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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 defined over a field with ring of integers , consider the Néron model of , which is a 'best possible' model of defined over . This model may be represented as a scheme over (cf. spectrum of a ring) for which the generic fibre constructed by means of the morphismgives back . The Néron model is a smooth group scheme, so we can consider , 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 , is a group variety over , hence an extension of an abelian variety by a linear group. If

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  • Semistable abelian variety
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  • 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 defined over a field with ring of integers , consider the Néron model of , which is a 'best possible' model of defined over . This model may be represented as a scheme over (cf. spectrum of a ring) for which the generic fibre constructed by means of the morphismgives back . The Néron model is a smooth group scheme, so we can consider , 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 , is a group variety over , hence an extension of an abelian variety by a linear group. If
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  • 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 defined over a field with ring of integers , consider the Néron model of , which is a 'best possible' model of defined over . This model may be represented as a scheme over (cf. spectrum of a ring) for which the generic fibre constructed by means of the morphismgives back . The Néron model is a smooth group scheme, so we can consider , 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 , is a group variety over , hence an extension of an abelian variety by a linear group. If this linear group is an algebraic torus, so that is a semiabelian variety, then has semistable reduction at the prime corresponding to . If is a global field, then is semistable if it has good or semistable reduction at all primes. The semistable reduction theorem of Alexander Grothendieck states that an abelian variety acquires semistable reduction over a finite extension of .
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