arXiv subjects
Publications and source records attributed to Torsten Ekedahl.
In this paper we find an explicit formula for the number of topologically different ramified coverings $C\to\CP^1$ (C is a compact Riemann surface of genus g) with only one complicated branching point in terms of Hodge integrals over the moduli space of genus g curves with marked points.
The enumeration of points on (or off) the union of some linear or affine subspaces over a finite field is dealt with in combinatorics via the characteristic polynomial and in algebraic geometry via the zeta function. We discuss the basic relations between these two points of view. Counting points is also related to the $\ell$-adic cohomology of the arrangement (as a variety). We describe the eigenvalues of the Frobenius map acting on this cohomology, which corresponds to a finer decomposition of the zeta function. The $\ell$-adic cohomology groups and their decomposition into eigenspaces are shown to be fully determined by combinatorial data. Finally, it is shown that the zeta function is determined by the topology of the corresponding complex variety in some important cases.
We introduce the notion of a variety (or more generally a motive) of CM-type which generalises the well known notion of abelian variety of CM-type. Just as in that particular case it will turn out that the cohomology of the variety is determined by purely combinatorial data; the type of the variety. As applications we will show that the ł-adic representations are given by algebraic Hecke characters whose algebraic parts are determined by the type and give a method for computing the discriminant of the Néron-Severi group of super-singular Fermat surfaces.
Actions of finite groups on stable curves are studied. They appear naturally at the boundary of a moduli space of smooth curves with group actions. Those actions which can equivariantly smoothed are characterised. A description of topological types of those actions in terms of the quotient curve and mappings from a kind of fundamental group to the given finite group is given, analogous to the well known case of actions on smooth curves.