y^ 2-ax^{-s}=0, \] where \(p=3\), \(a,b\) are non-zero elements of \(k\) and \(s\equiv 0(11)\) in the last two equations. The case of the equation \(y^{10}-ax^{-s}y^ 2-y-1=0\) is studied also, and this gives an unramified covering of the punctured affine line \(L_{k,1}\) which is tamely ramified at the puncture. -- Most of these examples belong to classes of coverings which have already received attention in the literature, and enlarge the range of the computed cases in these classes. Note that the Galois groups that come out in these examples are the group \(M_{11}\) of the sharply 4-transitive permutations of degree 11, two different ``incarnations'' of \(M_{11}\), \(M^*_{11}\) and \(\widehat M_{11}\), both isomorphic to \(M_{11}\) as abstract groups but not as permutation group, the one point stabilizers of \(M_{11}\) (in the punctured case) and of \(\widehat M_{11}\). -- Some observations about what happens when one drops the hypothesis ``\(k\) algebraically closed'' can also be found.
Galois groups of coverings of affine line
scientific article; zbMATH DE number 179217
scientific article