Residue field in the context of "Local class field theory"

In mathematics, a local field is a certain type of topological field: by definition, a local field is a locally compact Hausdorff non-discrete topological field. Local fields find many applications in algebraic number theory, where they arise naturally as completions of global fields. Further, tools like integration and Fourier analysis are available for functions defined on local fields.

Given a local field, an absolute value can be defined on it which gives rise to a complete metric that generates its topology. There are two basic types of local field: those called Archimedean local fields in which the absolute value is Archimedean, and those called non-Archimedean local fields in which it is not. The non-Archimedean local fields can also be defined as those fields which are complete with respect to a metric induced by a discrete valuation v whose residue field is finite.

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 ${\displaystyle A}$ defined over a field ${\displaystyle F}$ with ring of integers ${\displaystyle R}$, consider the Néron model of ${\displaystyle A}$, which is a 'best possible' model of ${\displaystyle A}$ defined over ${\displaystyle R}$. This model may be represented as a scheme over ${\displaystyle \mathrm {Spec} (R)}$ (cf. spectrum of a ring) for which the generic fibre constructed by means of the morphism${\displaystyle \mathrm {Spec} (F)\to \mathrm {Spec} (R)}$gives back ${\displaystyle A}$. The Néron model is a smooth group scheme, so we can consider ${\displaystyle A^{0}}$, 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 ${\displaystyle k}$, ${\displaystyle A_{k}^{0}}$ is a group variety over ${\displaystyle k}$, hence an extension of an abelian variety by a linear group. If this linear group is an algebraic torus, so that ${\displaystyle A_{k}^{0}}$ is a semiabelian variety, then ${\displaystyle A}$ has semistable reduction at the prime corresponding to ${\displaystyle k}$. If ${\displaystyle F}$ is a global field, then ${\displaystyle A}$ is semistable if it has good or semistable reduction at all primes.