Concept

Semistable abelian variety

Summary
In algebraic geometry, a semistable abelian variety is an abelian variety defined over a global or local field, which is characterized by how it reduces at the primes of the field. For an abelian variety A defined over a field F with ring of integers R, consider the Néron model of A, which is a 'best possible' model of A defined over R. This model may be represented as a scheme over \mathrm{Spec}(R) (cf. spectrum of a ring) for which the generic fibre constructed by means of the morphism \mathrm{Spec}(F) \to \mathrm{Spec}(R) gives back A. The Néron model is a smooth group scheme, so we can consider A^0, the connected component of the Néron model which contains the identity for the group law. This is an open subgroup scheme of the Néron model. For a residue field k, A^0_k is a group variety over k, hence an extension
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