Semistable Abelian Variety
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High Quality Content by WIKIPEDIA articles! High Quality Content by WIKIPEDIA articles! 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. A semistable elliptic curve may be described more concretely as an elliptic curve that has bad reduction only of multiplicative type. Suppose E is an elliptic curve defined...
High Quality Content by WIKIPEDIA articles! High Quality Content by WIKIPEDIA articles! 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. A semistable elliptic curve may be described more concretely as an elliptic curve that has bad reduction only of multiplicative type. Suppose E is an elliptic curve defined over the rational number field Q. It is known that there is a finite, non-empty set S of prime numbers p for which E has bad reduction modulo p. The latter means that the curve Ep obtained by reduction of E to the prime field with p elements has a singular point. Roughly speaking, the condition of multiplicative reduction amounts to saying that the singular point is a double point, rather than a cusp. Deciding whether this condition holds is effectively computable according to Tate's algorithm.
