511 research outputs found

    Some Definability Results in Abstract Kummer Theory

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    Let SS be a semiabelian variety over an algebraically closed field, and let XX be an irreducible subvariety not contained in a coset of a proper algebraic subgroup of SS. We show that the number of irreducible components of [n]1(X)[n]^{-1}(X) is bounded uniformly in nn, and moreover that the bound is uniform in families XtX_t. We prove this by purely Galois-theoretic methods. This proof applies in the more general context of divisible abelian groups of finite Morley rank. In this latter context, we deduce a definability result under the assumption of the Definable Multiplicity Property (DMP). We give sufficient conditions for finite Morley rank groups to have the DMP, and hence give examples where our definability result holds.Comment: 21 pages; minor notational fixe

    Generic Automorphisms and Green Fields

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    We show that the generic automorphism is axiomatisable in the green field of Poizat (once Morleyised) as well as in the bad fields which are obtained by collapsing this green field to finite Morley rank. As a corollary, we obtain "bad pseudofinite fields" in characteristic 0. In both cases, we give geometric axioms. In fact, a general framework is presented allowing this kind of axiomatisation. We deduce from various constructibility results for algebraic varieties in characteristic 0 that the green and bad fields fall into this framework. Finally, we give similar results for other theories obtained by Hrushovski amalgamation, e.g. the free fusion of two strongly minimal theories having the definable multiplicity property. We also close a gap in the construction of the bad field, showing that the codes may be chosen to be families of strongly minimal sets.Comment: Some minor changes; new: a result of the paper (Cor 4.8) closes a gap in the construction of the bad fiel

    Model theory of special subvarieties and Schanuel-type conjectures

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    We use the language and tools available in model theory to redefine and clarify the rather involved notion of a {\em special subvariety} known from the theory of Shimura varieties (mixed and pure)

    Involutive automorphisms of NN_\circ^\circ groups of finite Morley rank

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    We classify a large class of small groups of finite Morley rank: NN_\circ^\circ-groups which are the infinite analogues of Thompson's NN-groups. More precisely, we constrain the 22-structure of groups of finite Morley rank containing a definable, normal, non-soluble, NN_\circ^\circ-subgroup

    Chai's Conjecture and Fubini properties of dimensional motivic integration

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    We prove that a conjecture of Chai on the additivity of the base change conductor for semi-abelian varieties over a discretely valued field is equivalent to a Fubini property for the dimensions of certain motivic integrals. We prove this Fubini property when the valued field has characteristic zero.Comment: 22 page

    Expansions of fields by angular functions

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