Etale coverings of \(p\)-adic disks (Q2776768)

Let \(K\) be an algebraically closed field which is complete with respect to a non-archimedean valuation. Any étale covering of the (closed) unit disk in \(K\) splits totally over a small disk around 0, and any étale covering of an annulus \(\{z\in K:r_1 \leq|z|\leq r_2\}\) is of ``Kummer type'' over a sufficiently small annulus. \textit{W. Lütkebohmert} showed [Invent. Math. 111, No. 2, 309-330 (1993; Zbl 0780.32005)] that if char\( (K)=0\) there is a lower bound for the radius of that disk (respectively annulus) which only depends on the degree of the covering (and the residue characteristic). The main technical tool in Lütkebohmert's proof is the discriminant of a finite, generically étale covering of a disk or an annulus (which is defined using the discriminant of a finite separable field extension). NEWLINENEWLINENEWLINEIn the present thesis the author systematically studies this discriminant as a function on (an interval of) the value group of \(K\). In particular, he shows that it is continuous, and that Lütkebohmert's bound on the radius can be expressed using this function (which also gives the corresponding result in positive characteristic).NEWLINENEWLINENEWLINEThe second chapter of this thesis is devoted to the genus of an affinoid curve, which is defined using rigid analytic reduction. The author then proves the following criterion: A finite étale covering of the punctured disk can be extended to such a covering of the whole disk if and only if the genus of affinoid curves contained in the covering space is bounded. The paper ends with an example in positive characteristic where such an extension is not possible.