The period-index problem in WC-groups IV: a local transition theorem

Pete L. Clark

Displaying similar documents to “The period-index problem in WC-groups IV: a local transition theorem”

The Tate pairing for Abelian varieties over finite fields

Peter Bruin (2011)

Journal de Théorie des Nombres de Bordeaux

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In this expository note, we describe an arithmetic pairing associated to an isogeny between Abelian varieties over a finite field. We show that it generalises the Frey–Rück pairing, thereby giving a short proof of the perfectness of the latter.

On elementary equivalence, isomorphism and isogeny

Pete L. Clark (2006)

Journal de Théorie des Nombres de Bordeaux

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Motivated by recent work of Florian Pop, we study the connections between three notions of equivalence of function fields: isomorphism, elementary equivalence, and the condition that each of a pair of fields can be embedded in the other, which we call isogeny. Some of our results are purely geometric: we give an isogeny classification of Severi-Brauer varieties and quadric surfaces. These results are applied to deduce new instances of “elementary equivalence implies isomorphism”: for...

Explicit bounds for split reductions of simple abelian varieties

Jeffrey D. Achter (2012)

Journal de Théorie des Nombres de Bordeaux

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Let $X / K$ be an absolutely simple abelian variety over a number field; we study whether the reductions $X_{𝔭}$ tend to be simple, too. We show that if $End (X)$ is a definite quaternion algebra, then the reduction $X_{𝔭}$ is geometrically isogenous to the self-product of an absolutely simple abelian variety for $𝔭$ in a set of positive density, while if $X$ is of Mumford type, then $X_{𝔭}$ is simple for almost all $𝔭$ . For a large class of abelian varieties with commutative absolute endomorphism ring, we give an explicit...

Wintenberger’s functor for abelian extensions

Kevin Keating (2009)

Journal de Théorie des Nombres de Bordeaux

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Let $k$ be a finite field. Wintenberger used the field of norms to give an equivalence between a category whose objects are totally ramified abelian $p$ -adic Lie extensions $E / F$ , where $F$ is a local field with residue field $k$ , and a category whose objects are pairs $(K, A)$ , where $K ≅ k ((T))$ and $A$ is an abelian $p$ -adic Lie subgroup of ${Aut}_{k} (K)$ . In this paper we extend this equivalence to allow $Gal (E / F)$ and $A$ to be arbitrary abelian pro- $p$ groups.

Integral canonical models of Shimura varieties

Mark Kisin (2009)

Journal de Théorie des Nombres de Bordeaux

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The aim of these notes is to provide an introduction to the subject of integral canonical models of Shimura varieties, and then to sketch a proof of the existence of such models for Shimura varieties of Hodge and, more generally, abelian type. For full details the reader is refered to [Ki 3].