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Canonical static concept record

Variety violating the Hasse principle

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Summary

A variety over a global field that has a point over every completion but has no rational point over the global field.

Record metadata

Canonical name: Variety violating the Hasse principle
Concept ID: failure_hasse_principle
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

points over all completions

Axioms / constraints

absence of a global rational point

Canonically induces

An obstruction to naive local–global inference

Concept sources

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

Incoming relations (arrows to this concept)

Each relation below ends at this concept.

Everywhere locally soluble variety Variety violating the Hasse principle

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_failure_hasse_principle_impose_axiom
Relation type: Impose axiom impose-axiom
Direction: source → target
Endpoint roles: source: Builds toward; target: Built from
Authored annotation: also require no global rational point

Authored explanation

An everywhere locally soluble variety violates the Hasse principle precisely when $X(K)$ is empty.

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

Outgoing relations (arrows from this concept)

Each relation below starts at this concept.

No direct relations are authored in this direction.

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