Atlas of Fundamental Concepts

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Canonical static concept record

Local point

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Summary

For a variety XX over a global field KK and a place vv, a local point is a KvK_v-rational point of the base change XKvX_{K_v}.

Record metadata

Canonical name
Local point
Concept ID
local_point
Content version
2.3.0
Primary field
Mathematics
Fields
Mathematics
Primary domain
Number theory
Domains
Number theory, Algebraic geometry
Alternate terminology
No alternate terminology is currently authored for this record.

Carrier(s)

Data

Canonically induces

Concept sources

Incoming relations (arrows to this concept)

Each relation below ends at this concept.

Everywhere locally soluble varietyLocal point

Permalink to relation

This is an authored directed relation from the source endpoint to the target endpoint.

Relation ID
story_everywhere_locally_soluble_variety_to_local_point_induced_forgotten
Relation type
Induced / forgotten induced-forgotten
Direction
source β†’ target
Endpoint roles
source: Yields by induction / forgetting; target: Obtained by induction / forgetting from
Authored annotation
retain a solution at one place

Authored explanation

Choosing one place vv and forgetting the other local solutions leaves a KvK_v-rational point.

How to interpret this relation type

Pass canonically from a stronger object to structure it determines, or forget part of the data while retaining a valid weaker structure. The carrier may change under a canonical induced construction.

Relation sources

variety + completion + chosen local solution(construction junction)Local point

Permalink to relation

This is an authored directed relation from the source endpoint to the target endpoint.

Relation ID
j_local_point__out
Relation type
Combine compatible structures combine-compatible
Direction
source β†’ target
Endpoint roles
source: Builds toward; target: Built from
Authored annotation
choose a point after base change

Authored explanation

A KvK_v-valued point on the base-changed variety is a local point at vv.

How to interpret this relation type

Feed several structures into a construction junction and impose compatibility between them.

Relation sources

Rational pointLocal point

Permalink to relation

This is an authored directed relation from the source endpoint to the target endpoint.

Relation ID
story_rational_point_to_local_point_canonical_embedding
Relation type
Canonical embedding canonical-embedding
Direction
source β†’ target
Endpoint roles
source: Embeds canonically into; target: Contains a canonical copy of
Authored annotation
embed globally into every completion

Authored explanation

A global KK-point induces a KvK_v-point for every place vv.

How to interpret this relation type

Map one structure injectively into another by the standard structure-preserving inclusion or representation. This records a canonical copy, not necessarily literal set containment.

Relation sources

Outgoing relations (arrows from this concept)

Each relation below starts at this concept.

No direct relations are authored in this direction.