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Canonical static concept record

Everywhere locally soluble variety

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Summary

A variety $X$ over a global field $K$ such that $X(K_v)$ is nonempty for every place $v$ of $K$ .

Record metadata

Canonical name: Everywhere locally soluble variety
Concept ID: everywhere_locally_soluble_variety
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)

variety over a global field

Data

a local point over every completion

Axioms / constraints

$X(K_v)\ne\varnothing$ for every place $v$

Canonically induces

The local-solubility premise in a Hasse-principle question

Notes

Everywhere local solubility does not in general imply a global rational point.

Concept sources

Wikipedia — Hasse principle — Hasse principle · encyclopedic reference · source ID wp-local-point
Poonen — Rational Points on Varieties — Rational Points on Varieties · monograph · source ID poonen-rational-points-varieties
Milne — Algebraic Number Theory — Algebraic Number Theory · course text · source ID milne-algebraic-number-theory

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Algebraic variety Everywhere locally soluble variety

Permalink to relation

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

Relation ID: story_algebraic_variety_to_everywhere_locally_soluble_variety_impose_axiom
Relation type: Impose axiom impose-axiom
Direction: source → target
Endpoint roles: source: Builds toward; target: Built from
Authored annotation: require a point over every completion

Authored explanation

Restrict to varieties $X/K$ satisfying $X(K_v)\ne\varnothing$ at every place $v$ of the global field.

How to interpret this relation type

Keep the existing data and select the subclass satisfying an additional law, existence condition, finiteness condition, or other property.

Relation sources

Poonen — Rational Points on Varieties — Rational Points on Varieties · monograph · source ID poonen-rational-points-varieties
Milne — Algebraic Number Theory — Algebraic Number Theory · course text · source ID milne-algebraic-number-theory

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Each relation below starts at this concept.