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Semistable reduction and torsion subgroups of abelian varieties

Alice Silverberg, Yuri G. Zarhin (1995)

Annales de l'institut Fourier

The main result of this paper implies that if an abelian variety over a field F has a maximal isotropic subgroup of n -torsion points all of which are defined over F , and n 5 , then the abelian variety has semistable reduction away from n . This result can be viewed as an extension of Raynaud’s theorem that if an abelian variety and all its n -torsion points are defined over a field F and n 3 , then the abelian variety has semistable reduction away from n . We also give information about the Néron models...

Shimura varieties with Γ 1 ( p ) -level via Hecke algebra isomorphisms: the Drinfeld case

Thomas J. Haines, Michael Rapoport (2012)

Annales scientifiques de l'École Normale Supérieure

We study the local factor at  p of the semi-simple zeta function of a Shimura variety of Drinfeld type for a level structure given at  p 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.

Sieve methods for varieties over finite fields and arithmetic schemes

Bjorn Poonen (2007)

Journal de Théorie des Nombres de Bordeaux

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 𝔽 2 is smooth is asymptotically 21 / 64 as its degree tends to infinity. Much of this paper is an exposition...

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