Semi-abelian Schemes and Heights of Cycles in Moduli Spaces of abelian Varieties
The main result of this paper implies that if an abelian variety over a field has a maximal isotropic subgroup of -torsion points all of which are defined over , and , then the abelian variety has semistable reduction away from . This result can be viewed as an extension of Raynaud’s theorem that if an abelian variety and all its -torsion points are defined over a field and , then the abelian variety has semistable reduction away from . We also give information about the Néron models...
We study the local factor at of the semi-simple zeta function of a Shimura variety of Drinfeld type for a level structure given at by the pro-unipotent radical of an Iwahori subgroup. Our method is an adaptation to this case of the Langlands-Kottwitz counting method. We explicitly determine the corresponding test functions in suitable Hecke algebras, and show their centrality by determining their images under the Hecke algebra isomorphisms of Goldstein, Morris, and Roche.
Classical sieve methods of analytic number theory have recently been adapted to a geometric setting. In the new setting, the primes are replaced by the closed points of a variety over a finite field or more generally of a scheme of finite type over . We will present the method and some of the surprising results that have been proved using it. For instance, the probability that a plane curve over is smooth is asymptotically as its degree tends to infinity. Much of this paper is an exposition...